N.B. This summary does not include routines which have been superseded and are scheduled for withdrawal at a future mark.
SUBROUTINE A00AAF
C MARK 15 RE-ISSUE. NAG COPYRIGHT 1991.
C
C Writes information about the particular implementation of the
C NAG Library in use.
C
C The output channel is given by a call to X04ABF.
C
SUBROUTINE A02AAF(XXR,XXI,YR,YI)
C MARK 2A RELEASE. NAG COPYRIGHT 1973
C MARK 4.5 REVISED
C MARK 5C REVISED
C MARK 11C REVISED. IER-467 (MAR 1985)
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C COMPUTES THE SQUARE ROOT OF A COMPLEX NUMBER
C
DOUBLE PRECISION FUNCTION A02ABF(XXR,XXI)
C NAG COPYRIGHT 1975
C MARK 4.5 REVISED
C MARK 5C REVISED
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C RETURNS THE ABSOLUTE VALUE OF A COMPLEX NUMBER VIA ROUTINE
C NAME
C
SUBROUTINE A02ACF(XXR,XXI,YYR,YYI,ZR,ZI)
C MARK 2A RELEASE. NAG COPYRIGHT 1973
C MARK 4.5 REVISED
C MARK 5C REVISED
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C DIVIDES ONE COMPLEX NUMBER BY A SECOND
C
SUBROUTINE C02AFF(A,N,SCALE,Z,WORK,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C MARK 15B REVISED. IER-944 (NOV 1991).
C
C C02AFF ATTEMPTS TO FIND ALL THE ROOTS OF THE NTH ORDER COMPLEX
C POLYNOMIAL EQUATION
C
C N
C SUM [A(1,k)+A(2,k)*I] * Z**(N-k) = 0.
C k=0
C
C THE ZEROS OF POLYNOMIALS OF DEGREE 1 AND 2 ARE CALCULATED BY
C CAREFULLY EVALUATING THE "STANDARD" CLOSED FORMULAS
C Z = -B/A AND
C Z = (-B +/- SQRT(B*B-4*A*C))/(2*A) RESPECTIVELY, WHERE
C A = CMPLX(A(1,0),A(2,0))
C B = CMPLX(A(1,1),A(2,1)) AND
C C = CMPLX(A(1,2),A(2,2)).
C FOR N >= 3, THE ROOTS ARE LOCATED ITERATIVELY USING A VARIANT OF
C LAGUERRE'S METHOD, WHICH IS CUBICALLY CONVERGENT FOR ISOLATED
C ZEROS AND LINEARLY CONVERGENT FOR MULTIPLE ZEROS.
C
C C02AFF ITSELF IS ESSENTIALLY A DUMMY ROUTINE WHOSE FUNCTION IS TO
C PARTITION THE WORK ARRAY WORK FOR USE BY C02AFZ.
C WORK IS PARTITIONED INTO 2 ARRAYS EACH OF SIZE 2*(N + 1).
C
SUBROUTINE C02AGF(A,N,SCALE,Z,WORK,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C MARK 15B REVISED. IER-945 (NOV 1991).
C
C C02AGF ATTEMPTS TO FIND ALL THE ROOTS OF THE NTH ORDER REAL
C POLYNOMIAL EQUATION
C
C A(0)*Z**N + A(1)*Z**(N-1) + ... + A(N-1)*Z + A(N) = 0.
C
C THE ZEROS OF POLYNOMIALS OF DEGREE 1 AND 2 ARE CALCULATED USING
C THE "STANDARD" CLOSED FORMULAS
C Z = -A(1)/A(0) AND
C Z = (-A(1) +/- SQRT(DISC))/(2*A(0)) RESPECTIVELY, WHERE
C DISC = A(1)**2 - 4*A(0)*A(2).
C FOR N >= 3, THE ROOTS ARE LOCATED ITERATIVELY USING A VARIANT OF
C LAGUERRE'S METHOD, WHICH IS CUBICALLY CONVERGENT FOR ISOLATED
C ZEROS (REAL OR COMPLEX) AND LINEARLY CONVERGENT FOR MULTIPLE
C ZEROS.
C
C C02AGF ITSELF IS ESSENTIALLY A DUMMY ROUTINE WHOSE FUNCTION IS TO
C PARTITION THE WORK ARRAY WORK FOR USE BY C02AGZ.
C WORK IS PARTITIONED INTO 2 ARRAYS EACH OF SIZE (N + 1).
C
SUBROUTINE C02AHF(AR,AI,BR,BI,CR,CI,ZSM,ZLG,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C C02AHF DETERMINES THE ROOTS OF THE QUADRATIC EQUATION
C A*Z**2 + B*Z + C = 0
C WHERE A = CMPLX(AR,AI), B = CMPLX(BR,BI) AND C = CMPLX(CR,CI)
C ARE COMPLEX COEFFICIENTS, AND ZSM AND ZLG ARE THE SMALLEST AND
C LARGEST ROOT IN MAGNITUDE RESPECTIVELY.
C
SUBROUTINE C02AJF(A,B,C,ZSM,ZLG,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C C02AJF DETERMINES THE ROOTS OF THE QUADRATIC EQUATION
C A*Z**2 + B*Z + C = 0
C WHERE A, B AND C ARE REAL COEFFICIENTS, AND ZSM AND ZLG
C ARE THE SMALLEST AND LARGEST ROOT IN MAGNITUDE RESPECTIVELY.
C
SUBROUTINE C05ADF(A,B,EPS,ETA,F,X,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 10C REVISED. IER-422 (JUL 1983).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C DRIVER FOR C05AZF
SUBROUTINE C05AGF(X,HH,EPS,ETA,F,A,B,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 9 REVISED. IER-300 (SEP 1981).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C DRIVER FOR C05AVF AND C05AZF
SUBROUTINE C05AJF(X,EPS,ETA,F,NFMAX,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C DRIVER FOR C05AXF
SUBROUTINE C05AVF(X,FX,H,BOUNDL,BOUNDU,A,C,IND,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 10B REVISED. IER-398 (JAN 1983).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C FINDS INTERVAL [A,X] IN WHICH LIES ROOT OF F.
C [BOUNDL,BOUNDU] CONTAINS [A,X],X IS INITIAL GUESS
C AND H IS STARTING STEP.
SUBROUTINE C05AXF(X,FX,TOL,IR,SCALE,C,IND,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 9 REVISED. IER-301 (SEP 1981).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C USES METHOD OF OF SWIFT AND LINDFIELD,
C C.J.VOL 21. MINK=C(1),MAXK=C(2).
SUBROUTINE C05AZF(X,Y,FX,TOLX,IR,C,IND,IFAIL)
C MARK 8 RE-ISSUE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 12A REVISED. IER-496 (AUG 1986).
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
SUBROUTINE C05NBF(FCN,N,X,FVEC,TOL,WA,LWA,IFAIL)
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
C **********
C
C SUBROUTINE C05NBF (based on MINPACK routine HYBRD1)
C
C The purpose of C05NBF is to find a zero of a system of
C N nonlinear functions in N variables by a modification
C of the Powell Hybrid method. This is done by using the
C more general nonlinear equation solver C05NCF. The user
C must provide a subroutine which calculates the functions.
C The Jacobian is then calculated by a forward-difference
C approximation.
C
C Argonne National Laboratory. MINPACK Project. March 1980.
C Burton S. Garbow, Kenneth E. Hillstrom, Jorge J. More
C **********
C
C Revised to output explanatory messages.
C P.J.D. Mayes, NAG Central Office, December 1987
C
SUBROUTINE C05NCF(FCN,N,X,FVEC,XTOL,MAXFEV,ML,MU,EPSFCN,DIAG,MODE,
* FACTOR,NPRINT,NFEV,FJAC,LDFJAC,R,LR,QTF,W,IFAIL)
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
C **********
C
C SUBROUTINE C05NCF
C
C The purpose of C05NCF is to interface to C05NCS.
C The latter is based on MINPACK routine HYBRD.
C
C **********
C
C Revised to output explanatory messages.
C P.J.D. Mayes, NAG Central Office, December 1987.
C
SUBROUTINE C05NDF(IREVCM,N,X,FVEC,XTOL,ML,MU,EPSFCN,DIAG,MODE,
* FACTOR,FJAC,LDFJAC,R,LR,QTF,W,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C **********
C
C SUBROUTINE C05NDF
C
C The purpose of C05NDF is to interface to C05NDS.
C The latter is based on MINPACK routine HYBRD.
C
C **********
C
C Revised to output explanatory messages.
C P.J.D. Mayes, NAG Central Office, December 1987.
C Revised to reverse communication.
C M.S. Derakhshan, NAG Central Office, September 1988.
C
SUBROUTINE C05PBF(FCN,N,X,FVEC,FJAC,LDFJAC,TOL,WA,LWA,IFAIL)
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
C **********
C
C SUBROUTINE C05PBF (based on MINPACK routine HYBRJ1)
C
C The purpose of C05PBF is to find a zero of a system of
C N nonlinear functions in N variables by a modification
C of the Powell Hybrid method. This is done by using the
C more general nonlinear equation solver C05PCF. The user
C must provide a subroutine which calculates the functions
C and the Jacobian.
C
C **********
C
C Revised to output explanatory messages.
C P.J.D. Mayes, NAG Central Office, December 1987.
C
SUBROUTINE C05PCF(FCN,N,X,FVEC,FJAC,LDFJAC,XTOL,MAXFEV,DIAG,MODE,
* FACTOR,NPRINT,NFEV,NJEV,R,LR,QTF,W,IFAIL)
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
C **********
C
C SUBROUTINE C05PCF
C
C The purpose of C05PCF is to interface to C05PCZ.
C The latter is based upon MINPACK routine HYBRJ.
C
C **********
C
C Revised to output explanatory messages.
C P.J.D. Mayes, NAG Central Office, December 1987.
C
SUBROUTINE C05PDF(IREVCM,N,X,FVEC,FJAC,LDFJAC,XTOL,DIAG,MODE,
* FACTOR,R,LR,QTF,W,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C **********
C
C SUBROUTINE C05PDF
C
C The purpose of C05PDF is to interface to C05PDZ.
C The latter is based upon MINPACK routine HYBRJ.
C
C **********
C
C Revised to output explanatory messages.
C P.J.D. Mayes, NAG Central Office, December 1987.
C Revised to reverse communication.
C M.S. Derakhshan, NAG Central Office, September 1988.
C
C ..Subroutine Arguments ..
C
SUBROUTINE C05ZAF(M,N,X,FVEC,FJAC,LDFJAC,XP,FVECP,MODE,ERR)
C MARK 9 RELEASE. NAG COPYRIGHT 1981
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C **********
C
C SUBROUTINE C05ZAF(BASED ON MINPACK ROUTINE CHKDER)
C
C THIS SUBROUTINE CHECKS THE GRADIENTS OF M NONLINEAR FUNCTIONS
C IN N VARIABLES, EVALUATED AT A POINT X, FOR CONSISTENCY WITH
C THE FUNCTIONS THEMSELVES. THE USER MUST CALL C05ZAF TWICE,
C FIRST WITH MODE = 1 AND THEN WITH MODE = 2.
C
C MODE = 1. ON INPUT, X MUST CONTAIN THE POINT OF EVALUATION.
C ON OUTPUT, XP IS SET TO A NEIGHBORING POINT.
C
C MODE = 2. ON INPUT, FVEC MUST CONTAIN THE FUNCTIONS AND THE
C ROWS OF FJAC MUST CONTAIN THE GRADIENTS
C OF THE RESPECTIVE FUNCTIONS EACH EVALUATED
C AT X, AND FVECP MUST CONTAIN THE FUNCTIONS
C EVALUATED AT XP.
C ON OUTPUT, ERR CONTAINS MEASURES OF CORRECTNESS OF
C THE RESPECTIVE GRADIENTS.
C
C THE SUBROUTINE DOES NOT PERFORM RELIABLY IF CANCELLATION OR
C ROUNDING ERRORS CAUSE A SEVERE LOSS OF SIGNIFICANCE IN THE
C EVALUATION OF A FUNCTION. THEREFORE, NONE OF THE COMPONENTS
C OF X SHOULD BE UNUSUALLY SMALL (IN PARTICULAR, ZERO) OR ANY
C OTHER VALUE WHICH MAY CAUSE LOSS OF SIGNIFICANCE.
C
C **********
C
C EPSMCH IS THE MACHINE PRECISION.
C
SUBROUTINE C06BAF(SEQN,NCALL,RESULT,ABSERR,WORK,IWORK,IFAIL)
C MARK 10 RELEASE. NAG COPYRIGHT 1982.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C MARK 14C REVISED. IER-870 (NOV 1990).
C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C
C MODIFIED VERSION OF ROUTINE D01AJY (EPSILON ALGORITHM)
C BASED ON QUADPACK ROUTINE EPSALG
C
C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C
C PURPOSE
C THE ROUTINE TRANSFORMS A GIVEN SEQUENCE OF
C APPROXIMATIONS, BY MEANS OF THE EPSILON
C ALGORITHM OF P. WYNN, AND PREDICTS A LIMIT
C FOR THE SEQUENCE.
C AN ESTIMATE OF THE ABSOLUTE ERROR IS ALSO GIVEN.
C THE CONDENSED EPSILON TABLE IS COMPUTED. ONLY THOSE
C ELEMENTS NEEDED FOR THE COMPUTATION OF THE
C NEXT DIAGONAL ARE PRESERVED.
C
C PARAMETERS
C SEQN - THE NEW ELEMENT IN THE FIRST COLUMN
C OF THE EPSILON TABLE.
C
C NCALL - NUMBER OF CALLS MADE TO THE ROUTINE.
C MUST BE SET TO ZERO BEFORE FIRST ENTRY
C AND UNCHANGED BETWEEN CALLS.
C
C RESULT - RESULTING APPROXIMATION TO THE LIMIT
C OF THE SEQUENCE.
C
C ABSERR - ESTIMATE OF THE ABSOLUTE ERROR COMPUTED FROM
C RESULT AND THE 3 PREVIOUS /RESULTS/
C
C WORK - ONE DIMENSION ARRAY OF SIZE IWORK
C
C IWORK - DIMENSION OF WORK. MUST BE .GT. 8
C PREFERABLY IWORK.GE.(MAX NUMBER OF SEQN)+6
C
C IFAIL - ON EXIT =1 IF NCALL .LT. ZERO
C =2 IF IWORK .LT. 9
C
C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C
C MODIFICATIONS
C THE TWO ARRAYS EPSTAB AND RES3LA HAVE BEEN COMBINED
C TO ONE WORK ARRAY - RES3LA(1),(2),(3) IS NOW
C WORK (2),(3),(4)
C AND EPSTAB(1),(2),...,(N) IS NOW
C WORK (5),(6),...,(N+4).
C SEQN HAD TO BE ASSIGNED TO SOME ELEMENT OF EPSTAB
C BEFORE CALLING THIS ROUTINE IN ORIGINAL FORM, IS NOW
C ASSIGNED TO WORK(I) INSIDE THIS ROUTINE WHERE INFO
C NECESSARY TO DETERMINE I IS PASSED VIA WORK(1).
C
C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C
C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C
C /LIMEXP/ IS THE MAXIMUM NUMBER OF ELEMENTS THE EPSILON
C TABLE CAN CONTAIN. IF THIS NUMBER IS REACHED, THE UPPER
C DIAGONAL OF THE EPSILON TABLE IS DELETED.
C
DOUBLE PRECISION FUNCTION C06DBF(X,C,N,S)
C MARK 6 RELEASE. NAG COPYRIGHT 1977.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C THIS FUNCTION EVALUATES A CHEBYSHEV SERIES OF ONE OF THREE
C FORMS ACCORDING TO THE VALUE OF S
C S=1- 0.5*C(1)+SUM FROM J=2 TO N OF C(J)*T(J-1)(X),
C S=2- 0.5*C(1)+SUM FROM J=2 TO N OF C(J)*T(2J-2)(X),
C S=3- SUM FROM J=1 TO N OF C(J)*T(2J-1)(X),
C WHERE X LIES IN THE RANGE -1.0 .LE. X .LE. 1.0
C
SUBROUTINE C06EAF(X,PTS,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C REAL FOURIER TRANSFORM
SUBROUTINE C06EBF(X,PTS,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C HERMITE FOURIER TRANSFORM
SUBROUTINE C06ECF(X,Y,PTS,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C COMPLEX FOURIER TRANSFORM
SUBROUTINE C06EKF(JOB,X,Y,N,IFAIL)
C MARK 11 RELEASE. NAG COPYRIGHT 1984.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C IF JOB = 1, CONVOLUTION OF 2 REAL VECTORS.
C IF JOB = 2, CORRELATION OF 2 REAL VECTORS.
C
C (NO WORK ARRAY REQUIRED)
C
SUBROUTINE C06FAF(X,PTS,WORK,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C REAL FOURIER TRANSFORM
SUBROUTINE C06FBF(X,PTS,WORK,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C HERMITE FOURIER TRANSFORM
SUBROUTINE C06FCF(X,Y,PTS,WORK,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C COMPLEX FOURIER TRANSFORM
SUBROUTINE C06FFF(NDIM,L,ND,N,X,Y,WORK,LWORK,IFAIL)
C MARK 11 RELEASE. NAG COPYRIGHT 1984.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C DISCRETE FOURIER TRANSFORM OF ONE VARIABLE IN A
C MULTI-VARIABLE SEQUENCE OF COMPLEX DATA VALUES
C
SUBROUTINE C06FJF(NDIM,ND,N,X,Y,WORK,LWORK,IFAIL)
C MARK 11 RELEASE. NAG COPYRIGHT 1984.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C MULTI-DIMENSIONAL DISCRETE FOURIER TRANSFORM OF A
C MULTI-DIMENSIONAL SEQUENCE OF COMPLEX DATA VALUES
C
SUBROUTINE C06FKF(JOB,X,Y,N,WORK,IFAIL)
C MARK 11 RELEASE. NAG COPYRIGHT 1984.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C IF JOB = 1, CONVOLUTION OF 2 REAL VECTORS.
C IF JOB = 2, CORRELATION OF 2 REAL VECTORS.
C
C (USING WORK ARRAY FOR EXTRA SPEED)
C
SUBROUTINE C06FPF(M,N,X,INIT,TRIG,WORK,IFAIL)
CVD$R NOVECTOR
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE C06FQF(M,N,X,INIT,TRIG,WORK,IFAIL)
CVD$R NOVECTOR
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE C06FRF(M,N,X,Y,INIT,TRIG,WORK,IFAIL)
CVD$R NOVECTOR
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE C06FUF(M,N,X,Y,INIT,TRIGM,TRIGN,WORK,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C
C C06FUF computes a two-dimensional Fast Fourier Transform of
C a complex array, by calling the vectorized multiple
C one-dimensional routine C06FRF together with an explicit
C transposition of the data using the auxiliary routine C06FUZ.
C
SUBROUTINE C06GBF(X,PTS,IFAIL)
CVD$R VECTOR
CVD$R NOLSTVAL
CVD$R STRIP
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C HERMITIAN CONJUGATE
SUBROUTINE C06GCF(Y,PTS,IFAIL)
CVD$R VECTOR
CVD$R NOLSTVAL
CVD$R STRIP
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C COMPLEX CONJUGATE
SUBROUTINE C06GQF(M,N,X,IFAIL)
CVD$R VECTOR
CVD$R NOLSTVAL
CVD$R STRIP
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE C06GSF(M,N,X,U,V,IFAIL)
CVD$R VECTOR
CVD$R NOLSTVAL
CVD$R STRIP
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
SUBROUTINE C06HAF(M,N,X,INIT,TRIG,WORK,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C MARK 14C REVISED. IER-871 (NOV 1990).
C MARK 15 REVISED. IER-894 (APR 1991).
C
C C06HAF computes multiple Fourier sine transforms of sequences
C of real data using the multiple real transform kernel C06FPX,
C and pre- and post-processing steps described by Swarztrauber.
C
SUBROUTINE C06HBF(M,N,X,INIT,TRIG,WORK,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C MARK 14C REVISED. IER-872 (NOV 1990).
C MARK 15 REVISED. IER-895 (APR 1991).
C
C C06HBF computes multiple Fourier cosine transforms of sequences
C of real data using the multiple real transform kernel C06FPX,
C and pre- and post-processing steps described by Swarztrauber.
C
SUBROUTINE C06HCF(DIRECT,M,N,X,INIT,TRIG,WORK,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C MARK 13B REVISED. IER-650 (AUG 1988).
C MARK 14C REVISED. IER-873 (NOV 1990).
C MARK 15 REVISED. IER-896 (APR 1991).
C
C C06HCF computes multiple quarter-wave Fourier sine transforms
C of sequences of real data using the multiple real and
C Hermitian transform kernels C06FPX and C06FQX, and pre- and
C post-processing steps described by Swarztrauber.
C
SUBROUTINE C06HDF(DIRECT,M,N,X,INIT,TRIG,WORK,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C MARK 13B REVISED. IER-651 (AUG 1988).
C MARK 15 REVISED. IER-897 (APR 1991).
C
C C06HDF computes multiple quarter-wave Fourier cosine transforms
C of sequences of real data using the multiple real and
C Hermitian transform kernels C06FPX and C06FQX, and pre- and
C post-processing steps described by Swarztrauber.
C
SUBROUTINE C06LAF(FUN,N,T,VALINV,ERREST,RELERR,ALPHAB,TFAC,MXTERM,
* NTERMS,NA,ALOW,AHIGH,NFEVAL,WORK,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C Estimates values of the inverse Laplace transform of a given
C function using a Fourier series approximation.
C Real and imaginary parts of the function, and a bound on the
C exponential order of the inverse, are required.
C
SUBROUTINE C06LBF(F,SIGMA0,SIGMA,B,EPSTOL,MMAX,M,ACOEF,ERRVEC,
* IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C MARK 15B REVISED. IER-946 (NOV 1991).
C
C C06LBF computes a set of Laguerre expansion coefficients for the
C inverse Laplace transform of a user-specified analytic function.
C The value of the inverse Laplace transform at a specified argument
C can be obtained by a subsequent call to C06LCF.
C
C C06LBF is derived from the subroutine MODUL1 in the package WEEKS
C by B.S. Garbow, G. Giunta, J.N. Lyness and A. Murli, Algorithm
C 662: A Fortran software package for the numerical inversion of the
C Laplace Transform based on Weeks' method, ACM Trans. Math.
C Software, 14, pp 171-176 (1988).
C
C INPUT arguments
C
C F - procedure - name of function subprogram for the complex
C valued Laplace transform to be inverted. F must
C be declared EXTERNAL in the calling program.
C sigma0 - real - the abscissa of convergence of the Laplace
C transform.
C sigma - real - the first parameter of the Laguerre expansion.
C If sigma is not greater than sigma0, it defaults
C and is reset to (sigma0 + 0.7).
C b - real - the second parameter of the Laguerre expansion.
C If b is less than 2.0*(sigma - sigma0), it
C defaults and is reset to 2.5*(sigma - sigma0).
C epstol - real - the required absolute uniform pseudo accuracy for
C the coefficients and inverse Laplace transform
C values.
C mmax - integer - an upper limit on the number of coefficients
C to be computed. Note that the maximum number of
C Laplace transform evaluations is (mmax/2 + 2).
C
C OUTPUT arguments
C
C m - integer - the number of coefficients actually computed.
C acoef - real(mmax) - the array of Laguerre coefficients.
C errvec - real(8) - an 8-component vector of diagnostic
C information.
C All components are functions of Laguerre coefficients acoef.
C (1) = Overall estimate of the pseudo-error = (2) + (3) + (4).
C Pseudo-error = absolute error / exp(sigma*t) .
C (2) = Estimate of the discretisation pseudo-error.
C (3) = Estimate of the truncation pseudo-error.
C (4) = Estimate of the conditioning pseudo-error, on the basis
C of minimal noise levels in function values.
C (5) = K - Coefficient of the decay function for acoef.
C (6) = R - Base of the decay function for acoef.
C abs(acoef(j+1)) .le. K/R**j for j .ge. m/2 .
C (7) = ALPHA - Logarithm of the largest acoef.
C (8) = BETA - Logarithm of the smallest nonzero acoef.
C ifail - integer - the output state parameter, takes one of 6
C values:
C 0 => Normal termination, estimated error less than epstol.
C 1 => MMAX < 8.
C 2 => Normal termination, but with estimated error bounds
C slightly larger than epstol. Note, however, that the
C actual errors on the final results may be smaller than
C epstol as bounds independent of t are pessimistic.
C 3 => The round-off level makes it impossible to achieve the
C required accuracy.
C 4 => The decay rate of the coefficients is too small.
C It may improve results to increase mmax.
C 5 => The decay rate of the coefficients is too small and the
C truncation error too large because of round-off error.
C 6 => No error bounds are returned as the behavior of the
C coefficients does not enable reasonable prediction.
C Check the value of sigma0.
C In this case, (errvec(j),j=1,5) are each set to -1.0.
C NOTE - When ifail is 3, 4, 5 or 6, changing b and sigma
C may help. If not, the method should be abandoned.
C
SUBROUTINE C06LCF(T,SIGMA,B,M,ACOEF,ERRVEC,FINV,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C C06LCF evaluates, for a specified nonnegative t, the inverse
C finv of the prescribed Laplace transform using the series below,
C denoted as SUM, whose coefficients are provided by C06LBF. The
C values of sigma and b are those returned by C06LBF.
C
C finv = exp(sigma*t)*SUM
C
C where
C
C SUM = summation of (acoef(j)*exp(-b*t/2)*L(j,b*t),j=1,m),
C L(j,b*t) denotes the Laguerre polynomial of degree j-1,
C exp(-b*t/2)*L(j,b*t) is the associated Laguerre function.
C
C When t is nonpositive, the evaluation approximates
C the analytic continuation of the inverse Laplace transform,
C becoming progressively poorer as t becomes more negative.
C
C Note that this routine is overflow/underflow(destructive) free
C and can be used even when the value exp(sigma*t) overflows
C or exp(-bt/2) underflows.
C
C C06LCF is derived from the subroutine MODUL2 in the package WEEKS
C by B.S. Garbow, G. Giunta, J.N. Lyness and A. Murli, Algorithm
C 662: A Fortran software package for the numerical inversion of the
C Laplace Transform based on Weeks' method, ACM Trans. Math.
C Software, 14, pp 171-176 (1988).
C
C INPUT arguments
C
C t - real - the point where the inverse Laplace transform
C is to be computed.
C m - integer - the number of terms of the Laguerre expansion.
C acoef - real(m) - the coefficients of the Laguerre expansion.
C sigma - real - the first parameter of the Laguerre expansion.
C It must have the same value as returned by C06LBF.
C b - real - the second parameter of the Laguerre expansion.
C It must have the same value as returned by C06LBF.
C errvec - real(8) - the vector of diagnostic information from
C C06LBF. Only components 1,7 and 8 are used in
C C06LCF. (If C06LCF is used independently of
C C06LBF, store ALPHA,BETA into (7),(8) and set
C errvec(1) = 0.0 .)
C
C OUTPUT arguments
C
C finv - real - the value of the inverse Laplace transform at t.
C ifail - integer - the error indicator.
C 0 => Normal termination.
C 1 => The value of the inverse Laplace transform is found to
C be too large to be representable - finv is set to 0.0.
C 2 => The value of the inverse Laplace transform is found to
C be too small to be representable - finv is set to 0.0.
C
DOUBLE PRECISION FUNCTION D01AHF(A,B,EPR,NPTS,RELERR,F,NL,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 8A REVISED. IER-254 (AUG 1980).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 12B REVISED. IER-525 (FEB 1987).
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C MARK 14 REVISED. IER-819 (DEC 1989).
C
C THIS FUNCTION ROUTINE PERFORMS AUTOMATIC INTEGRATION OVER A
C FINITE INTERVAL USING THE BASIC INTEGRATION ALGORITHMS D01AHY
C AND D01AHX, TOGETHER WITH, IF NECESSARY, AN ADAPTIVE
C SUBDIVISION PROCESS.
C
C INPUT ARGUMENTS
C ----- ----------
C A,B - LOWER AND UPPER INTEGRATION LIMITS.
C EPR - REQUIRED RELATIVE ACCURACY.
C NL - APPROXIMATE LIMIT ON NUMBER OF INTEGRAND
C EVALUATIONS. IF SET NEGATIVE OR ZERO THE
C DEFAULT IS 10000.
C F - THE USER NAMED AND PREPARED FUNCTION F(X)
C GIVES THE VALUE OF THE INTEGRAND AT X.
C IFAIL INTEGER VARIABLE
C - 0 FOR HARD FAIL REPORT
C - 1 FOR SOFT FAIL REPORT
C
C OUTPUT ARGUMENTS
C ------ ----------
C NPTS - NUMBER OF INTEGRAND EVALUATIONS USED IN OBTAINING
C THE RESULT.
C RELERR - ROUGH ESTIMATE OF RELATIVE ACCURACY ACHIEVED.
C IFAIL - VALUE INDICATES THE OUTCOME OF THE INTEGRATION -
C IFAIL = 0 CONVERGED
C IFAIL = 1 INTEGRAND EVALUATIONS EXCEEDED NL.
C THE RESULT WAS OBTAINED BY CONTINUING
C BUT IGNORING ANY NEED TO SUBDIVIDE.
C RESULT LIKELY TO BE INACCURATE.
C IFAIL = 2 DURING THE SUBDIVISION PROCESS
C THE STACK BECAME FULL
C (PRESENTLY SET TO HOLD 20
C LEVELS OF INFORMATION. MAY BE
C INCREASED BY ALTERING ISMAX
C AND THE DIMENSIONS OF STACK
C AND ISTACK). RESULT IS
C OBTAINED BY CONTINUING BUT
C IGNORING CONVERGENCE FAILURES
C ON INTERVALS WHICH CANNOT BE
C ACCOMMODATED ON THE STACKS.
C RESULT LIKELY TO BE
C INACCURATE.
C IFAIL = 3 INVALID ACCURACY REQUEST.
C
C THE SUBDIVISION STRATEGY IS AS FOLLOWS -
C AT EACH STAGE AN INTERVAL IS PRESENTED FOR SUBDIVISION
C (INITIALLY THE WHOLE INTERVAL). THE POINT OF SUBDIVISION IS
C DETERMINED BY THE RELATIVE GRADIENT OF THE INTEGRAND
C AT THE END POINTS (SEE D01AHZ) AND MAY BE IN THE
C RATIO 1/2, 1/1 OR 2/1.D01AHY IS THEN APPLIED TO EACH
C SUBINTERVAL. SHOULD IT FAIL TO CONVERGE ON THE LEFT
C SUBINTERVAL THE SUBINTERVAL IS STACKED FOR FUTURE
C EXAMINATION AND THE RIGHT SUBINTERVAL IMMEDIATELY
C EXAMINED. SHOULD IT FAIL ON THE RIGHT SUBINTERVAL
C SUBDIVISION IS IMMEDIATELY PERFORMED AND THE WHOLE
C PROCESS REPEATED. EACH CONVERGED RESULT IS
C ACCUMULATED AS THE PARTIAL VALUE OF THE INTEGRAL.
C WHEN THE LEFT AND RIGHT SUBINTERVALS BOTH CONVERGE
C THE INTERVAL LAST STACKED IS SUBDIVIDED AND THE
C PROCESS REPEATED.
C A NUMBER OF REFINEMENTS ARE INCLUDED. ATTEMPTS ARE MADE TO
C DETECT LARGE VARIATIONS IN THE INTEGRAND AND
C TRANSFORMATIONS ARE MADE IF ENDPOINT VARIATION IS
C EXTREME. THIS DEPENDS ON THE RATE OF CONVERGENCE OF
C D01AHX AND ON THE END POINT RELATIVE GRADIENTS OF THE
C INTEGRAND FOR THE NON-SUBDIVIDED INTERVAL. RANDOM
C TRANSFORMATIONS ARE ALSO APPLIED TO IMPROVE THE
C RELIABILITY. THE RELATIVE ACCURACY REQUESTED ON
C EACH SUBINTERVAL IS ADJUSTED IN ACCORDANCE WITH ITS
C LIKELY CONTRIBUTION TO THE TOTAL INTEGRAL.
C
SUBROUTINE D01AJF(F,A,B,EPSABS,EPSREL,RESULT,ABSERR,WORK,LWORK,
* IWORK,LIWORK,IFAIL)
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
C
C D01AJF IS A GENERAL PURPOSE INTEGRATOR WHICH CALCULATES
C AN APPROXIMATION TO THE INTEGRAL OF A FUNCTION OVER A FINITE
C INTERVAL (A,B)
C
C D01AJF ITSELF IS ESSENTIALLY A DUMMY ROUTINE WHOSE FUNCTION IS TO
C PARTITION THE WORK ARRAYS WORK AND IWORK FOR USE BY D01AJV.
C WORK IS PARTITIONED INTO 4 ARRAYS EACH OF SIZE LIMIT, WHERE
C LIMIT = MIN(LWORK/4, LIWORK).
C IWORK IS A SINGLE ARRAY IN D01AJV OF SIZE LIMIT.
C
SUBROUTINE D01AKF(F,A,B,EPSABS,EPSREL,RESULT,ABSERR,WORK,LWORK,
* IWORK,LIWORK,IFAIL)
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
C
C D01AKF IS AN ADAPTIVE INTEGRATOR, ESPECIALLY SUITED TO
C STRONGLY OSCILLATORY NON-SINGULAR INTEGRANDS, WHICH CALCULATES
C AN APPROXIMATION TO THE INTEGRAL OF A FUNCTION OVER A FINITE
C INTERVAL (A,B)
C
C D01AKF ITSELF IS ESSENTIALLY A DUMMY ROUTINE WHOSE FUNCTION IS TO
C PARTITION THE WORK ARRAYS WORK AND IWORK FOR USE BY D01AKV.
C WORK IS PARTITIONED INTO 4 ARRAYS EACH OF SIZE LIMIT, WHERE
C LIMIT = MIN(LWORK/4, LIWORK).
C IWORK IS A SINGLE ARRAY IN D01AKV OF SIZE LIMIT.
C
SUBROUTINE D01ALF(F,A,B,NPTS,POINTS,EPSABS,EPSREL,RESULT,ABSERR,
* WORK,LWORK,IWORK,LIWORK,IFAIL)
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
C
C D01ALF IS A GENERAL PURPOSE INTEGRATOR WHICH CALCULATES AN
C APPROXIMATION TO THE INTEGRAL OVER THE INTERVAL (A,B) OF A
C FUNCTION WHICH MAY HAVE A LOCAL SINGULAR BEHAVIOUR AT A
C FINITE NUMBER OF POINTS WITHIN THE INTEGRATION INTERVAL.
C
C D01ALF ITSELF IS ESSENTIALLY A DUMMY ROUTINE WHOSE FUNCTION
C IS TO PARTITION THE WORK ARRAYS WORK AND LWORK FOR USE BY D01ALV.
C WORK IS PARTITIONED INTO 4 ARRAYS OF SIZE LIMIT, WHERE
C LIMIT = MIN((LWORK-2*NPTS-4)/4, (LIWORK-NPTS-2)/2) AND
C 2 ARRAYS OF SIZE NPTS + 2. IWORK IS PARTITIONED INTO 2 ARRAYS OF
C SIZE LIMIT AND 1 ARRAY OF SIZE NPTS + 2.
C
SUBROUTINE D01AMF(F,BOUND,INF,EPSABS,EPSREL,RESULT,ABSERR,WORK,
* LWORK,IWORK,LIWORK,IFAIL)
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
C BASED ON THE QUADPACK ROUTINE QAGI.
C
C D01AMF CALCULATES AN APPROXIMATION TO THE INTEGRAL OF A FUNCTION
C OVER THE INTERVAL (A,B) , WHERE A AND/OR B IS INFINITE.
C
C D01AMF ITSELF IS ESSENTIALLY A DUMMY ROUTINE WHOSE FUNCTION IS TO
C PARTITION THE WORK ARRAYS WORK AND IWORK FOR USE BY D01AMV.
C WORK IS PARTITIONED INTO 4 ARRAYS EACH OF SIZE LIMIT, WHERE
C LIMIT = MIN(LWORK/4, LIMIT).
C IWORK IS A SINGLE ARRAY IN D01AMV OF SIZE LIMIT.
C
SUBROUTINE D01ANF(F,A,B,OMEGA,KEY,EPSABS,EPSREL,RESULT,ABSERR,
* WORK,LWORK,IWORK,LIWORK,IFAIL)
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
C
C D01ANF CALCULATES AN APPROXIMATION TO THE COSINE OR SINE
C TRANSFORM OF A FUNCTION OVER AN INTERVAL (A,B)
C
C D01ANF ITSELF IS ESSENTIALLY A DUMMY ROUTINE WHOSE FUNCTION
C IS TO PARTITION THE WORK ARRAYS WORK AND LWORK FOR USE BY
C D01ANV. WORK IS PARTITIONED INTO 4 ARRAYS EACH OF SIZE LIMIT,
C WHERE LIMIT = MIN(LWORK/4, LIWORK/2) .
C IWORK IS PARTITIONED INTO 2 ARRAYS EACH OF SIZE LIMIT.
C
C MAXP1 - AN UPPER BOUND ON THE NUMBER OF CHEBYSHEV MOMENTS
C WHICH CAN BE STORED (SEE D01ANV).
C ICALL - ENABLES THE CHEBYSHEV MOMENTS ALREADY COMPUTED
C (WITH ICALL = 1) TO BE RE-USED ON SUBSEQUENT CALLS
C (ICALL > 1) (SEE D01ANV).
SUBROUTINE D01APF(F,A,B,ALFA,BETA,KEY,EPSABS,EPSREL,RESULT,ABSERR,
* WORK,LWORK,IWORK,LIWORK,IFAIL)
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
C
C D01APF IS AN ADAPTIVE INTEGRATOR WHICH CALCULATES AN
C APPROXIMATION TO THE INTEGRAL OF A FUNCTION G(X)W(X) OVER
C (A,B) WHERE THE WEIGHT FUNCTION W HAS END-POINT
C SINGULARITIES OF ALGEBRAICO-LOGARITHMIC TYPE.
C
C D01APF ITSELF IS ESSENTIALLY A DUMMY ROUTINE WHOSE FUNCTION IS TO
C PARTITION THE WORK ARRAYS WORK AND IWORK FOR USE BY D01APV.
C WORK IS PARTITIONED INTO 4 ARRAYS EACH OF SIZE LIMIT, WHERE
C LIMIT = MIN(LWORK/4, LIWORK).
C IWORK IS A SINGLE ARRAY IN D01APV OF SIZE LIMIT.
C
SUBROUTINE D01AQF(F,A,B,C,EPSABS,EPSREL,RESULT,ABSERR,WORK,LWORK,
* IWORK,LIWORK,IFAIL)
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
C
C D01AQF CALCULATES AN APPROXIMATION TO THE HILBERT TRANSFORM
C OF A FUNCTION G(X) OVER (A,B)
C
C D01AQF ITSELF IS ESSENTIALLY A DUMMY ROUTINE WHOSE FUNCTION IS TO
C PARTITION THE WORK ARRAYS WORK AND IWORK FOR USE BY D01AQV.
C WORK IS PARTITIONED INTO 4 ARRAYS EACH OF SIZE LIMIT, WHERE
C LIMIT = MIN(LWORK/4, LIWORK).
C IWORK IS A SINGLE ARRAY IN D01AQV OF SIZE LIMIT.
C
SUBROUTINE D01ARF(A,B,F,RELACC,ABSACC,MAXRUL,IPARM,ACC,ANS,N,
* ALPHA,IFAIL)
C MARK 10 RELEASE. NAG COPYRIGHT 1982.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 13 X02 FUNCTIONS (APR 1988).
C MARK 13 REVISED. IER-576 (MAR 1988).
C
C DEFINITE AND INDEFINITE INTEGRATION OVER A FINITE INTERVAL
C
C THE NUMERICAL INTEGRATION IS PERFORMED USING A SEQUENCE OF
C RULES BASED ON THE REPEATED OPTIMAL EXTENSIONS OF THE 3-POINT
C GAUSS RULE (SEE PATTERSON,T.N.L.,MATHS.COMP.,22,847-857,1968).
C UP TO 9 RULES MAY BE APPLIED USING 1,3,7,15,31,63,127,255 AND
C 511 POINTS OF RESPECTIVE POLYNOMIAL INTEGRATING DEGREE 1,5,11
C 23,47,95,191,383 AND 767. EACH RULE HAS THE INTERLACING
C PROPERTY THAT THE NEXT IN SEQUENCE INCLUDES ALL THE POINTS OF
C ITS PREDECESSOR. IN CARRYING OUT THE INTEGRATION THE RESULTS
C OF SUCCESSIVE RULES ARE COMPARED. WHEN THE LATEST TWO RESULTS
C DIFFER ABSOLUTELY BY NOT MORE THAN A SPECIFIED TOLERANCE THE
C RESULT OF THE LAST RULE USED IS TAKEN AS THE VALUE OF THE
C INTEGRAL. THE PROCEDURE IS NON-ADAPTIVE IN THAT IF THE REQUIRED
C TOLERANCE IS NOT ACHIEVED AFTER APPLYING ALL RULES NO ATTEMPT
C IS MADE TO CONTINUE, THE BEST AVAILABLE RESULT IS RETURNED
C WITH A WARNING.
C
C THE CALL TO THE SUBROUTINE TAKES THE FORM-
C
C CALL D01ARF(A,B,F,RELACC,ABSACC,MAXRUL,IPARM,
C ACC,ANS,N,ALPHA,IFAIL)
C
C THE ACTION PERFORMED IS CONTROLLED BY THE PARAMETER IPARM
C AS FOLLOWS-
C
C IPARM=0 -THE INTEGRAND F(X) IS INTEGRATED TO A PRESCRIBED
C ACCURACY OVER THE FINITE INTERVAL A.LE.X.LE.B.
C
C IPARM=1 -AS FOR IPARM=0 BUT ADDITIONALLY THE EXPANSION OF
C INTEGRAND OVER THE INTERVAL (A,B) IN LEGENDRE
C POLYNOMIALS IS CALCULATED USING THE SAME VALUES
C OF THE INTEGRAND AS USED TO CALCULATE THE INTEGRAL.
C THE EXPANSION COEFFICIENTS MAY BE USED LATER IN
C COMPUTING INDEFINITE INTEGRALS.
C
C IPARM=2 -INDEFINITE INTEGRATION -THE INTEGRAND F(X) IS
C INTEGRATED ANALYTICALLY OVER THE INTERVAL
C A.LE.X.LE.B USING THE LEGENDRE EXPANSION OF THE
C INTEGRAND. NO FURTHER EVALUATIONS OF THE INTEGRAND
C ARE REQUIRED. THE SUBROUTINE MUST PREVIOUSLY
C HAVE BEEN CALLED WITH IPARM=1 AND THE INTERVAL
C (A,B) MUST LIE WITHIN THE INTERVAL PRESCRIBED FOR
C THAT PREVIOUS CALL.
C
C -FORMAL ARGUMENT SPECIFICATIONS-
C
C INPUT ARGUMENTS-
C
C A,B -RESPECTIVE LOWER AND UPPER LIMITS TO THE INTERVAL
C OF INTEGRATION. REAL VARIABLES OR CONSTANTS.
C
C F -USER SUPPLIED AND DEFINED REAL FUNCTION F(X) TO
C CALCULATE THE VALUE OF THE INTEGRAND AT X (REAL
C VARIABLE). (IPARM=0 OR 1 ONLY).
C
C RELACC, -RESPECTIVE RELATIVE AND ABSOLUTE ACCURACIES
C ABSACC REQUIRED. THE REQUIRED ACCURACY IS ASSUMED TO HAVE
C BEEN ACHIEVED WHEN THE RESULTS OF TWO SUCCESSIVE
C RULES DIFFER ABSOLUTELY BY NOT MORE THAN
C AMAX1(ABS(ABSACC),ABS(RELACC*ANS))
C WHERE ANS IS THE RESULT OBTAINED BY THE LAST RULE
C USED. IF BOTH RELACC AND ABSACC ARE ZERO THEN
C RELACC IS ASSUMED TO HAVE VALUE 10.0*EPMACH, WHERE
C EPMACH IS THE SMALLEST POSITIVE REAL NUMBER SUCH
C THAT
C 1.0+EPMACH .GT. 1.0
C TO MACHINE ACCURACY. IF ALL RULES ARE EXHAUSTED
C WITHOUT ACHIEVING THE REQUIRED ACCURACY, ERROR
C IFAIL=1 IS RAISED AND ANS IS SET TO THE VALUE
C OBTAINED BY THE LAST RULE AS THE BEST
C APPROXIMATION POSSIBLE. REAL VARIABLES OR
C CONSTANTS.
C (IPARM=0 OR 1 ONLY).
C
C MAXRUL -MAXIMUM NUMBER OF SUCCESSIVE RULES TO BE USED.
C 1.LE.MAXRUL.LE.9. IF OUTSIDE THESE LIMITS MAXRUL
C DEFAULTS TO THE VALUE 9. INTEGER VARIABLE OR
C CONSTANT. (IPARM=0 OR 1 ONLY).
C
C IPARM -THIS CONTROLS THE ACTION OF THE SUBROUTINE AND IS
C SET TO ONE OF THE THREE POSSIBLE VALUES 0,1 OR 2.
C IPARM=0 EVALUATES THE INTEGRAL ONLY.
C =1 EVALUATES THE INTEGRAL AND THE EXPANSION
C OF THE INTEGRAND IN LEGENDRE POLYNOMIALS.
C =2 CARRIES OUT INDEFINITE INTEGRATION.
C IN THIS CASE ONLY THE ARGUMENTS A,B,
C IPARM, ANS, ALPHA AND IFAIL ARE RELEVANT
C WITH THE OTHERS SUPPLIED SIMPLY AS DUMMY
C ARGUMENTS.
C INTEGER VARIABLE OR CONSTANT.
C FULLER DETAILS ON IPARM HAVE BEEN GIVEN EARLIER.
C
C ALPHA -REAL ARRAY
C THIS ARRAY SHOULD HAVE BEEN GENERATED BY A
C PREVIOUS CALL WITH IPARM=1. CONTENTS AS DESCRIBED
C UNDER OUTPUT ARGUMENTS BELOW.
C (IPARM=2 ONLY)
C
C IFAIL -NAG ERROR CONTROL. INTEGER VARIABLE.
C IFAIL=0 HARD FAIL.
C =1 SOFT FAIL.
C
C
C OUTPUT ARGUMENTS-
C
C ACC -ESTIMATED ABSOLUTE ERROR IN RESULT. THIS IS TAKEN
C AS THE VALUE OF THE ABSOLUTE DIFFERENCE BETWEEN
C THE RESULTS OF THE LAST TWO RULES USED.
C REAL VARIABLE. (IPARM=0 OR 1 ONLY).
C
C ANS -APPROXIMATE VALUE OF THE INTEGRAL. REAL VARIABLE.
C
C N -NUMBER OF EVALUATIONS OF THE INTEGRAND USED.
C INTEGER VARIABLE. (IPARM=0 OR 1 ONLY).
C
C ALPHA -REAL ARRAY
C THIS ARRAY CONTAINS
C ALPHA(1) TO ALPHA(NPTS) = THE NPTS COEFFICIENTS OF
C THE APPROXIMATION TO F(X) OVER (A,B) GIVEN BY-
C
C F(X) = SUM( I=1 TO NPTS ) ALPHA(I)*P(I-1,T)
C
C WHERE X = (B+A+(B-A)*T)/2 AND P(I,T) IS THE
C LEGENDRE POLYNOMIAL OF DEGREE I OVER (-1,1). THIS
C APPROXIMATION IS INTEGRATED TO OBTAIN THE VALUE OF
C THE INDEFINITE INTEGRAL.
C NPTS=3*(N+1)/4 FOR N=1,3 OR 7
C NPTS=3*N/4 OTHERWISE.
C
C ALPHA ALSO HOLDS THE FOLLOWING INFORMATION
C REQUIRED FOR CALLS FOR INDEFINITE INTEGRATION
C ALPHA(390)=NPTS
C ALPHA(389)=1 IF ALPHA USABLE FOR INDEF INTEGRATION
C (IE. IPARM=1 AND RANGE NON-TRIVIAL)
C =0 OTHERWISE
C ALPHA(388)=IFAIL (IF ALPHA(389)=1)
C UNDEFINED OTHERWISE
C ALPHA(387)=B
C ALPHA(386)=A
C ALPHA SHOULD BE DECLARED IN THE CALLING
C PROGRAM TO HAVE AT LEAST 390 ELEMENTS.
C (IPARM=1 ONLY)
C ALPHA IS UNCHANGED WHEN IPARM=2
C
C IFAIL -REPORTS ERRORS AS FOLLOWS- (INTEGER VARIABLE)
C IFAIL=0 NO ERRORS DETECTED.
C =1 ALL MAXRUL RULES HAVE BEEN APPLIED
C WITHOUT ACHIEVING THE REQUIRED ACCURACY.
C ANS IS SET TO THE VALUE OBTAINED BY THE
C LAST RULE USED.
C =2 IPARM FAILS TO SATISFY 0.LE.IPARM.LE.2.
C =3 SUBROUTINE HAS BEEN CALLED WITH IPARM=2
C BUT A PREVIOUS CALL WITH IPARM=1 HAS BEEN
C OMITTED OR WAS INVOKED WITH AN
C INTEGRATION INTERVAL OF LENGTH ZERO.
C (IPARM=2 ONLY).
C =4 THE INTERVAL FOR INDEFINITE INTEGRATION
C IS NOT CONTAINED WITHIN THE INTERVAL
C SPECIFIED WHEN THE SUBROUTINE WAS
C PREVIOUSLY CALLED WITH IPARM=1.
C (IPARM=2 ONLY).
C
SUBROUTINE D01ASF(F,A,OMEGA,KEY,EPSABS,RESULT,ABSERR,LIMLST,LST,
* ERLST,RSLST,IERLST,WORK,LWORK,IWORK,LIWORK,
* IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C
C D01ASF CALCULATES AN APPROXIMATION TO THE COSINE OR SINE
C TRANSFORM OF A FUNCTION OVER AN INTERVAL (A,INFINITY)
C
C D01ASF ITSELF IS ESSENTIALLY A DUMMY ROUTINE WHOSE FUNCTION
C IS TO PARTITION THE WORK ARRAYS WORK AND LWORK FOR USE BY
C D01ASV. WORK IS PARTITIONED INTO 4 ARRAYS EACH OF SIZE LIMIT,
C WHERE LIMIT = MIN(LWORK/4, LIWORK/2).
C IWORK IS PARTITIONED INTO 2 ARRAYS EACH OF SIZE LIMIT.
C
C MAXP1 - AN UPPER BOUND ON THE NUMBER OF CHEBYSHEV MOMENTS
C WHICH CAN BE STORED, WHERE MAXP1.GE.1 (SEE D01ASV).
SUBROUTINE D01ATF(F,A,B,EPSABS,EPSREL,RESULT,ABSERR,WORK,LWORK,
* IWORK,LIWORK,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C MARK 13B REVISED. IER-653 (AUG 1988).
C
C D01ATF IS A GENERAL PURPOSE INTEGRATOR WHICH CALCULATES
C AN APPROXIMATION TO THE INTEGRAL OF A FUNCTION OVER A FINITE
C INTERVAL (A,B)
C
C D01ATF ITSELF IS ESSENTIALLY A DUMMY ROUTINE WHOSE FUNCTION IS TO
C PARTITION THE WORK ARRAYS WORK AND IWORK FOR USE BY D01ATV.
C WORK IS PARTITIONED INTO 4 ARRAYS EACH OF SIZE LIMIT, WHERE
C LIMIT = MIN(LWORK/4, LIWORK).
C IWORK IS A SINGLE ARRAY IN D01ATV OF SIZE LIMIT.
C
SUBROUTINE D01AUF(F,A,B,KEY,EPSABS,EPSREL,RESULT,ABSERR,WORK,
* LWORK,IWORK,LIWORK,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C
C D01AUF IS AN ADAPTIVE INTEGRATOR, ESPECIALLY SUITED TO
C STRONGLY OSCILLATORY NON-SINGULAR INTEGRANDS, WHICH CALCULATES
C AN APPROXIMATION TO THE INTEGRAL OF A FUNCTION OVER A FINITE
C INTERVAL (A,B)
C
C D01AUF ITSELF IS ESSENTIALLY A DUMMY ROUTINE WHOSE FUNCTION IS TO
C PARTITION THE WORK ARRAYS WORK AND IWORK FOR USE BY D01AUV.
C WORK IS PARTITIONED INTO 4 ARRAYS EACH OF SIZE LIMIT, WHERE
C LIMIT = MIN(LWORK/4, LIWORK).
C IWORK IS A SINGLE ARRAY IN D01AUV OF SIZE LIMIT.
C
DOUBLE PRECISION FUNCTION D01BAF(WTFUN,A,B,NPTS,FUN,IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 8A REVISED. IER-250 (JULY 1980).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C EVALUATES INTEGRAL OF FUNCTION FUN BY NPTS GAUSS FORMULA OF
C TYPE WTFUN
C IFAIL = 1 - THE NPTS RULE IS NOT AMONG THOSE STORED
C ( ANSWER EVALUATED FOR LARGEST VALID NPTS LESS THAN REQUESTED
C VALUE)
C IFAIL = 2 - VALUES OF A OR B INVALID
C ( ANSWER RETURNED AS ZERO)
C
C WTFUN
SUBROUTINE D01BBF(WTFUN,A,B,ITYPE,NPTS,WEIGHT,ABSCIS,IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C RETURNS WEIGHTS AND PIVOTS FOR ONE GAUSS-WTFUN FORMULA IF
C STORED
C IFAIL = 1 - THE NPTS RULE IS NOT AMONG THOSE STORED
C ( WEIGHT,ABSCIS EVALUATED FOR LARGEST VALID NPTS LESS THAN
C REQUESTED VALUE)
C IFAIL = 2 - VALUES OF A OR B INVALID
C ( ALL WEIGHTS AND ABSCISSAE RETURNED AS ZERO)
C IFAIL = 3 - UNDERFLOW IN EVALUATING LAGUERRE OR HERMITE
C NORMAL WEIGHTS
C ( THE UNDERFLOWING WEIGHTS ARE RETURNED AS ZERO)
C
C THE WEIGHTS AND ABSCISSAE RETURNED DEPEND ON A AND B.
C
C WTFUN
SUBROUTINE D01BCF(ITYPE,AA,BB,CC,DD,NPNTS,WEIGHT,ABSCIS,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 9C REVISED. IER-370 (JUN 1982).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C MARK 14A REVISED. IER-677 (DEC 1989).
C MARK 14B REVISED. IER-840 (MAR 1990).
C SUBROUTINE FOR THE DETERMINATION OF GAUSSIAN QUADRATURE RULES
C **************************************************************
C
C INPUT PARAMETERS
C
C ITYPE INTEGER WHICH SPECIFIES THE RULE TYPE CHOSEN
C WEIGHT W(X) INTERVAL RESTRICTIONS
C 0 1 A,B B.GT.A
C 1 (B-X)**C*(X-A)**D A,B B.GT.A,C,D.GT.-1
C 2 ABS(X-0.5*(A+B))**C A,B C.GT.-1,B.GT.A
C 3 ABS(X-A)**C*EXP(-B*X) A,INF C.GT.-1,B.GT.0
C 3 ABS(X-A)**C*EXP(-B*X) -INF,A C.GT.-1,B.LT.0
C 4 ABS(X-A)**C*EXP(-B*(X-A)**2) -INF,INF C.GT.-1,B.GT.0
C 5 ABS(X-A)**C/ABS(X+B)**D A,INF A.GT.-B,C.GT.-1,D.GT.C+1
C 5 ABS(X-A)**C/ABS(X+B)**D -INF,A A.LT.-B,C.GT.-1,D.GT.C+1
C ABS(ITYPE) MUST BE LESS THAN 6. IF ITYPE IS GIVEN LESS THAN
C ZERO THEN THE ADJUSTED WEIGHTS ARE CALCULATED. IF NPNTS IS
C ODD AND ITYPE EQUALS -2 OR -4 AND C IS NOT ZERO, THERE MAY BE
C PROBLEMS.
C
C AA REAL PARAMETER USED TO SPECIFY RULE TYPE. SEE ITYPE.
C
C BB REAL PARAMETER USED TO SPECIFY RULE TYPE. SEE ITYPE.
C
C CC REAL PARAMETER USED TO SPECIFY RULE TYPE. SEE ITYPE.
C
C DD REAL PARAMETER USED TO SPECIFY RULE TYPE. SEE ITYPE.
C
C IFAIL NAG FAILURE PARAMETER. SEE NAG DOCUMENTATION.
C
C NPNTS INTEGER THAT DETERMINES DIMENSION OF WEIGHT AND ABSCIS
C
C OUTPUT PARAMETERS
C
C WEIGHT REAL ARRAY OF DIMENSION NPNTS WHICH CONTAINS
C RULE WEIGHTS
C
C ABSCIS REAL ARRAY OF DIMENSION NPNTS WHICH CONTAINS
C RULE ABSCISSAE
C
C IFAIL INTEGER NAG FAILURE PARAMETER
C IFAIL=0 FOR NORMAL EXIT
C IFAIL=1 FOR FAILURE IN NAG ROUTINE F02AVF
C IFAIL=2 FOR PARAMETER NPNTS OR ITYPE OUT OF RANGE
C IFAIL=3 FOR PARAMETER AA OR BB OR CC OR DD OUT OF
C ALLOWED RANGE
C IFAIL=4 FOR OVERFLOW IN CALCULATION OF WEIGHTS
C IFAIL=5 FOR UNDERFLOW IN CALCULATION OF WEIGHTS
C IFAIL=6 FOR ITYPE=-2 OR -4, NPNTS ODD, C NOT ZERO
C
C **************************************************************
SUBROUTINE D01BDF(F,A,B,EPSABS,EPSREL,RESULT,ABSERR)
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
C BASED ON QUADPACK ROUTINE QNG.
SUBROUTINE D01DAF(YA,YB,PHI1,PHI2,F,ABSACC,ANS,NPTS,IFAIL)
C
C EVALUATES A DOUBLE INTEGRAL BY REPEATED APPLICATIONS
C OF PATTERSON'S METHOD.
C NAG COPYRIGHT 1975
C MARK 5 RELEASE
C MARK 7 REVISED IER-138 (DEC 1978)
C MARK 11.5(F77) REVISED. (SEPT 1985.)
SUBROUTINE D01EAF(NDIM,A,B,MINCLS,MAXCLS,NFUN,FUNSUB,ABSREQ,
* RELREQ,LENWRK,WORK,FINEST,ABSEST,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C ADAPTIVE MULTIDIMENSIONAL INTEGRATION SUBROUTINE
C
C AUTHOR - ALAN GENZ, COMPUTER SCIENCE DEPARTMENT,
C WASHINGTON STATE UNIVERSITY,
C PULLMAN, WASHINGTON 99164
C
C THIS SUBROUTINE COMPUTES AN APPROXIMATION TO THE INTEGRAL
C
C B(1) B(2) B(NDIM)
C I I ... I (F ,F ,...,F ) DX(NDIM)...DX(2)DX(1),
C A(1) A(2) A(NDIM) 1 2 NFUN
C
C WHERE F = F (X ,X ,...,X ), I = 1,2,...,NFUN.
C I I 1 2 NDIM
C
C ********* PARAMETERS FOR D01EAF *******************************
C INPUT PARAMETERS
C NDIM NUMBER OF VARIABLES
C A REAL ARRAY OF LOWER LIMITS, WITH DIMENSION(NDIM)
C B REAL ARRAY OF UPPER LIMITS, WITH DIMENSION(NDIM)
C MINCLS MINIMUM NUMBER OF FUNSUB CALLS TO BE ALLOWED,
C MINCLS MUST NOT EXCEED MAXCLS. IF MINCLS.LT.0 THEN THE
C ROUTINE ASSUMES A PREVIOUS CALL HAS BEEN MADE WITH THE
C SAME INTEGRANDS AND CONTINUES THAT CALCULATION.
C MAXCLS MAXIMUM NUMBER OF FUNSUB CALLS TO BE ALLOWED,
C WHICH MUST BE AT LEAST IRCLS, WHERE
C IRCLS = 2**NDIM+2*NDIM**2+2*NDIM+1, IF NDIM .LT. 11 OR
C IRCLS = 1+NDIM*(12+(NDIM-1)*(6+(NDIM-2)*4))/3, OTHERWISE.
C NFUN NUMBER OF INTEGRANDS
C FUNSUB EXTERNALLY DECLARED USER DEFINED INTEGRAND SUBROUTINE
C IT MUST HAVE PARAMETERS (NDIM,Z,NFUN,F), WHERE Z IS A REAL
C ARRAY OF DIMENSION(NDIM) AND F IS A REAL ARRAY OF
C DIMENSION (NFUN). F(I) MUST GIVE THE VALUE OF INTEGRAND
C I AT Z.
C ABSREQ REQUIRED ABSOLUTE ACCURACY
C RELREQ REQUIRED RELATIVE ACCURACY
C LENWRK LENGTH OF ARRAY WORK OF WORKING STORAGE, THE ROUTINE
C NEEDS LENWRK AT LEAST AS LARGE AS 8*NDIM + 11*NFUN + 3.
C FOR MAXIMUM EFFICIENCY LENWRK SHOULD BE APPROXIMATELY
C 6*NDIM+9*NFUN+(NDIM+NFUN+2)*(1+MAXCLS/IRCLS) IF
C MAXCLS FUNCTION CALLS ARE USED. IF LENWRK IS SIGNIFICANTLY
C LESS THAN THIS D01EAF WILL WORK LESS EFFICIENTLY.
C
C OUTPUT PARAMETERS
C MINCLS ACTUAL NUMBER OF FUNSUB CALLS USED BY D01EAF
C WORK REAL ARRAY OF WORKING STORAGE OF DIMENSION (LENWRK).
C FINEST REAL ARRAY OF DIMENSION(NFUN) OF ESTIMATED VALUES OF
C INTEGRALS.
C ABSEST REAL ARRAY DIMENSION(NFUN) ESTIMATED ABSOLUTE ACCURACIES.
C ABSEST(I) GIVES THE ESTIMATED ACCURACY FOR FINEST(I).
C IFAIL FAILURE INDICATOR
C IFAIL=0 FOR NORMAL EXIT, WHEN MAX(ABSEST) .LE. ABSREQ OR
C MAX(ABSEST) .LE. MAX(ABS(FINEST))*RELREQ WITH
C MAXCLS OR LESS FUNSUB CALLS MADE.
C IFAIL=1 IF MAXCLS WAS TOO SMALL FOR D01EAF TO OBTAIN THE
C REQUIRED RELATIVE ACCURACY EPS. IN THIS CASE
C D01EAF RETURNS VALUES OF FINEST WITH ESTIMATED
C ABSOLUTE ACCURACIES ABSEST.
C IFAIL=2 IF LENWRK TOO SMALL FOR MAXCLS FUNCTION CALLS.
C IN THIS CASE D01EAF RETURNS VALUES OF FINEST WITH
C ESTIMATED ACCURACIES ABSEST USING THE WORKING
C STORAGE AVAILABLE.
C IFAIL=3 RE-ENTRY WITH MAXCLS TOO SMALL TO MAKE ANY
C PROGRESS
C IFAIL=4 IF MINCLS .GT. MAXCLS OR MAXCLS .LT. IRCLS
C OR LENWRK .LT. 8*NDIM + 11*NFUN + 3.
C ******************************************************************
C
DOUBLE PRECISION FUNCTION D01FBF(NDIM,NPTVEC,LWA,WEIGHT,ABSCIS,
* FUN,IFAIL)
C *** MULTIDIMENSIONAL INTEGRATION RULE EVALUATION
C **************************************************************
C
C INPUT PARAMETERS
C NDIM INTEGER WHICH SPECIFIES THE NUMBER OF VARIABLES
C NDIM MUST BE GREATER THAN ONE AND LESS THAN 21.
C NPTVEC INTEGER ARRAY OF DIMENSION NDIM. NPTVEC(J) SPECIFIES
C THE NUMBER OF FUNCTION EVALUATION POINTS IN THE
C INTEGRATION RULE FOR VARIABLE J.
C LWA INTEGER WHICH SPECIFIES THE DIMENSION OF THE ARRAYS
C WEIGHT AND ABSCIS. LWA .GE. THE SUM OF THE COMPONENTS
C OF NPTVEC.
C WEIGHT REAL ARRAY OF DIMENSION LWA. THIS ARRAY MUST
C CONTAIN ALL THE INTEGRATION RULE WEIGHTS IN
C THE CORRECT ORDER.
C ABSCIS REAL ARRAY OF DIMENSION LWA. THIS ARRAY MUST
C CONTAIN ALL THE INTEGRATION RULE ABSCISSAE IN
C THE CORRECT ORDER.
C FUN EXTERNALLY DECLARED USER DEFINED INTEGRAND
C WHICH MUST HAVE PARAMETERS NDIM AND A REAL
C ARRAY OF DIMENSION NDIM.
C IFAIL INTEGER NAG FAILURE PARAMETER. SEE NAG DOCUMENTATION.
C ON ENTRY IFAIL SHOULD ALWAYS BE SET TO ZERO
C
C OUTPUT PARAMETERS
C D01FBF REAL VALUE WHICH GIVES THE VALUE OF THE
C INTEGRATION RULE WHEN APPLIED TO THE INTEGRAND
C FUN.
C IFAIL INTEGER NAG FAILURE PARAMETER
C IFAIL=0 FOR NORMAL EXIT
C IFAIL=1 IF NDIM.LT.1 OR NDIM.GT.20 OR LWA TOO SMALL
C **************************************************************
SUBROUTINE D01FCF(NDIM,A,B,MINPTS,MAXPTS,FUNCTN,EPS,ACC,LENWRK,
* WRKSTR,FINVAL,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 10B REVISED. IER-399 (JAN 1983).
C MARK 11C REVISED. IER-464 (MAR 1985).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 12A REVISED. IER-498 (AUG 1986).
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C
C ADAPTIVE MULTIDIMENSIONAL INTEGRATION SUBROUTINE
C
C ********* PARAMETERS FOR D01FCF ****************************
C
C INPUT PARAMETERS
C
C NDIM INTEGER NUMBER OF VARIABLES, MUST EXCEED 1 BUT
C NOT EXCEED 15.
C
C A REAL ARRAY OF LOWER LIMITS, WITH DIMENSION NDIM
C
C B REAL ARRAY OF UPPER LIMITS, WITH DIMENSION NDIM
C
C MINPTS INTEGER MINIMUM NUMBER OF INTEGRAND VALUES TO BE
C ALLOWED, WHICH MUST NOT EXCEED MAXPTS.
C
C MAXPTS INTEGER MAXIMUM NUMBER OF INTEGRAND VALUES TO BE
C ALLOWED, WHICH MUST BE AT LEAST
C 2**NDIM+2*NDIM**2+2*NDIM+1.
C
C FUNCTN EXTERNALLY DECLARED USER DEFINED REAL FUNCTION
C INTEGRAND. IT MUST HAVE PARAMETERS (NDIM,Z),
C WHERE Z IS A REAL ARRAY OF DIMENSION NDIM.
C
C EPS REAL REQUIRED RELATIVE ACCURACY, MUST BE GREATER
C THAN ZERO
C
C LENWRK INTEGER LENGTH OF ARRAY WRKSTR, MUST BE AT LEAST
C 2*NDIM+4.
C
C IFAIL INTEGER NAG FAILURE PARAMETER
C IFAIL=0 FOR HARD FAIL
C IFAIL=1 FOR SOFT FAIL
C
C OUTPUT PARAMETERS
C
C MINPTS INTEGER NUMBER OF INTEGRAND VALUES USED BY THE
C ROUTINE
C
C WRKSTR REAL ARRAY OF WORKING STORAGE OF DIMENSION (LENWRK).
C
C ACC REAL ESTIMATED RELATIVE ACCURACY OF FINVAL
C
C FINVAL REAL ESTIMATED VALUE OF INTEGRAL
C
C IFAIL IFAIL=0 FOR NORMAL EXIT, WHEN ESTIMATED RELATIVE
C LESS INTEGACCURACY RAND VALUES USED.
C
C IFAIL=1 IF NDIM.LT.2, NDIM.GT.15, MINPTS.GT.MAXPTS,
C MAXPTS.LT.2**NDIM+2*NDIM*(NDIM+1)+1, EPS.LE.0
C OR LENWRK.LT.2*NDIM+4.
C
C IFAIL=2 IF MAXPTS WAS TOO SMALL FOR D01FCF TO OBTAIN THE
C REQUIRED RELATIVE ACCURACY EPS. IN THIS
C CASE D01FCF RETURNS A VALUE OF FINVAL
C WITH ESTIMATED RELATIVE ACCURACY ACC.
C
C IFAIL=3 IF LENWRK TOO SMALL FOR MAXPTS INTEGRAND
C VALUES. IN THIS CASE D01FCF RETURNS A
C VALUE OF FINVAL WITH ESTIMATED ACCURACY
C ACC USING THE WORKING STORAGE
C AVAILABLE, BUT ACC WILL BE GREATER
C THAN EPS.
C
C **************************************************************
C
SUBROUTINE D01FDF(N,F,SIGMA,REGION,LIMIT,R0,U,RESULT,NPTS,IFAIL)
C MARK 10 RELEASE. NAG COPYRIGHT 1982.
C MARK 10C REVISED. IER-425 (JUL 1983).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C
C MULTIPLE INTEGRATION
C
C THIS SUBROUTINE CALCULATES AN APPROXIMATION TO-
C D1 DN
C I ......I F(X1,...,XN) DX1 .. DXN (SIGMA .LT. 0)
C C1 CN
C
C OR
C
C I ......I F(X1,...,XN) DX1 .. DXN (SIGMA .GE. 0)
C N-SPHERE RADIUS SIGMA
C
C MULTIPLE INTEGRATION BASED ON THE METHOD OF SAG AND SZEKERES,
C NUMERICAL EVALUATION OF HIGH DIMENSIONAL INTEGRALS -
C MATHS.COMP. V18,245-253,1964.
C
C INPUT ARGUMENTS
C ----- ----------
C
C N - INTEGER NUMBER OF DIMENSIONS (1 .LE. N .LE. 30)
C F - EXTERNALLY DECLARED REAL USER FUNCTION INTEGRAND
C HAVING ARGUMENTS (N,X) WHERE X IS A REAL ARRAY
C OF DIMENSION N WHOSE ELEMENTS ARE THE VARIABLE
C VALUES.
C SIGMA - THIS DETERMINES THE DOMAIN OF INTEGRATION AS FOLLOWS
C (A) SIGMA .LT. 0.
C THE INTEGRATION IS CARRIED OUT OVER THE PRODUCT
C REGION DETERMINED BY THE USER SUPPLIED SUBROUTINE
C REGION.
C (B) SIGMA .GE. 0.
C THE INTEGRATION IS CARRIED OUT OVER THE N-SPHERE
C OF RADIUS SIGMA. IN THIS CASE SUBROUTINE REGION IS
C NOT CALLED AND IS SIMPLY SUPPLIED AS A DUMMY ROUTINE.
C REGION- EXTERNALLY DECLARED USER SUBROUTINE WITH ARGUMENTS
C (N,X,J,C,D) WHICH CALCULATES THE LOWER LIMIT C
C AND THE UPPER LIMIT D CORRESPONDING TO THE ARRAY
C VARIABLE X(J). C AND D MAY DEPEND ON X(1)..X(J-1).
C LIMIT - APPROXIMATE NO. OF INTEGRAND EVALUATIONS TO BE USED.
C IT MUST BE GREATER THAN 100. IN VERY LOW DIMENSIONS
C IT MAY NOT BE POSSIBLE TO ATTAIN THIS AS AN UPPER
C LIMIT DUE TO INTERNAL STORAGE LIMITATIONS (YY(30)).
C R0 - ADJUSTABLE PARAMETER, TYPICAL VALUE 0.8,
C (0 .LT. R0 .LE. 1)
C U - ADJUSTABLE PARAMETER, TYPICAL VALUE 1.5,(U .GT. 0)
C IFAIL - INTEGER, NAG FAILURE PARAMETER
C IFAIL=0, HARD FAIL
C IFAIL=1, SOFT FAIL
C
C OUTPUT ARGUMENTS
C ------ ----------
C
C RESULT- APPROXIMATE VALUE OF INTEGRAL
C NPTS - ACTUAL NUMBER OF INTEGRAND EVALUATIONS.
C IFAIL - IFAIL=0, NORMAL EXIT
C IFAIL=1, INVALID N VALUE (N .GT. 30)
C IFAIL=2, INVALID LIMIT VALUE
C IFAIL=3, INVALID R0 VALUE
C IFAIL=4, INVALID U VALUE
C
C REGION
SUBROUTINE D01GAF(X,Y,N,ANS,ER,IFAIL)
C
C THIS SUBROUTINE INTEGRATES A FUNCTION (Y) SPECIFIED
C NUMERICALLY AT N POINTS (X), WHERE N IS AT LEAST 4,
C OVER THE RANGE X(1) TO X(N). THE POINTS NEED NOT BE
C EQUALLY SPACED, BUT SHOULD BE DISTINCT AND IN ASCENDING
C OR DESCENDING ORDER. AN ERROR ESTIMATE IS RETURNED.
C THE METHOD IS DUE TO GILL AND MILLER.
C
C NAG COPYRIGHT 1975
C MARK 5 RELEASE
C MARK 7 REVISED IER-154 (DEC 1978)
C MARK 11.5(F77) REVISED. (SEPT 1985.)
SUBROUTINE D01GBF(NUMVAR,A,B,MINPTS,MAXPTS,FUNCTN,RELEPS,RELERR,
* LENWRK,WRKSTR,FINVAL,IFAIL)
C MARK 10 RELEASE. NAG COPYRIGHT 1982.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C ADAPTIVE MONTE-CARLO MULTIDIMENSIONAL INTEGRATION SUBROUTINE
C THIS SUBROUTINE COMPUTES AN APPROXIMATION TO THE INTEGRAL
C
C B(1) B(2) B(NUMVAR)
C I I ... I FUNCTN(NUMVAR,X) DX(NUMVAR)...DX(2)DX(1)
C A(1) A(2) A(NUMVAR)
C
C ***** INPUT PARAMETERS
C
C NUMVAR INTEGER NUMBER OF VARIABLES, MUST EXCEED 0
C A REAL ARRAY OF LOWER LIMITS, WITH DIMENSION NUMVAR
C B REAL ARRAY OF UPPER LIMITS, WITH DIMENSION NUMVAR
C MINPTS INTEGER MINIMUM NUMBER OF FUNCTION EVALUATIONS TO BE
C ALLOWED. MINPTS MUST BE LESS THAN MAXPTS.
C IF MINPTS IS NEGATIVE, THIS SPECIFIES
C A)A CONTINUATION OF THE PREVIOUS CALCULATION USING
C THE SAME INTEGRAND, OR
C B)THE BEGINNING OF A NEW CALCULATION WITH A SIMILAR
C INTEGRAND BUT USING THE SUBDIVISION THAT WAS USED
C AT THE END OF THE PREVIOUS CALL OF D01GBF
C MAXPTS INTEGER MAXIMUM NUMBER OF FUNCTION EVALUATIONS TO BE
C ALLOWED, WHICH MUST EXCEED MINPTS AND MUST BE AT LEAST
C 4*NUMVAR+4. IN THE CONTINUATION CASE THIS IS THE
C NUMBER OF NEW FUNCTION EVALUATIONS TO BE ALLOWED.
C FUNCTN REAL EXTERNALLY DECLARED USER DEFINED FUNCTION
C INTEGRAND. IT MUST HAVE PARAMETERS (NUMVAR,X), WHERE X
C IS A REAL ARRAY OF DIMENSION NUMVAR.
C RELEPS REAL REQUIRED RELATIVE ACCURACY
C LENWRK INTEGER LENGTH OF ARRAY WRKSTR OF WORKING STORAGE,
C WHICH MUST BE AT LEAST 10*NUMVAR WITH RECOMMENDED
C VALUE 3*NUMVAR*((MAXPTS/4)**(1/NUMVAR))+7*NUMVAR
C WRKSTR REAL ARRAY OF WORKING STORAGE OF DIMENSION(LENWRK).
C WHEN MINPTS IS SET NEGATIVE FOR CASE B), THEN
C WRKSTR(LENWRK) MUST BE SET TO ZERO. IN CASES A) AND
C B) OTHER VALUES OF WRKSTR MUST NOT BE CHANGED BETWEEN
C SUCCESSIVE CALLS.
C IFAIL INTEGER NAG FAILURE PARAMETER
C
C ***** OUTPUT PARAMETERS
C
C WRKSTR REAL ARRAY OF WORKING STORAGE OF DIMENSION (LENWRK),
C ON EXIT THIS CONTAINS INFORMATION WHICH COULD BE USED
C IN LATER CALLS OF D01GBF WITH THE SAME OR A SIMILAR
C INTEGRAND.
C RELERR REAL ESTIMATED RELATIVE ACCURACY OF FINVAL
C FINVAL REAL ESTIMATED VALUE OF INTEGRAL. IN THE CONTINUATION
C CASE A FINVAL MUST NOT BE CHANGED BETWEEN SUCCESSIVE
C CALLS.
C IFAIL IFAIL=0 FOR NORMAL EXIT, WHEN ESTIMATED RELATIVE
C ACCURACY IS LESS THAN RELEPS WITH FEWER THAN
C MAXPTS FUNCTION CALLS MADE.
C IFAIL=1 IF NUMVAR.LT.1, MINPTS.GE.MAXPTS,
C LENWRK.LT.10*NUMVAR OR MAXPTS.LT.4*(NUMVAR+1)
C OR RELEPS.LT.0.0.
C IFAIL=2 IF MAXPTS WAS TOO SMALL FOR D01GBF TO OBTAIN
C THE REQUIRED ACCURACY. IN THIS CASE D01GBF
C RETURNS A VALUE OF FINVAL WITH ESTIMATED
C RELATIVE ACCURACY RELERR.
C
C FAIL FOR TOO MANY OR TOO FEW VARIABLES OR LENWRK TOO SMALL OR
C MINPTS.GE.MAXPTS OR MAXPTS.LT.4*NUMVAR+4
C
SUBROUTINE D01GCF(N,F,REGION,NPTS,VK,NRAND,ITRANS,RES,ERR,IFAIL)
C MARK 10 RELEASE. NAG COPYRIGHT 1982.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C MULTIPLE INTEGRATION
C
C THIS SUBROUTINE CALCULATES AN APPROXIMATION TO
C D1 DN
C I ......I F(X1,...,XN) DX1 .. DXN
C C1 CN
C
C USING THE KOROBOV-CONROY NUMBER-THEORETIC METHOD.
C (N.M.KOROBOV,NUMBER THEORETIC METHODS IN APPROXIMATE ANALYSIS,
C FIZMATGIZ,MOSCOW,1963, H.CONROY,J.CHEM.PHYS.47,(1967),
C 5307-5813)
C
C INPUT ARGUMENTS
C
C N -INTEGER SPECIFYING THE NUMBER OF DIMENSIONS.
C 1 .LE. N .LE. 20.
C
C F -EXTERNALLY DECLARED REAL USER FUNCTION INTEGRAND
C HAVING ARGUMENTS (N,X) WHERE X IS A REAL ARRAY
C OF DIMENSION N WHOSE ELEMENTS ARE THE VARIABLE
C VALUES.
C
C REGION -EXTERNALLY DECLARED USER SUBROUTINE WITH ARGUMENTS
C (N,X,J,C,D) WHICH CALCULATES THE LOWER LIMIT C
C AND THE UPPER LIMIT D CORRESPONDING TO THE ARRAY
C VARIABLE X(J). C AND D MAY DEPEND ON X(1)..X(J-1).
C
C NPTS -INTEGER VARIABLE WHICH SPECIFIES THE KOROBOV RULE
C TO BE USED. THERE ARE TWO ALTERNATIVES DEPENDING
C ON THE VALUE OF NPTS
C (A) 1 .LE. NPTS .LE. 6
C IN THIS CASE ONE OF THE SIX PRESET RULES IS
C CHOSEN USING 2129, 5003, 10007, 20011, 40009 OR
C 80021 POINTS ACCORDING TO THE RESPECTIVE VALUE
C OF NPTS.
C (B) NPTS .GT. 6
C NPTS IS THE NUMBER OF ACTUAL POINTS TO BE USED
C WITH CORRESPONDING OPTIMAL COEFFICIENTS SUPPLIED
C BY THE USER IN ARRAY VK.
C
C VK -REAL ARRAY CONTAINING N OPTIMAL COEFFICIENTS
C FOR THE CASE NPTS .GT. 6.
C
C NRAND -INTEGER NUMBER SPECIFYING THE NUMBER OF RANDOM
C SAMPLES TO BE GENERATED IN THE ERROR ESTIMATION.
C (GENERALLY A SMALL VALUE ,SAY 3 TO 5, IS SUFFICIENT)
C THE TOTAL NUMBER OF INTEGRAND EVALUATIONS WILL BE
C NRAND*NPTS.
C
C ITRANS -THIS SHOULD BE SET TO 0 OR 1. THE NORMAL SETTING
C IS 0. WHEN SET TO 1 THE PERIODIZING TRANSFORMATION
C IS SUPPRESSED (TO COVER CASES WHERE THE INTEGRAND IS
C ALREADY PERIODIC OR WHERE THE USER DESIRES TO
C SPECIFY A PARTICULAR TRANSFORMATION IN THE
C DEFINITION OF F).
C
C IFAIL -INTEGER VARIABLE SPECIFYING THE ERROR REPORTING
C OPTION. IFAIL IS SET TO 0 FOR HARD FAIL AND TO 1 FOR
C SOFT FAIL REPORT.
C
C
C OUTPUT ARGUMENTS
C
C VK -WHEN THE SUBROUTINE IS CALLED WITH 1 .LE. NPTS
C .LE. 6 THEN N ELEMENTS OF ARRAY VK WILL BE SET
C TO THE OPTIMAL COEFFICIENTS USED BY THE PRESET RULE.
C
C RES -APPROXIMATION TO THE VALUE OF THE INTEGRAL.
C
C ERR -STANDARD ERROR AS COMPUTED FROM NRAND SAMPLE VALUES.
C IF NRAND IS SET TO 1 THEN ERR WILL BE RETURNED WITH
C VALUE 0.
C
C IFAIL -THIS REPORTS THE ERROR CONDITIONS AS FOLLOWS
C IFAIL=0 NO ERROR DETECTED.
C =1 N FAILS TO SATISFY 1 .LE. N .LE. 20
C =2 NPTS .LT. 1.
C =3 NRAND .LT. 1.
C
C ************* IMPLEMENTATION-DEPENDENT CODING ****************
C THIS ROUTINE REQUIRES AT ONE POINT TO PERFORM EXACT INTEGER
C ARITHMETIC WITH AT LEAST 32 BITS AVAILABLE FOR INTERMEDIATE
C RESULTS. THE INTEGERS ARE STORED AS FLOATING-POINT VARIABLES
C (IN ORDER TO AVOID INTEGER OVERFLOW ON SOME MACHINES). IF THE
C MACHINE DOES NOT PROVIDE AT LEAST 32 BITS OF BASIC FLOATING-
C POINT PRECISION, THEN ONE STATEMENT IN THE CODE MUST BE
C CONVERTED TO ADDITIONAL PRECISION (SEE COMMENTS IN CODE).
C **************************************************************
C
C REGION
SUBROUTINE D01GDF(N,VECFUN,VECREG,NPTS,VK,NRAND,ITRANS,RES,ERR,
* IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
SUBROUTINE D01GYF(N,NPTS,VK,IFAIL)
C MARK 10 RELEASE. NAG COPYRIGHT 1982.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C
C CALCULATION OF THE KOROBOV OPTIMAL COEFFICIENTS
C FOR A PRIME NUMBER OF POINTS.
C
C SEE KOROBOV,NUMBER THEORETIC METHODS IN APPROXIMATE ANALYSIS,
C FIZMATGIZ,MOSCOW,1963.
C
C INPUT ARGUMENTS
C
C N -INTEGER NUMBER OF DIMENSIONS.
C
C NPTS -PRIME INTEGER SPECIFYING THE NUMBER OF POINTS
C TO BE USED.
C
C IFAIL -INTEGER VARIABLE SPECIFYING THE ERROR
C REPORTING OPTION. IFAIL IS SET TO 0 FOR HARD
C FAIL AND TO 1 FOR SOFT FAIL REPORT
C
C OUTPUT ARGUMENTS
C
C VK -REAL ARRAY WHOSE FIRST N ELEMENTS CONTAIN
C THE OPTIMAL COEFFICIENTS.
C
C IFAIL -THIS REPORTS THE ERROR CONDITIONS AS FOLLOWS
C IFAIL=0 NO ERROR REPORTED
C IFAIL=1 N .LT. 1
C IFAIL=2 NPTS .LT. 5
C IFAIL=3 NPTS IS NOT PRIME
C IFAIL=4 INSUFFICIENT PRECISION
C
C VALIDITY CHECK
SUBROUTINE D01GZF(N,NP1,NP2,VK,IFAIL)
C MARK 10 RELEASE. NAG COPYRIGHT 1982.
C MARK 10C REVISED. IER-426 (JUL 1983).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C
C CALCULATION OF THE KOROBOV OPTIMAL COEFFICIENTS
C FOR NUMBERS OF POINTS WHICH ARE PRODUCTS OF PRIMES
C
C SEE KOROBOV,NUMBER THEORETIC METHODS IN APPROXIMATE ANALYSIS,
C FIZMATGIZ,MOSCOW,1963.
C
C INPUT ARGUMENTS
C
C N -INTEGER NUMBER OF DIMENSIONS.
C
C NP1,NP2 -PRIME INTEGERS SPECIFYING THE NUMBER OF POINTS
C TO BE USED AS NP1*NP2.
C
C IFAIL -INTEGER VARIABLE SPECIFYING THE ERROR
C REPORTING OPTION. IFAIL IS SET TO 0 FOR HARD
C FAIL AND TO 1 FOR SOFT FAIL REPORT.
C
C OUTPUT ARGUMENTS
C
C VK -REAL ARRAY WHOSE FIRST N ELEMENTS CONTAIN
C THE OPTIMAL COEFFICIENTS.
C
C IFAIL -THIS REPORTS THE ERROR CONDITIONS AS FOLLOWS
C IFAIL=0 NO ERROR DETECTED
C IFAIL=1 N .LT. 1
C IFAIL=2 NP1.LT.5 OR NP2.LT.2 OR NP2.GE.NP1
C IFAIL=3 NP1*NP2 OVERFLOWS
C IFAIL=4 NP1 IS NOT PRIME
C IFAIL=5 NP2 IS NOT PRIME
C IFAIL=6 INSUFFICIENT PRECISION
C
C VALIDITY CHECK
SUBROUTINE D01JAF(F,N,RADIUS,EPSA,EPSR,METHOD,ICOORD,RESULT,
* ESTERR,EVALS,IFAIL)
C MARK 10 RELEASE. NAG COPYRIGHT 1982.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C
C **************************************************************
C
C AUTOMATIC INTEGRATION OVER A SPHERE
C -----------------------------------
C
C STANDARD FORTRAN SUBROUTINE
C
C PURPOSE
C -------
C
C THE ROUTINE CALCULATES AN APPROXIMATION RESULT TO THE
C INTEGRAL
C
C J = I ... I F(X1, ..., XN) DX1 ... DXN
C S
C
C WHERE S IS THE HYPERSPHERE (X1**2 + ...XN**2)**0.5.LE.RADIUS.
C THE INTEGRAND FUNCTION MAY ALSO BE DEFINED IN SPHERICAL
C COORDINATES (SEE PARAMETER ICOORD).
C
C IF EVERYTHING GOES WELL,
C ABS(RESULT-J).LE.MAX(EPSA,EPSR*ABS(J)).
C
C PARAMETERS
C ----------
C
C ON ENTRY
C --------
C F - REAL FUNCTION, SUPPLIED BY THE USER, IT
C EVALUATES THE INTEGRAND AT THE POINT (X(1),...,X(N))
C (IN CARTESIAN COORDINATES) OR (RO,THETA(1),...,
C THETA(N-1)) (IN SPHERICAL COORDINATES) (SEE
C PARAMETER ICOORD). F MUST BE DECLARED EXTERNAL
C IN THE CALLING (SUB)PROGRAM.
C CARE SHOULD BE TAKEN TO AVOID UNDERFLOW AND/OR
C OVERFLOW PROBLEMS IN THE INTEGRAND FUNCTION
C SUBPROGRAM, BECAUSE SOME OF THE INTEGRATION NODES
C USED BY D01JAF ARE VERY CLOSE TO THE SURFACE
C OR TO THE CENTRE OF THE SPHERE.
C F IS SPECIFIED AS
C
C REAL FUNCTION F(N,X)
C C INTEGER N
C C REAL X(N)
C
C N - DIMENSION OF THE SPHERE (N.LE.4, FOR N.GT.4 SEE
C DIMENSION INFORMATION BELOW)
C
C RADIUS - RADIUS OF THE SPHERE
C
C EPSA - REQUESTED ABSOLUTE ACCURACY
C
C EPSR - REQUESTED RELATIVE ACCURACY
C
C THE ROUTINE ATTEMPTS TO CALCULATE AN APPROXIMATION
C RESULT TO THE INTEGRAL J, HOPEFULLY SATISFYING
C ABS(RESULT-J).LE.MAX(EPSA,EPSR*ABS(J)).
C
C METHOD - KEY FOR CHOOSING THE TRANSFORMATION TO BE
C USED BY THE ROUTINE, DEPENDING ON THE BEHAVIOUR OF
C THE INTEGRAND FUNCTION AND THE REQUESTED ACCURACY
C - DEFAULT VALUE METHOD = 0
C - WELL-BEHAVED FUNCTION OR FUNCTION WITH SINGULARITY
C ONLY ON THE SURFACE OF THE SPHERE-
C - LOW ACCURACY REQUESTED, METHOD = 1
C - HIGH ACCURACY REQUESTED, METHOD = 2
C - FUNCTION WITH STRONG SINGULARITY ON THE SURFACE-
C - LOW ACCURACY REQUESTED, METHOD = 3
C - HIGH ACCURACY REQUESTED, METHOD = 4
C IN THIS CASE THE INTEGRAND MUST BE DEFINED IN THE
C SPECIAL SPHERICAL COORDINATES (SEE ICOORD).
C - FUNCTION WITH SINGULARITY IN THE CENTRE OF THE
C SPHERE (SINGULARITY ON THE SURFACE IS ALLOWED
C AS WELL)-
C - LOW ACCURACY REQUESTED, METHOD = 5
C - HIGH ACCURACY REQUESTED, METHOD = 6
C WHEN METHOD = 0, EPSR.LE.1.E-6 IS CONSIDERED AS A
C HIGH, EPSR.GT.1.E-6 AS A LOW ACCURACY. HOWEVER, THE
C VALUE OF EPSR WHICH DISTINGUISHES HIGH FROM LOW
C TOLERANCES DEPENDS ALSO ON THE INTEGRAND. WHEN USING
C METHOD.GT.0, THIS VALUE MAY IN GENERAL BE DECREASED
C WHEN THE INTEGRAND IS SMOOTH, INCREASED WHEN IT
C IS BADLY BEHAVED.
C
C ICOORD - INDICATES WHICH KIND OF COORDINATES
C ARE USED IN THE FUNCTION SUBPROGRAM F(N,X)-
C ICOORD = 0- X(I) I = 1...N- CARTESIAN COORDINATES
C ICOORD = 1- X- SPHERICAL COORDINATES-
C X(1) = RO
C X(I) = THETA(I-1) I = 2...N
C ICOORD = 2- IF METHOD IS SET TO 3 OR 4, THE USER HAS
C TO DEFINE THE FUNCTION IN SPHERICAL
C COORDINATES, WITH THE ADDITIONAL
C TRANSFORMATION RO = RADIUS - LAMBDA.
C IN THIS CASE THE ROUTINE DOES ITS
C COMPUTATIONS WITH LAMBDA INSTEAD OF WITH
C RO, AND THUS
C X(1) = LAMBDA
C X(I) = THETA(I-1) I = 2...N.
C THE USER HAS TO TAKE CARE THAT THE
C FUNCTION SUBPROGRAM DOES NOT CONTAIN
C EXPRESSIONS OF THE FORM
C RADIUS - LAMBDA.
C REMARKS - THE CARTESIAN COORDINATES ARE RELATED
C TO THE SPHERICAL COORDINATES BY
C X(1) = RO*SIN(THETA(1))*...
C *SIN(THETA(N-2))*SIN(THETA(N-1))
C X(2) = RO*SIN(THETA(1))*...
C *SIN(THETA(N-2))*COS(THETA(N-1))
C X(3) = RO*SIN(THETA(1))*...
C *COS(THETA(N-2))
C X(N) = RO*COS(THETA(1))
C WHERE 0.LT.THETA(I).LE.PI I = 1...N-2
C 0.LT.THETA(N-1).LE.2.E0*PI
C
C ON EXIT
C -------
C RESULT - APPROXIMATION TO THE INTEGRAL
C
C ESTERR - ESTIMATE OF THE (MODULUS OF THE) ABSOLUTE ERROR
C
C EVALS - NUMBER OF FUNCTION EVALUATIONS NEEDED TO OBTAIN
C THE RESULT
C
C IFAIL - ERROR FLAG
C IFAIL= 0- NORMAL AND RELIABLE TERMINATION OF THE
C ROUTINE, IT IS ASSUMED THAT THE TOLERANCE
C CRITERIA HAVE BEEN SATISFIED.
C IFAIL= 1- THE REQUESTED ACCURACY CANNOT BE ACHIEVED
C WITHIN A LIMITING NUMBER OF FUNCTION
C EVALUATIONS (WHICH IS SET BY THE ROUTINE).
C IFAIL= 2- THE PRESENCE OF ROUNDOFF ERROR PREVENTS
C THE REQUESTED ACCURACY FROM BEING OBTAINED.
C IFAIL= 3- THE MAXIMUM ACCURACY WITH RESPECT TO THE
C MACHINE DEPENDENT CONSTANTS EPMACH AND
C UFLOW HAS BEEN ACHIEVED. IF THIS MAXIMUM
C ACCURACY IS RATHER LOW, IFAIL=3 MEANS THAT
C A SEVERE SINGULARITY ON THE BOUNDARY OR
C AT THE CENTRE OF THE SPHERE PREVENTS THE
C REQUESTED ACCURACY FROM BEING ACHIEVED.
C IF METHOD.LT.3, SETTING METHOD=3 OR 4 MAY
C HELP.
C IFAIL= 4- THE INPUT IS INVALID. NO CALCULATIONS HAVE
C BEEN PERFORMED (RESULT AND ESTERR ARE SET
C TO 0). THIS OCCURS WHEN
C N.LT.2 OR.GT.4,
C OR METHOD.LT.0 OR.GT.6,
C OR ICOORD.LT.0 OR.GT.2,
C OR METHOD.NE.3 OR 4 AND ICOORD = 2,
C OR METHOD = 3 OR 4 AND ICOORD.NE.2,
C OR RADIUS .LT. 0.0.
C
C SUBPROGRAMS CALLED
C ------------------
C
C - D01JAY
C - D01JAZ - D01JAX
C - D01JAW
C
C DIMENSION INFORMATION
C ---------------------
C
C TO USE THE ROUTINE FOR INTEGRATION OVER A SPHERE
C OF DIMENSION NDIM.GT.4, SET THE SIZE OF ALL THE
C ARRAYS IN THE ROUTINES D01JAF, D01JAZ, D01JAX AND D01JAW
C TO NDIM. FURTHERMORE DELETE THE TEST ON THE VALIDITY OF INPUT
C PARAMETER N.
C
C MACHINE DEPENDENT CONSTANTS
C ---------------------------
C
C EPMACH IS THE RELATIVE MACHINE ACCURACY.
C UFLOW IS ABOUT EQUAL TO, BUT MUST NOT BE LESS THAN
C THE SMALLEST POSITIVE REAL NUMBER REPRESENTABLE
C BY THE MACHINE.
C
C RMAX, RMIN AND RMID WILL BE CALCULATED AS FUNCTIONS OF THE
C ABOVE CONSTANTS. THEIR VALUES CAN INFLUENCE THE PERFORMANCE
C OF THE ROUTINE CONSIDERABLY.
C
C LABELLED COMMON AREAS
C ---------------------
C
C - AD01JA
C - BD01JA
C - CD01JA
C
C LIST OF MAJOR VARIABLES
C -----------------------
C
C X - THE CARTESIAN COORDINATES OF A GRID POINT
C
C ABSCIS - THE CARTESIAN OR SPHERICAL COORDINATES
C OF THE INTEGRATION NODE IN THE ORIGINAL REGION,
C CORRESPONDING TO X
C
C R - DISTANCE FROM THE CENTRE
C
C RO - DISTANCE FROM THE CENTRE IN THE ORIGINAL REGION,
C CORRESPONDING TO R
C
C IRAD2 - INPUT PARAMETER OF THE ROUTINE D01JAZ, IRAD2*H = R**2
C
C ITRANS - CODE FOR THE TRANSFORMATION TO BE USED, THE
C VALUE OF ITRANS IS DETERMINED BY THE INPUT PARAMETERS
C METHOD AND EPSR.
C
C EPSLIM - CRITICAL VALUE OF EPSR FOR CHOICE OF THE
C TRANSFORMATION WHEN METHOD = 0
C
C WEIGHT - THE JACOBIAN OF THE TRANSFORMATION IN THE GRID POINTS
C AT DISTANCE R EQUALS WEIGHT * RADIUS**N.
C
C RMAX - IF R.GT.RMAX- RADIUS*(1-EPMACH).LE.R0.LE.RADIUS, AND
C /OR TO PREVENT UNDERFLOW DURING THE CALCULATION OF
C THE WEIGHT.
C
C RMIN - IF METHOD = 5 OR 6 AND R.LT.RMIN, UNDERFLOW OCCURS
C DURING THE CALCULATION OF THE WEIGHT.
C
C RMID - IF R = RMID, THEN RO = RMAX / 2.
C
C BREAK1,- THE TRANSFORMED INTEGRAND IS NEGLIGIBLE WHEN
C BREAK2 R.GT.BREAK1 OR R.LT.BREAK2.
C
C TMIN - USED TO DETERMINE BREAK1 AND BREAK2
C
C SAVE - SAVEGUARD WITH RESPECT TO TRUNCATION ERROR
C
C H - STEPLENGTH USED IN EACH DIRECTION IN THE TRAPEZOIDAL
C RULE
C
C HFAK - THE INITIAL STEPLENGTH IS HFAK * RMAX / 2.
C
C ITRMAX - DETERMINES THE MAXIMUM NUMBER OF FUNCTION
C EVALUATIONS, THE STEPLENGTH H CANNOT BE HALVED
C MORE THAN ITRMAX TIMES.
C
C ERREXT - EXTRAPOLATED ERROR ESTIMATE
C
C RESFOR - APPROXIMATION TO THE INTEGRAL IN THE PREVIOUS
C STEP
C
C ERRDIF - SIMPLE ERROR ESTIMATE (=ABS(RESULT-RESFOR))
C
C ERREXT - EXTRAPOLATED ERROR ESTIMATE
C
C ERTRUN - TRUNCATION ERROR ESTIMATE
C
C RTRUN1 - GREATEST R (LESS THAN RMAX) USED IN THE PREVIOUS
C STEP
C
C RTRUN2 - IF METHOD=5 OR 6, THE SMALLEST R (GREATER THAN RMIN)
C USED IN THE PREVIOUS STEP
C
C STRUN1 - USED TO DETERMINE THE TRUNCATION ERROR ESTIMATE
C
C STRUN2 - USED TO DETERMINE THE TRUNCATION ERROR ESTIMATE
C
C SRAHN - USED TO COMPUTE TMIN- IN EACH STEP,
C SRAHN = SAVE / (RADIUS * H)**N.
C
C **************************************************************
C
SUBROUTINE D01PAF(NUMVAR,VERTEX,IV1,IV2,INTGND,MINORD,MAXORD,
* INTVLS,ESTERR,IFAIL)
C MARK 10 RELEASE. NAG COPYRIGHT 1982.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MULTIDIMENSIONAL SIMPLEX ROMBERG INTEGRATION SUBROUTINE
C SUBROUTINE FOR COMPUTING INTEGRALS OF THE TYPE
C
C I F(X) DX(1)DX(2)...DX(NUMVAR)
C S
C
C WHERE S IS A NUMVAR DIMENSIONAL SIMPLEX
C
C ************** PARAMETERS FOR D01PAF ***********************
C NUMVAR INTEGER NUMBER OF VARIABLES, MUST EXCEED 1.
C VERTEX REAL ARRAY OF SIMPLEX VERTICES WITH DIMENSION
C (IV1,IV2). VERTEX(I,J) IS THE JTH COORDINATE OF THE
C ITH VERTEX
C IV1 INTEGER FIRST DIMENSION OF VERTEX, MUST BE
C .GE. NUMVAR+1.
C IV2 INTEGER SECOND DIMENSION OF VERTEX, MUST BE
C .GE. 2*NUMVAR+2.
C INTGND EXTERNALLY DECLARED USER DEFINED REAL FUNCTION
C INTEGRAND.
C IT MUST HAVE PARAMETERS (NUMVAR,X), WHERE X IS A REAL
C ARRAY WITH DIMENSION NUMVAR.
C MINORD INTEGER MINIMUM ORDER PARAMETER, MUST BE AT LEAST 0
C AND LESS THAN MAXORD. WHEN MINORD IS NOT 0 THE
C SUBROUTINE ASSUMES A PREVIOUS CALL OR CALLS OF THE
C SUBROUTINE HAVE BEEN USED TO COMPUTE
C INTVLS(1), INTVLS(2),..., INTVLS(MINORD).
C ON EXIT, MINORD IS SET EQUAL TO MAXORD.
C MAXORD INTEGER MAXIMUM ORDER PARAMETER, MUST BE GREATER THAN
C MINORD THE SUBROUTINE COMPUTES INTVLS(MINORD+1),
C INTVLS(MINORD+2), ..., INTVLS(MAXORD).
C INTVLS REAL ARRAY OF DIMENSION(MAXORD). UPON SUCCESSFUL EXIT
C INTVLS(1), INTVLS(2),..., INTVLS(MAXORD) ARE
C APPROXIMATIONS TO THE INTEGRAL. INTVLS(J) WILL BE AN
C APPROXIMATION OF POLYNOMIAL DEGREE 2*J-1.
C ESTERR REAL ABSOLUTE ERROR ESTIMATE
C IFAIL INTEGER FAILURE OUTPUT PARAMETER
C IFAIL=0 FOR SUCCESSFUL TERMINATION OF THE SUBROUTINE
C IFAIL=1 WHEN PARAMETERS NUMVAR,MINORD,MAXORD,
C IV1 OR IV2 ARE OUT OF RANGE
C IFAIL=2 WHEN F03AAF FAILS FROM UNDERFLOW OR OVERFLOW
C IN THE COMPUTATION OF THE SIMPLEX VOLUME
C **************************************************************
SUBROUTINE D02AGF(H,ERROR,PARERR,PARAM,C,N,N1,M1,AUX,BCAUX,RAAUX,
* PRSOL,MAT,COPY,WSPACE,WSPAC1,WSPAC2,IFAIL)
C NAG COPYRIGHT 1975
C MARK 4.5 REVISED
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C MARK 15 REVISED. IER-898 (APR 1991). (LAPACK)
C
C SOLVES A GENERAL BOUNDARY VALUE
C PROBLEM FOR N DIFFERENTIAL EQUATIONS
C IN N1 PARAMETERS USING A SHOOTING
C AND MATCHING TECHNIQUE. EPS IS THE
C LARGEST REAL VARIABLE SUCH THAT 1+EPS=1
C ALL IMPLICITLY DECLARED REALS MAY BE USED DOUBLE-LENGTH
C THE ARRAY COPY IS REDUNDANT
SUBROUTINE D02BGF(X,XEND,N,Y,TOL,HMAX,M,VAL,FCN,W,IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C TO FIND POSITION AT WHICH A COMPONENT OF THE SOLUTION ATTAINS
C A GIVEN VALUE
C FCN
SUBROUTINE D02BHF(X,XEND,N,Y,TOL,IRELAB,HMAX,FCN,G,W,IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C CALCULATES THE POINT X,WHERE G(X,Y)=0
C FCN
SUBROUTINE D02CJF(T,TEND,NEQ,Y,FCN,TOL,RELABS,OUTPUT,G,RWORK,
* IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C MARK 14A REVISED. IER-680 (DEC 1989).
SUBROUTINE D02EJF(X,XEND,N,Y,FCN,PEDERV,TOL,RELABS,OUTPUT,G,W,IW,
* IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C MARK 12B REVISED. IER-551 (FEB 1987).
C MARK 13 REVISED. IER-586 (MAR 1988).
C MARK 13 REVISED. IER-611 (APR 1988).
C MARK 13 REVISED. IER-612 (APR 1988).
C MARK 13B REVISED. IER-654 (AUG 1988).
C MARK 16 REVISED. IER-971 (JUN 1993).
SUBROUTINE D02GAF(U,V,N,A,B,TOL,FCN,MNP,X,Y,NP,W,LW,IW,LIW,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 9 REVISED. IER-305 (SEP 1981).
C MARK 9C REVISED. IER-373 (JUN 1982)
C MARK 10 REVISED. IER-377 (JUN 1982).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C FCN
SUBROUTINE D02GBF(A,B,N,TOL,FCNF,FCNG,C,D,GAM,MNP,X,Y,NP,W,LW,IW,
* LIW,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 9 REVISED. IER-305 (SEP 1981).
C MARK 9C REVISED. IER-373 (JUN 1982)
C MARK 10 REVISED. IER-377 (JUN 1982).
C MARK 10A REVISED. IER-389 (OCT 1982).
C MARK 10B REVISED. IER-400 (JAN 1983).
C MARK 11 REVISED. IER-434 (FEB 1984).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. IER-606 (APR 1988).
C FCNF, FCNG
SUBROUTINE D02HAF(A,B,N,X,X1,TOL,FCN,SOLN,M1,W,IW,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C FCN
SUBROUTINE D02HBF(P,N1,PE,E,N,SOLN,M1,FCN,BC,RANGE,W,IW,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 9 REVISED. IER-306 (SEP 1981).
C MARK 9C REVISED. IER-366 (MAY 1982).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. IER-619 (APR 1988).
C BC, FCN, RANGE
SUBROUTINE D02JAF(N,CF,BC,X0,X1,K1,KP,C,W,LW,IW,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C BC
SUBROUTINE D02JBF(N,CF,BC,X0,X1,K1,KP,C,IC,W,LW,IW,LIW,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C BC
SUBROUTINE D02KAF(XL,XR,COEFFN,BCOND,K,TOL,ELAM,DELAM,MONIT,IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 8A REVISED. IER-257 (AUG 1980).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
SUBROUTINE D02KDF(XPOINT,NXP,COEFFN,BDYVAL,K,TOL,ELAM,DELAM,HMAX,
* MAXIT,MAXFUN,MONIT,IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 8A REVISED. IER-258 (AUG 1980).
C MARK 9 REVISED. IER-307 (SEP 1981).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C
C MEANING OF COMMON VARIABLES:
C ZER,ONE,TWO,PI - CONSTANTS WITH INDICATED VALUES
C LAMDA - CURRENT VALUE OF EIGENVALUE ESTIMATE. SET AND USED IN
C EIGODE. USED IN AUX,OPTSC.
C PSIGN - SAMPLE P(X) VALUE. SET IN PRUFER, USED IN AUX.
C MINSC - MINIMUM ALLOWED SCALEFACTOR. SET IN EIGODE, USED IN
C OPTSC.
C BP - USED BY THE POLYGONAL SCALING METHOD. THE CURRENT
C VALUE OF THE DERIVATIVE OF B(X). SET IN PRUFER, USED I
C
C BDYVAL, MONIT
SUBROUTINE D02KEF(XPOINT,NXP,IC1,COEFFN,BDYVAL,K,TOL,ELAM,DELAM,
* HMAX,MAXIT,MAXFUN,MONIT,REPORT,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C
C MEANING OF COMMON VARIABLES:
C ZER,ONE,TWO,PI - CONSTANTS WITH INDICATED VALUES
C LAMDA - CURRENT VALUE OF EIGENVALUE ESTIMATE. SET AND USED IN
C EIGODE. USED IN AUX,OPTSC.
C PSIGN - SAMPLE P(X) VALUE. SET IN PRUFER, USED IN AUX.
C MINSC - MINIMUM ALLOWED SCALEFACTOR. SET IN EIGODE, USED IN
C OPTSC.
C BP - USED BY THE POLYGONAL SCALING METHOD. THE CURRENT
C VALUE OF THE DERIVATIVE OF B(X). SET IN PRUFER, USED I
C
C BDYVAL, COEFFN, MONIT, REPORT
SUBROUTINE D02LAF(F,NEQ,T,TEND,Y,YP,YDP,RWORK,LRWORK,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C MARK 16A REVISED. IER-972 (JUN 1993).
C
C NOTE -
C THE ARRAYS IN COMMON ARE LARGE ENOUGH TO ACCOMODATE THE
C COEFFICIENTS OF THE 12(10) FORMULA PAIR. THIS ADDS CLARITY
C TO THE PROGRAM BUT SOME STORAGE COULD BE SAVED IF NECESSARY
C (BY ADDITIONAL PROGRAMMING EFFORT). TWO POSSIBLE WAYS OF
C SAVING STORAGE ARE:
C 1) STORE THE MATRIX OF COEFFICIENTS A IN STRICTLY LOWER
C TRIANGULAR FORM. THIS WOULD SAVE 153 LOCATIONS.
C 2) INCLUDE THE STORAGE REQUIREMENT FOR THE COEFFICIENTS
C IN THE WORKSPACE ARRAY RWORK.
C
C
C IF THE USE OF COMMON STATEMENTS IS TO BE AVOIDED, /AD02LA/ CAN
C BE REPLACED BY CALLING THE APPROPRIATE COEFFICIENT ROUTINE
C (D02LAY OR D02LAZ) ON EACH ENTRY TO D02LAF. SIMILARLY, THE
C VALUES IN /CD02LA/ COULD BE RECOMPUTED ON EACH ENTRY.
C OTHER MEANS WILL HAVE TO BE FOUND TO REPLACE /BD02LA/.
C
SUBROUTINE D02LXF(NEQ,H,TOL,THRES,THRESP,MAXSTP,START,ONESTP,HIGH,
* RWORK,LRWORK,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C
C THIS IS THE SET-UP ROUTINE FOR THE NYSTROM CODE D02LAF. IT
C MUST BE CALLED BEFORE THE FIRST CALL TO D02LAF AND CAN BE
C USED BETWEEN CONTINUATION CALLS TO CHANGE OPTIONAL INPUTS.
C
C THE ROUTINE USES THE MACHINE-DEPENDENT CONSTANTS UROUND
C AND THRDEF, WHICH IS CURRENTLY SET TO 50*UROUND.
C
C IF STORAGE IS AT A PREMIUM, THEN IN THE CALLING (SUB)PROGRAM THE
C ACTUAL ARGUMENT 'THRES' MAY BE THE ACTUAL ARGUMENT
C 'RWORK(17+4*NEQ)' AND SIMILARLY 'THRESP' MAY BE 'RWORK(17+5*NEQ)'
C
SUBROUTINE D02LYF(NEQ,HNEXT,HUSED,HSTART,NSUCC,NFAIL,NATT,THRES,
* THRESP,RWORK,LRWORK,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C
C THIS IS THE DIAGNOSTIC ROUTINE ASSOCIATED WITH THE NYSTROM
C SOLVER D02LAF. THE VALUES OF THE PARAMETERS IN THE SUBROUTINE
C CALL ARE TAKEN OUT OF RWORK.
C
C SEE COMMENT IN D02LXF ABOUT STORAGE REDUCTION FOR THRES AND
C THRESP.
C
SUBROUTINE D02LZF(NEQ,T,Y,YP,NWANT,TWANT,YWANT,YPWANT,RWORK,
* LRWORK,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C
C THIS ROUTINE USES THE THIRD FORMULA OF THE RKN6(4)6 TRIPLE
C TO RETURN THE VALUES OF Y AND YP AT THE POINT TWANT, WHICH
C SHOULD BE BETWEEN T-H AND T.
C
SUBROUTINE D02MVF(NEQMAX,NY2DIM,MAXORD,CONST,TCRIT,HMIN,HMAX,H0,
* MAXSTP,MXHNIL,NORM,RWORK,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C***********************************************************************
C THIS ROUTINE INITIALISES THE DASSL IMPLENTATION OF B.D.F
C METHODS AS USED IN THE D02N SUB CHAPTER
C INPUT PARAMETERS
C
C NEQMAX - INTEGER
C THE MAXIMUM NUMBER OF EQUATIONS
C NY2DIM THE SECOND DIMENSION OF THE YSAVE ARRAY , IT SHOULD BE
C SET TO MAXORD + 3, UNLESS MAXORD IS ZERO ,SEE BELOW.
C MAXORD THE MAXIMUM ORDER TO BE USED. THIS SHOULD BE POSITIVE AND
C THIS SHOULD BE LESS THAN 6 . WHEN MAXORD IS SET
C TO ZERO THE DEFAULT VALUES OF 5
C WILL BE USED BY THE CODE (NY2DIM MUST BE LARGE ENOUGH
C FOR THESE DEFAULT VALUES).
C
C
C N.B. THE SIZE OF THE YSAVE ARRAY PASSED INTO NITE3 MUST BE
C YSAVE(NYH, MAXORD + 3). WHERE NYH IS THE NUMBER OF
C DIFFERENTIAL ALGEBRAIC EQUATIONS OR AN UPPER BOUND ON
C THIS NUMBER .
C----------------------------------------------------------------------
SUBROUTINE D02MZF(TSOL,SOL,M,NEQMAX,NEQ,YSAVE,NY2DIM,RWORK,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
SUBROUTINE D02NBF(NEQ,NEQMAX,T,TOUT,Y,YDOTI,RWORK,RTOL,ATOL,ITOL,
* INFORM,FCN,YSAVE,NY2DIM,JAC,WKJAC,NWKJAC,MONITR,
* ITASK,ITRACE,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE D02NCF(NEQ,NEQMAX,T,TOUT,Y,YDOTI,RWORK,RTOL,ATOL,ITOL,
* INFORM,FCN,YSAVE,NY2DIM,JAC,WKJAC,NWKJAC,JACPVT,
* NJCPVT,MONITR,ITASK,ITRACE,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE D02NDF(NEQ,NEQMAX,T,TOUT,Y,YDOTI,RWORK,RTOL,ATOL,ITOL,
* INFORM,FCN,YSAVE,NY2DIM,JAC,WKJAC,NWKJAC,JACPVT,
* NJCPVT,MONITR,ITASK,ITRACE,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE D02NGF(NEQ,NEQMAX,T,TOUT,Y,YDOTI,RWORK,RTOL,ATOL,ITOL,
* INFORM,RESID,YSAVE,NY2DIM,JAC,WKJAC,NWKJAC,
* MONITR,LDERIV,ITASK,ITRACE,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE D02NHF(NEQ,NEQMAX,T,TOUT,Y,YDOTI,RWORK,RTOL,ATOL,ITOL,
* INFORM,RESID,YSAVE,NY2DIM,JAC,WKJAC,NWKJAC,
* JACPVT,NJCPVT,MONITR,LDERIV,ITASK,ITRACE,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE D02NJF(NEQ,NEQMAX,T,TOUT,Y,YDOTI,RWORK,RTOL,ATOL,ITOL,
* INFORM,RESID,YSAVE,NY2DIM,JAC,WKJAC,NWKJAC,
* JACPVT,NJCPVT,MONITR,LDERIV,ITASK,ITRACE,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE D02NMF(NEQ,NEQMAX,T,TOUT,Y,YDOTI,RWORK,RTOL,ATOL,ITOL,
* INFORM,YSAVE,NY2DIM,WKJAC,NWKJAC,JACPVT,NJCPVT,
* IMON,INLN,IRES,IREVCM,ITASK,JTRACE,IFAIL)
C MARK 14 RE-ISSUE. NAG COPYRIGHT 1989.
C MARK 14C REVISED. IER-874 (NOV 1990).
C MARK 16A REVISED. IER-974 (JUN 1993).
C-----------------------------------------------------------------------
C THE FOLLOWING INTERNAL COMMON BLOCKS ARE DEFINED AS FOLLOWS;
C AD02NM CONTAINS TRACE LEVEL AND CHANNEL FOR DEBUGGING
C BD02NM CONTAINS THE INTEGRATION STATISTICS PARAMETERS SUCH
C NUMBER OF JACOBIAN EVALUATIONS AND STEPS. THE PARAMETER
C NINTER CONTAINS THE EFFECTIVE SIZE OF THE MEMORY ARRAY
C YH(NEQMAX,NINTER) IF INTERPOLATION IS TO BE USED ON IT.
C DD02NM CONTAINS THE VARIABLES USED IN THE TIME MANAGEMENT
C SCHEME AND THE WORKSPACE POINTERS.
C ED02NM CONTAINS THE COMMUNICATION POINTERS USED IN REVERSE
C COMMUNICATION BETWEEN THE TIME MANAGEMENT SCHEME AND
C THE MODULES CALLED BY IT.
C FD02NM CONTAINS THE IFIP TRANSPORTABLE NUMERICAL SOFTWARE
C PARAMETERS WHICH ARE INITIALISED IN THE BLOCK DATA .
C GD02NM CONTAINS INFORMATION USED BY THE NONLINEAR EQUATIONS
C SOLVER CAN BE USED BY THE STEP MODULE TO IMPLEMENT
C A DAMPED NEWTON METHOD OR TO GOVERN THE NUMBER OF
C ITERATIONS PERFORMED.
C HD02NM PASSES THE TYPE OF NORM TO BE USED TO THE FUNCTION D02ZAF.
C IMPORTANT
C ---------
C THE RESIDUAL, STEP, NONLINEAR EQUATIONS AND LINEAR ALGEBRA
C MODULES ALL COMMUNICATE BY USING REVERSE COMMUNICATION THROUGH
C THE TIME MANAGEMENT SCHEME IN D02NMF. ACCESS TO THESE MODULES IS
C ONLY THROUGH THE PARAMETER LIST USED IN D02NMF. INTERNAL
C COMMUNICATION BETWEEN THE GROUPS OF ROUTINES WHICH MAKE UP A
C MODULE IS BY COMMON BLOCKS. FOR EXAMPLE THERE ARE THREE ROUTINES
C IN THE B.D.F./ ADAMS METHOD TIME INTEGRATION MODULE :BDFSET, D02NMX
C AND D02XJY. INTERCOMMUNICATION BETWEEN THESE ROUTINES IS VIA COMMON
C BLOCKS.
C-----------------------------------------------------------------------
SUBROUTINE D02NNF(NEQ,NEQMAX,T,TOUT,Y,YDOTI,RWORK,RTOL,ATOL,ITOL,
* INFORM,YSAVE,NY2DIM,WKJAC,NWKJAC,JACPVT,NJCPVT,
* IMON,INLN,IRES,IREVCM,LDERIV,ITASK,JTRACE,IFAIL)
C MARK 14 RE-ISSUE. NAG COPYRIGHT 1989.
C MARK 15A REVISED. IER-938 (APR 1991).
C MARK 15 REVISED. IER-939 (APR 1991).
C MARK 16A REVISED. IER-977 (JUN 1993).
C-----------------------------------------------------------------------
C THE FOLLOWING INTERNAL COMMON BLOCKS ARE DEFINED AS FOLLOWS;
C AD02NM CONTAINS TRACE LEVEL AND CHANNEL FOR DEBUGGING
C BD02NM CONTAINS THE INTEGRATION STATISTICS PARAMETERS SUCH
C NUMBER OF JACOBIAN EVALUATIONS AND STEPS. THE PARAMETER
C NINTER CONTAINS THE EFFECTIVE SIZE OF THE MEMORY ARRAY
C YH(NEQMAX,NINTER) IF INTERPOLATION IS TO BE USED ON IT.
C DD02NM CONTAINS THE VARIABLES USED IN THE TIME MANAGEMENT
C SCHEME AND THE WORKSPACE POINTERS.
C ED02NM CONTAINS THE COMMUNICATION POINTERS USED IN REVERSE
C COMMUNICATION BETWEEN THE TIME MANAGEMENT SCHEME AND
C THE MODULES CALLED BY IT.
C FD02NM CONTAINS THE IFIP TRANSPORTABLE NUMERICAL SOFTWARE
C PARAMETERS WHICH ARE INITIALISED IN THE BLOCK DATA .
C GD02NM CONTAINS INFORMATION USED BY THE NONLINEAR EQUATIONS
C SOLVER CAN BE USED BY THE STEP MODULE TO IMPLEMENT
C A DAMPED NEWTON METHOD OR TO GOVERN THE NUMBER OF
C ITERATIONS PERFORMED.
C HD02NM PASSES THE TYPE OF NORM TO BE USED TO THE FUNCTION D02ZAF.
C IMPORTANT
C ---------
C THE RESIDUAL, STEP, NONLINEAR EQUATIONS AND LINEAR ALGEBRA
C MODULES ALL COMMUNICATE BY USING REVERSE COMMUNICATION THROUGH
C THE TIME MANAGEMENT SCHEME IN D02NNF. ACCESS TO THESE MODULES IS
C ONLY THROUGH THE PARAMETER LIST USED IN D02NNF. INTERNAL
C COMMUNICATION BETWEEN THE GROUPS OF ROUTINES WHICH MAKE UP A
C MODULE IS BY COMMON BLOCKS. FOR EXAMPLE THERE ARE THREE ROUTINES
C IN THE B.D.F./ ADAMS METHOD TIME INTEGRATION MODULE :BDFSET, D02NMX
C AND D02XJZ. INTERCOMMUNICATION BETWEEN THESE ROUTINES IS VIA COMMON
C BLOCKS.
C-----------------------------------------------------------------------
SUBROUTINE D02NRF(J,IPLACE,INFORM)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE D02NSF(NEQ,NEQMAX,JCEVAL,NWKJAC,RWORK,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C SETUP ROUTINE FOR FULL LINEAR ALGEBRA OPTION
C
SUBROUTINE D02NTF(NEQ,NEQMAX,JCEVAL,ML,MU,NWKJAC,NJCPVT,RWORK,
* IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C SETUP ROUTINE FOR BANDED LINEAR ALGEBRA OPTION
C
SUBROUTINE D02NUF(NEQ,NEQMAX,JCEVAL,NWKJAC,IA,NIA,JA,NJA,JACPVT,
* NJCPVT,SENS,U,ETA,LBLOCK,ISPLIT,RWORK,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C MARK 12B REVISED. IER-528 (FEB 1987).
SUBROUTINE D02NVF(NEQMAX,NY2DIM,MAXORD,METHOD,PETZLD,CONST,TCRIT,
* HMIN,HMAX,H0,MAXSTP,MXHNIL,NORM,RWORK,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C SETUP ROUTINE FOR INTEGRATOR EMPLOYING B.D.F. METHOD
C
SUBROUTINE D02NWF(NEQMAX,NY2DIM,MAXORD,CONST,TCRIT,HMIN,HMAX,H0,
* MAXSTP,MXHNIL,NORM,RWORK,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C MARK 12B REVISED. IER-530 (FEB 1987).
C
C SETUP ROUTINE FOR INTEGRATOR EMPLOYING A BLEND METHOD
C
SUBROUTINE D02NXF(ICALL,LIWREQ,LIWUSD,LRWREQ,LRWUSD,NLU,NNZ,NGP,
* ISPLIT,IGROW,LBLOCK,NBLOCK,INFORM)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C MARK 14 REVISED. IER-712 (DEC 1989).
SUBROUTINE D02NYF(NEQ,NEQMAX,HU,H,TCUR,TOLSF,RWORK,NST,NRE,NJE,
* NQU,NQ,NITER,IMXER,ALGEQU,INFORM,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C MARK 12B REVISED. IER-531 (FEB 1987).
SUBROUTINE D02NZF(NEQMAX,TCRIT,H,HMIN,HMAX,MAXSTP,MAXHNL,RWORK,
* IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C MARK 12B REVISED. IER-532 (FEB 1987).
SUBROUTINE D02PCF(F,TWANT,TGOT,YGOT,YPGOT,YMAX,WORK,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
C$$$$ SUBROUTINE UT $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
C
C If you are not familiar with the code D02PCF and how it is used in
C conjunction with D02PVF to solve initial value problems, you should
C study the document file RKSUITE.DOC carefully before proceeding
C further. The following "Brief Reminder" is intended only to remind
C you of the meaning, type, and size requirements of the arguments.
C
C NAME DECLARED IN AN EXTERNAL STATEMENT IN THE CALLING PROGRAM:
C
C F - name of the subroutine for evaluating the differential
C equations.
C
C The subroutine F must have the form
C
C SUBROUTINE F(T,Y,YP)
C DOUBLE PRECISION T,Y(*),YP(*)
C Given input values of the independent variable T and the solution
C components Y(*), for each L = 1,2,...,NEQ evaluate the differential
C equation for the derivative of the Ith solution component and place
C the value in YP(L). Do not alter the input values of T and Y(*).
C RETURN
C END
C
C INPUT VARIABLE
C
C TWANT - DOUBLE PRECISION
C The next value of the independent variable where a
C solution is desired.
C
C Constraints: TWANT must lie between the previous value
C of TGOT (TSTART on the first call) and TEND. TWANT can
C be equal to TEND, but it must be clearly
C distinguishable from the previous value of TGOT
C (TSTART on the first call) in the precision available.
C
C OUTPUT VARIABLES
C
C TGOT - DOUBLE PRECISION
C A solution has been computed at this value of the
C independent variable.
C YGOT(*) - DOUBLE PRECISION array of length NEQ
C Approximation to the true solution at TGOT. Do not
C alter the contents of this array
C YPGOT(*) - DOUBLE PRECISION array of length NEQ
C Approximation to the first derivative of the true
C solution at TGOT.
C YMAX(*) - DOUBLE PRECISION array of length NEQ
C YMAX(L) is the largest magnitude of YGOT(L) computed
C at any time in the integration from TSTART to TGOT.
C Do not alter the contents of this array.
C
C WORKSPACE
C
C WORK(*) - DOUBLE PRECISION array as used in D02PVF
C Do not alter the contents of this array.
C
C OUTPUT VARIABLE
C
C IFAIL - INTEGER
C
C SUCCESS. TGOT = TWANT.
C = 0 - Complete success.
C
C "SOFT" FAILURES
C = 2 - Warning: You are using METHOD = 3 inefficiently
C by computing answers at many values of TWANT. If
C you really need answers at so many specific
C points, it would be more efficient to compute
C them with METHOD = 2. To do this you would need
C to restart from TGOT, YGOT(*) by a call to
C D02PVF. If you wish to continue as you are, you
C may.
C = 3 - Warning: A considerable amount of work has been
C expended. If you wish to continue on to TWANT,
C just call D02PCF again.
C = 4 - Warning: It appears that this problem is
C "stiff". You really should change to another
C code that is intended for such problems, but if
C you insist, you can continue with D02PCF by
C calling it again.
C
C "HARD" FAILURES
C = 5 - You are asking for too much accuracy. You cannot
C continue integrating this problem.
C = 6 - The global error assessment may not be reliable
C beyond the current point in the integration. You
C cannot continue integrating this problem.
C
C "CATASTROPHIC" FAILURES
C = 1 - The nature of the catastrophe is reported on
C the standard output channel. Unless special
C provision was made in advance (see RKSUITE.DOC),
C the computation then comes to a STOP.
C
C$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
SUBROUTINE D02PDF(F,TNOW,YNOW,YPNOW,WORK,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
C$$$$ SUBROUTINE CT $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
C
C If you are not familiar with the code D02PDF and how it is used in
C conjunction with D02PVF to solve initial value problems, you should
C study the document file RKSUITE.DOC carefully before attempting to
C use the code. The following "Brief Reminder" is intended only to
C remind you of the meaning, type, and size requirements of the
C arguments.
C
C NAME DECLARED IN AN EXTERNAL STATEMENT IN THE CALLING PROGRAM:
C
C F - name of the subroutine for evaluating the differential
C equations.
C
C The subroutine F must have the form
C
C SUBROUTINE F(T,Y,YP)
C DOUBLE PRECISION T,Y(*),YP(*)
C Using the input values of the independent variable T and the
C solution components Y(*), for each L = 1,2,...,NEQ evaluate the
C differential equation for the derivative of the Lth solution
C component and place the value in YP(L). Do not alter the input
C values of T and Y(*).
C RETURN
C END
C
C OUTPUT VARIABLES
C
C TNOW - DOUBLE PRECISION
C Current value of the independent variable.
C YNOW(*) - DOUBLE PRECISION array of length NEQ
C Approximation to the true solution at TNOW.
C YPNOW(*) - DOUBLE PRECISION array of length NEQ
C Approximation to the first derivative of the
C true solution at TNOW.
C
C WORKSPACE
C
C WORK(*) - DOUBLE PRECISION array as used in D02PVF
C Do not alter the contents of this array.
C
C OUTPUT VARIABLE
C
C IFAIL - INTEGER
C
C SUCCESS. A STEP WAS TAKEN TO TNOW.
C = 0 - Complete success.
C
C "SOFT" FAILURES
C = 2 - Warning: You have obtained an answer by
C integrating to TEND (TNOW = TEND). You have
C done this at least 100 times, and monitoring of
C the computation reveals that this way of getting
C output has degraded the efficiency of the code.
C If you really need answers at so many specific
C points, it would be more efficient to get them
C with D02PXF. (If METHOD = 3, you would need to
C change METHOD and restart from TNOW, YNOW(*) by
C a call to D02PVF.) If you wish to continue as
C you are, you may.
C = 3 - Warning: A considerable amount of work has been
C expended. To continue the integration, just call
C D02PDF again.
C = 4 - Warning: It appears that this problem is
C "stiff". You really should change to another
C code that is intended for such problems, but if
C you insist, you can continue with D02PDF by
C calling it again.
C
C "HARD" FAILURES
C = 5 - You are asking for too much accuracy. You cannot
C continue integrating this problem.
C = 6 - The global error assessment may not be reliable
C beyond the current point in the integration.
C You cannot continue integrating this problem.
C
C "CATASTROPHIC" FAILURES
C = 1 - The nature of the catastrophe is reported on
C the standard output channel. Unless special
C provision was made in advance (see RKSUITE.DOC),
C the computation then comes to a STOP.
C
C$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
SUBROUTINE D02PVF(NEQ,TSTART,YSTART,TEND,TOL,THRES,METHOD,TASK,
* ERRASS,HSTART,WORK,LENWRK,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
C$$$$ SUBROUTINE SETUP $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
C
C If you are not familiar with the code D02PVF and how it is used in
C conjunction with D02PCF or D02PDF to solve initial value problems,
C you should study the document file RKSUITE.DOC carefully before
C attempting to use the code.
C The following "Brief Reminder" is intended only to remind you of the
C meaning, type, and size requirements of the arguments.
C
C The environmental parameters OUTCH, MCHEPS, and DWARF are used in the
C following description. To find out their values
C
C CALL D02PVX(OUTCH,MCHEPS,DWARF)
C
C INPUT VARIABLES
C
C NEQ - INTEGER
C The number of differential equations in the system.
C Constraint: NEQ >= 1
C TSTART - DOUBLE PRECISION
C The initial value of the independent variable.
C YSTART(*) - DOUBLE PRECISION array of length NEQ
C The vector of initial values of the solution
C components.
C TEND - DOUBLE PRECISION
C The integration proceeds from TSTART in the direction
C of TEND. You cannot go past TEND.
C Constraint: TEND must be clearly distinguishable from
C TSTART in the precision available.
C TOL - DOUBLE PRECISION
C The relative error tolerance.
C Constraint: 0.01D0 >= TOL >= 10*MCHEPS
C THRES(*) - DOUBLE PRECISION array of length NEQ
C THRES(L) is the threshold for the Ith solution
C component.
C Constraint: THRES(L) >= SQRT(DWARF)
C METHOD - INTEGER
C Specifies which Runge-Kutta pair is to be used.
C = 1 - use the (2,3) pair
C = 2 - use the (4,5) pair
C = 3 - use the (7,8) pair
C TASK - CHARACTER*(*)
C Only the first character of TASK is significant.
C TASK(1:1) = `U' or `u' - D02PCF is to be used
C = `C' or `c' - D02PDF is to be used
C Constraint: TASK(1:1) = `U'or `u' or`C' or `c'
C ERRASS - LOGICAL
C = .FALSE. - do not attempt to assess the true error.
C = .TRUE. - assess the true error. Costs roughly twice
C as much as the integration with METHODs 2
C and 3, and three times with METHOD = 1.
C HSTART - DOUBLE PRECISION
C 0.0D0 - select automatically the first step size.
C non-zero - try HSTART for the first step.
C
C WORKSPACE
C
C WORK(*) - DOUBLE PRECISION array of length LENWRK
C Do not alter the contents of this array after calling
C D02PVF.
C
C INPUT VARIABLES
C
C LENWRK - INTEGER
C Length of WORK(*): How big LENWRK must be depends
C on the task and how it is to be solved.
C
C LENWRK = 32*NEQ is sufficient for all cases.
C
C If storage is a problem, the least storage possible
C in the various cases is:
C
C If TASK = `U' or `u', then
C if ERRASS = .FALSE. and
C METHOD = 1, LENWRK must be at least 10*NEQ
C = 2 20*NEQ
C = 3 16*NEQ
C if ERRASS = .TRUE. and
C METHOD = 1, LENWRK must be at least 15*NEQ
C = 2 32*NEQ
C = 3 21*NEQ
C
C If TASK = `C' or `c', then
C if ERRASS = .FALSE. and
C METHOD = 1, LENWRK must be at least 10*NEQ
C = 2 14*NEQ
C = 3 16*NEQ
C if ERRASS = .TRUE. and
C METHOD = 1, LENWRK must be at least 15*NEQ
C = 2 26*NEQ
C = 3 21*NEQ
C
C Warning: To exploit the interpolation capability
C of METHODs 1 and 2, you have to call D02PXF. This
C subroutine requires working storage in addition to
C that specified here.
C
C In the event of a "catastrophic" failure to call D02PVF correctly,
C the nature of the catastrophe is reported on the standard output
C channel, regardless of the value of MESAGE. Unless special provision
C was made in advance (see RKSUITE.DOC), the computation then comes to
C STOP.
C
C$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
SUBROUTINE D02PWF(TENDNU,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
C$$$$ SUBROUTINE RESET $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
C
C If you are not familiar with the code D02PWF and how it is used in
C conjunction with D02PDF to solve initial value problems, you should
C study the document file RKSUITE.DOC carefully before attempting to
C use the code. The following "Brief Reminder" is intended only to
C remind you of the meaning, type, and size requirements of the
C arguments.
C
C The integration using D02PDF proceeds from TSTART in the direction of
C TEND, and is now at TNOW. To reset TEND to a new value TENDNU, just
C call D02PWF with TENDNU as the argument. You must continue
C integrating in the same direction, so the sign of (TENDNU - TNOW)
C must be the same as the sign of (TEND - TSTART). To change direction
C you must restart by a call to D02PVF.
C
C INPUT VARIABLE
C
C TENDNU - DOUBLE PRECISION
C The new value of TEND.
C Constraint: TENDNU and TNOW must be clearly
C distinguishable in the precision used. The
C sign of (TENDNU - TNOW) must be the same as
C the sign of (TEND - TSTART).
C
C$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
SUBROUTINE D02PXF(TWANT,REQEST,NWANT,YWANT,YPWANT,F,WORK,WRKINT,
* LENINT,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
C$$$$ SUBROUTINE INTRP $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
C
C If you are not familiar with the code D02PXF and how it is used in
C conjunction with D02PDF to solve initial value problems, you should
C study the document file RKSUITE.DOC carefully before attempting to
C use the code. The following "Brief Reminder" is intended only to
C remind you of the meaning, type, and size requirements of the
C arguments.
C
C When integrating with METHOD = 1 or 2, answers may be obtained
C inexpensively between steps by interpolation. D02PXF is called after
C a step by D02PDF from a previous value of T, called TOLD below, to
C the current value of T to get an answer at TWANT. You can specify any
C value of TWANT you wish, but specifying a value outside the interval
C [TOLD,T] is likely to yield answers with unsatisfactory accuracy.
C
C INPUT VARIABLE
C
C TWANT - DOUBLE PRECISION
C The value of the independent variable where a solution
C is desired.
C
C The interpolant is to be evaluated at TWANT to approximate the
C solution and/or its first derivative there. There are three
C cases:
C
C INPUT VARIABLE
C
C REQEST - CHARACTER*(*)
C Only the first character of REQEST is significant.
C REQEST(1:1) = `S' or `s'- compute approximate
C = `D' or `d'- compute approximate first
C `D'erivative of the solution
C only.
C = `B' or `b'- compute `B'oth approximate
C solution and first
C derivative.
C Constraint: REQEST(1:1) must be `S',`s',`D',`d',`B' or
C `b'.
C
C If you intend to interpolate at many points, you should arrange for
C the the interesting components to be the first ones because the code
C approximates only the first NWANT components.
C
C INPUT VARIABLE
C
C NWANT - INTEGER
C Only the first NWANT components of the answer are to
C be computed.
C Constraint: NEQ >= NWANT >= 1
C
C OUTPUT VARIABLES
C
C YWANT(*) - DOUBLE PRECISION array of length NWANT
C Approximation to the first NWANT components of the
C true solution at TWANT when REQESTed.
C YPWANT(*) - DOUBLE PRECISION array of length NWANT
C Approximation to the first NWANT components of the
C first derivative of the true solution at TWANT when
C REQESTed.
C
C NAME DECLARED IN AN EXTERNAL STATEMENT IN THE PROGRAM CALLING D02PXF:
C
C F - name of the subroutine for evaluating the differential
C equations as provided to D02PDF.
C
C WORKSPACE
C
C WORK(*) - DOUBLE PRECISION array as used in D02PVF and D02PDF
C Do not alter the contents of this array.
C
C WRKINT(*) - DOUBLE PRECISION array of length LENINT
C Do not alter the contents of this array.
C
C LENINT - INTEGER
C Length of WRKINT. If
C METHOD = 1, LENINT must be at least 1
C = 2, LENINT must be at least
C NEQ+MAX(NEQ,5*NWANT)
C = 3--CANNOT BE USED WITH THIS SUBROUTINE
C
C$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
SUBROUTINE D02PYF(TOTFCN,STPCST,WASTE,STPSOK,HNEXT,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
C$$$$ SUBROUTINE STAT $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
C
C If you are not familiar with the code D02PYF and how it is used in
C conjunction with the integrators D02PDF and D02PCF, you should study
C the document file RKSUITE.DOC carefully before attempting to use the
C code. The following "Brief Reminder" is intended only to remind you
C of the meaning, type, and size requirements of the arguments.
C
C D02PYF is called to obtain some details about the integration.
C
C OUTPUT VARIABLES
C
C TOTFCN - INTEGER
C Total number of calls to F in the integration so far
C -- a measure of the cost of the integration.
C STPCST - INTEGER
C Cost of a typical step with this METHOD measured in
C calls to F.
C WASTE - DOUBLE PRECISION
C The number of attempted steps that failed to meet the
C local error requirement divided by the total number of
C steps attempted so far in the integration.
C STPSOK - INTEGER
C The number of accepted steps.
C HNEXT - DOUBLE PRECISION
C The step size the integrator plans to use for the next
C step.
C
C$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
SUBROUTINE D02PZF(RMSERR,ERRMAX,TERRMX,WORK,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
C$$$$ SUBROUTINE GLBERR $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
C
C If you are not familiar with the code D02PZF and how it is used in
C conjunction with D02PCF and D02PDF to solve initial value problems,
C you should study the document file RKSUITE.DOC carefully before
C attempting to use the code. The following "Brief Reminder" is
C intended only to remind you of the meaning, type, and size
C requirements of the arguments.
C
C If ERRASS was set .TRUE. in the call to D02PVF, then after any call
C to D02PCF or D02PDF to advance the integration to TNOW or TWANT, the
C subroutine D02PZF may be called to obtain an assessment of the true
C error of the integration. At each step and for each solution
C component Y(L), a more accurate "true" solution YT(L), an average
C magnitude "size(L)" of its size, and its error
C abs(Y(L) - YT(L))/max("size(L)",THRES(L)) are computed. The
C assessment returned in RMSERR(L) is the RMS (root-mean-square)
C average of the error in the Lth solution component over all steps of
C the integration from TSTART through TNOW.
C
C OUTPUT VARIABLES
C
C RMSERR(*) - DOUBLE PRECISION array of length NEQ
C RMSERR(L) approximates the RMS average of the true
C error of the numerical solution for the Ith solution
C component, L = 1,2,...,NEQ. The average is taken over
C all steps from TSTART to TNOW.
C ERRMAX - DOUBLE PRECISION
C The maximum (approximate) true error taken over all
C solution components and all steps from TSTART to TNOW.
C TERRMX - DOUBLE PRECISION
C First value of the independent variable where the
C (approximate) true error attains the maximum value
C ERRMAX.
C
C WORKSPACE
C
C WORK(*) - DOUBLE PRECISION array as used in D02PVF and D02PCF or
C D02PDF
C
C$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
SUBROUTINE D02QFF(F,NEQF,T,Y,TOUT,G,NEQG,ROOT,RWORK,LRWORK,IWORK,
* LIWORK,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE D02QGF(NEQF,T,Y,TOUT,NEQG,ROOT,IREVCM,TRVCM,YRVCM,
* YPRVCM,GRVCM,KGRVCM,RWORK,LRWORK,IWORK,LIWORK,
* IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE D02QWF(STATEF,NEQF,VECTOL,ATOL,LATOL,RTOL,LRTOL,ONESTP,
* CRIT,TCRIT,HMAX,MAXSTP,NEQG,ALTERG,SOPHST,RWORK,
* LRWORK,IWORK,LIWORK,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE D02QXF(NEQF,YP,TCURR,HLAST,HNEXT,ODLAST,ODNEXT,NSUCC,
* NFAIL,TOLFAC,BADCMP,RWORK,LRWORK,IWORK,LIWORK,
* IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE D02QYF(NEQG,INDEX,TYPE,EVENTS,RESIDS,RWORK,LRWORK,
* IWORK,LIWORK,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE D02QZF(NEQF,TWANT,NWANT,YWANT,YPWANT,RWORK,LRWORK,
* IWORK,LIWORK,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE D02RAF(M,NMAX,N,NUMBEG,NUMMIX,TOL,INIT,X,Y,IY,ABT,FCN,
* G,IJAC,JACOBF,JACOBG,DELEPS,JACEPS,JACGEP,WORK,
* LWORK,IWORK,LIWORK,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C MARK 14 REVISED. IER-821 (DEC 1989).
C FCN, G, JACEPS, JACGEP, JACOBF, JACOBG
SUBROUTINE D02SAF(P,M,N,N1,PE,PF,E,DP,NPOINT,WP,IWP,ICOUNT,RANGE,
* BC,FCN,EQN,CONSTR,YMAX,MONIT,PRSOL,W,IW1,IW2,
* IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C SOLVES A TWO POINT BVP
C BC, EQN, FCN, MONIT, PRSOL, RANGE
SUBROUTINE D02TGF(N,M,L,X0,X1,K1,KP,C,IC,COEFF,BDYC,W,LW,IW,LIW,
* IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C BDYC,COEFF
SUBROUTINE D02XJF(XSOL,SOL,M,W,NEQMAX,IW,NEQ,X,NQ,HU,H,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C MARK 12B REVISED. IER-545 (FEB 1987).
SUBROUTINE D02XKF(XSOL,SOL,M,W,NEQMAX,IW,W2,NEQ,X,NQ,HU,H,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C MARK 12B REVISED. IER-533 (FEB 1987).
DOUBLE PRECISION FUNCTION D02ZAF(N,V,W,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C OLD NAME VNORM
C
C-----------------------------------------------------------------------
C INORM = 1 MAXIMUM NORM
C = 2 AVERAGE L2 NORM
C THE CHOICE OF NORM IS DEFINED BY THE USER ON ENTRY TO SPRINT AND
C PASSED TO THIS FUNCTION BY THE COMMON BLOCK /HD02NM/.
C
C BOTH NORMS ARE WEIGHTED VECTOR NORMS OF
C THE VECTOR OF LENGTH N CONTAINED IN THE ARRAY V, WITH WEIGHTS
C CONTAINED IN THE ARRAY W OF LENGTH N.
C E.G. FOR THE AVERAGED L2 NORM
C D02ZAF = SQRT( (1/N) * SUM( V(I)*W(I) )**2 )
C THE L2 NORM CAN BE OBTAINED BY APPROPRIATE SCALING OF THE WEIGHTS
C FOR THE MAX NORM
C D02ZAF = MAX '' V(I)*W(I) ''
C I
C THE MODIFIED MAX NORM CAN BE OBTAINED , AS FOR THE L2 NORM, BY
C APPROPRIATE SCALING OF THE WEIGHTS.
C-----------------------------------------------------------------------
SUBROUTINE D03EAF(STAGE1,EXT,DORM,N,P,Q,X,Y,N1P1,PHI,PHID,ALPHA,C,
* IC,NP4,ICINT,NP1,IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 7G REVISED. IER-218 (FEB 1980)
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C
C THIS SUBROUTINE SOLVES DIRICHLET, NEUMANN AND MIXED BOUNDARY
C VALUE PROBLEMS FOR LAPLACES EQUATION IN TWO DIMENSIONS, BY AN
C INTEGRAL EQUATION METHOD BASED UPON GREENS FORMULA. IT IS
C APPLICABLE TO ANY DOMAIN BOUNDED, INTERNALLY AND/OR
C EXTERNALLY, BY CLOSED CONTOURS.
C
C
C THE NEXT TWO VARIABLES ARE MACHINE-DEPENDENT CONSTANTS. THE
C FIRST CONTAINS THE VALUE OF PI AND THE SECOND THE VALUE OF EPS
C WHICH IS THE SMALLEST NUMBER FOR WHICH 1.0 + EPS .GT. 1.0
C WITHIN THE MACHINE.
C
SUBROUTINE D03EBF(N1,N2,N1M,A,B,C,D,E,Q,T,APARAM,ITMAX,ITCOUN,
* ITUSED,NDIR,IXN,IYN,CONRES,CONCHN,RESIDS,CHNGS,
* WRKSP1,WRKSP2,WRKSP3,IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 8C REVISED. IER-267 (OCT. 1980).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C *************************************************************
C D03EBF OBTAINS THE SOLUTION TO A SYSTEM OF SIMULTANEOUS
C ALGEBRAIC EQUATIONS OF FIVE POINT MOLECULE FORM ON A
C TOPOLOGICALLY RECTANGULAR MESH.
C IT IS THE DRIVING ROUTINE FOR D03UAF FOR STANDARD
C APPLICATIONS.
C
C
C INPUTS
C
C N1 NUMBER OF NODES IN THE FIRST COORDINATE DIRECTION.
C N2 NUMBER OF NODES IN THE SECOND COORDINATE DIRECTION.
C N1M FIRST DIMENSION OF ALL THE TWO-DIMENSIONAL ARRAYS.
C A ARRAY OF DIMENSION (N1M,N2) STORES THE COEFFICIENT
C OF THE ITERATIVE UPDATE EQUATIONS AS SHOWN BELOW -
C B DITTO ... DIMENSION (N1M,N2) ...
C C DITTO ... DIMENSION (N1M,N2) ...
C D DITTO ... DIMENSION (N1M,N2) ...
C E DITTO ... DIMENSION (N1M,N2) ...
C
C A(I,J)*T(I,J-1)+B(I,J)*T(I-1,J)+C(I,J)*T(I,J)+D(I,J)*
C T(I+1,J)+E(I,J)*T(I,J+1) = Q(I,J)
C
C WITH I=1,2,...,N1 AND J=1,2,...,N2 AND WHERE T(I,J)
C IS THE ARRAY WHOSE VALUES ARE SOUGHT AND WHERE
C Q(I,J) IS THE ARRAY OF SOURCE TERMS. ANY VALUES OF
C T OUTSIDE THE PROBLEM AREA (1-N1,1-N2) ARE TAKEN
C AS ZERO.
C Q IS THE ARRAY OF SOURCE TERMS DIMENSION (N1M,N2)
C T ARRAY STORING THE APPROXIMATE SOLUTION OF T IN THE
C ABOVE EQUATION WHICH IS PASSED TO THE ROUTINE.
C IF NO BETTER APPROXIMATION IS KNOWN, AN ARRAY OF
C ZEROS CAN BE USED DIMENSIONED (N1M,N2).
C APARAM IS AN ITERATION ACCELERATION PARAMETER FACTOR
C TYPICALLY SET TO 1.0. IF CONVERGENCE IS SLOW, IT
C CAN BE DECREASED. IF DIVERGENCE IS OBTAINED, IT
C SHOULD BE INCREASED. IN EITHER CASE IT MUST NOT
C GO OUTSIDE THE BOUNDS PRESCRIBED (SEE IFAIL
C PARAMETER FOR D03UAF).
C ITMAX IS THE MAXIMUM NUMBER OF ITERATIONS TO BE USED.
C ITCOUN IS AN INDICATOR SET EQUAL TO 0 ON THE FIRST CALL
C TO D03EBF. FOR SUBSEQUENT CALLS FOR THE SAME
C PROBLEM, I.E. SAME N1 N2,BUT POSSIBLY DIFFERENT
C COEFFICIENTS AND/OR SOURCE TERMS, AS OCCUR WITH NON
C LINEAR SYSTEMS OR TIME DEPENDENT SYSTEMS, ITCOUN
C SHOULD BE SET TO THE NUMBER OF ITERATIONS USED SO
C FAR ON SUBSEQUENT CALLS TO ENSURE SUITABLE CYCLING
C OF THE ITERATION ACCELERATION PARAMETERS IN D03UAF.
C CHANGED ON OUTPUT TO THE NUMBER OF ITERATIONS USED
C SO FAR I.E. ITCOUN ON RETURN = ITCOUN ON INPUT +
C ITUSED ON RETURN
C NDIR IS AN INTEGER SET TO A NON-ZERO VALUE FOR SYSTEMS
C OF EQUATIONS WHICH HAVE A UNIQUE SOLUTION. FOR
C SYSTEMS DERIVED FROM, E.G. LAPLACES EQUATION WITH
C ALL NEUMANN BOUNDARY CONDITIONS, PROBLEMS WITH AN
C ARBITRARY CONSTANT CAN BE ADDED TO THE SOLUTION,
C NDIR SHOULD BE SET TO ZERO AND THE VALUES OF THE
C NEXT 2 PARAMETERS MUST BE SPECIFIED. FOR SUCH
C PROBLEMS THE ROUTINE SUBTRACTS THE VALUE OF THE
C FUNCTION DERIVED AT THE NODE (IXN,IYN) FROM THE
C WHOLE SOLUTION AFTER EACH ITERATION TO REDUCE THE
C POSSIBLITY OF LARGE ROUNDING ERRORS. THE USER MU
C ALSO ENSURE THAT FOR SUCH PROBLEMS THE APPROPRIATE
C CONSISTENCY CONDITION ON THE SOURCE TERMS IS
C SATISFIED.
C IXN,IYN NODAL INDECIES OF THE NODE AT WHICH THE SOLUTION IS
C TO BE REDUCED TO ZERO FOR PROBLEMS FOR WHICH NDIR IS
C SET TO ZERO THE NODE SHOULD NOT CORRESPOND TO A
C CORNER NODE OF THE REGION OF INTEREST.
C
C CONVERGENCE CRITERIA
C
C CONRES CONVERGENCE CRITERION ON THE MAXIMUM ABSOLUTE VALUE
C OF THE NORMALIZED RESIDUAL, THE LATTER BEING
C DEFINED AS THE RESIDUAL OF THE EQUATION DIVIDED BY
C THE CENTRAL COEFFICIENT WHEN THE LATTER IS NOT
C EQUAL TO O.O, AND DEFINED AS THE RESIDUAL WHEN THE
C CENTRAL COEFFICIENT IS 0.0.
C CONCHN CONVERGENCE CRITERION ON THE MAXIMUM ABSOLUTE VALUE
C OF THE CHANGE MADE AT EACH ITERATION OF THE ARRAY T
C
C CONVERGENCE IS ACHIEVED WHEN BOTH THE CRITERIA ARE SATISFIED
C
C RESIDS ARRAY OF DIMENSION ITMAX ... SEE OUTPUT
C CHNGS ARRAY OF DIMENSION ITMAX ... SEE OUTPUT
C
C WRKSP1 WORKSPACE ARRAY OF DIMENSION (N1M,N2) USED BY D03UAF
C WRKSP2 ... DITTO ...
C WRKSP3 WORKSPACE ARRAY OF DIMENSION (N1M,N2) USED TO PASS
C THE RESIDUALS TO D03UAF, AND IN WHICH THE UPDATE
C VALUES ARE RETURNED FROM D03UAF TO THIS ROUTINE.
C
C IFAIL IS AN ERROR PARAMETER INDICATOR SET BY THE USER TO
C INDICATE THE TYPE OF FAILURE IF AN ERROR IS
C ENCOUNTERED.
C
C PROCESS
C
C SET ERROR PARAMETER
C CHECK INPUT INTEGERS
C COMMENCEMENT OF CALCULATIONAL PROCEDURE
C START OF ITERATIVE SOLUTION PROCEDURE
C SET CONVERGENCE PARAMETER ACCUMULATORS TO ZERO
C CALCULATE THE RESIDUALS AND STORE IN ARRAY WRKSP3
C CALL SUBROUTINE D03UAF TO PERFORM 1 ITERATION OF S.I.P
C AND WHICH RETURNS THE ARRAY OF UPDATES IN THE ARRAY WRKSP3
C COMBINE THE UPDATES WITH THE OLD SOLUTION
C CHECK FOR CONVERGENCE
C REPEAT TO CONVERGENCE OR UNTIL MAXIMUM NUMBER OF ITERATIONS
C HAVE BEEN USED
C SET ERROR INDICATOR ACCORDINGLY
C RETURN
C
C OUTPUTS
C
C T ARRAY STORING THE SOLUTION OF THE SYSTEM OF EQUATIONS
C PROVIDED.
C ITUSED NUMBER OF ITERATIONS ACTUALLY USED THIS CALL.
C ITCOUN ACCUMULATED NUMBER OF ITERATIONS USED
C ITCOUN ON OUTPUT = ITCOUN ON INPUT + ITUSED ON OUTPUT
C RESIDS VECTOR STORING THE MAXIMUM ABSOLUTE RESIDUALS OF EACH
C ITERATION, DIMENSION ITMAX. ONLY THE FIRST ITUSED
C VALUES ARE SET ON RETURN.
C CHNGS ARRAY STORING THE MAXIMUM ABSOLUTE CHANGES OF EACH
C ITERATION, DIMENSION ITMAX. ONLY THE FIRST ITUSED
C VALUES ARE SET ON RETURN.
C IFAIL ERROR INDICATOR
C =0 CORRECT RETURN
C =1 EITHER N1.LE.1 OR N2.LE.1
C =2 CONVERGENCE NOT ACHIEVED. REFER TO VALUES IN
C RESIDS AND CHNGS TO DETERMINE IF THE ITERATION
C WAS CONVERGING. REFER TO GUIDE ON SETTING APARAM
C IF CONVERGENCE IS VERY SLOW.
C =3 APARAM.LE.0.0
C =4 APARAM.GE.((N1-1)**2+(N2-1)**2)/2.0
C
C ROUTINES USED
C
C D03UAF SINGLE ITERATION OF S.I.P. FOR A FIVE POINT MOLECULE
C SYSTEM.
C P01AAF ERROR HANDLING ROUTINE.
C
C **************************************************************
C
C
C SET ERROR PARAMETER
C
SUBROUTINE D03ECF(N1,N2,N3,N1M,N2M,A,B,C,D,E,F,G,Q,T,APARAM,ITMAX,
* ITCOUN,ITUSED,NDIR,IXN,IYN,IZN,CONRES,CONCHN,
* RESIDS,CHNGS,WRKSP1,WRKSP2,WRKSP3,WRKSP4,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C **************************************************************
C
C DAVID A. H. JACOBS 22/5/79
C D03ECF OBTAINS THE SOLUTION OF A SYSTEM OF SIMULTANEOUS
C ALGEBRAIC EQUATIONS OF SEVEN POINT MOLECULE FORM ON A
C THREE-DIMENSIONAL TOPOLOGICALLY RECTANGULAR MESH.
C
C IT IS THE DRIVING ROUTINE FOR ROUTINE D03UBF FOR
C STANDARD APPLICATIONS
C
C INPUTS
C ------
C
C N1 NUMBER OF NODES IN THE FIRST COORDINATE DIRECTION.
C
C N2 NUMBER OF NODES IN THE SECOND COORDINATE DIRECTION.
C
C N3 NUMBER OF NODES IN THE THIRD COORDINATE DIRECTION.
C
C N1M FIRST DIMENSION OF ALL THREE DIMENSIONAL ARRAYS.
C
C N2M SECOND DIMENSION OF ALL THREE DIMENSIONAL ARRAYS.
C
C A,B,C, ARRAYS ALL OF DIMENSION (N1M,N2M,N3) STORING
C D,E,F,G THE COEFFICIENTS OF THE ITERATIVE UPDATE
C EQUATIONS -
C
C A(I,J,K)*T(I,J,K-1)+B(I,J,K)*T(I,J-1,K)+C(I,J,K)*T(I-1,J,K)+
C D(I,J,K)*T(I,J,K)+E(I,J,K)*T(I+1,J,K)+F(I,J,K)*T(I,J+1,K)+
C G(I,J,K)*T(I,J,K+1)=Q(I,J,K)
C
C WITH I=1,2,...,N1 , J=1,2,...,N2 AND K=1,2,...,N3
C AND WHERE T(I,J,K) IS THE ARRAY WHOSE VALUES
C ARE SOUGHT (AND WHICH OVERWRITES THE INITIAL
C APPROXIMATION STORED ON INPUT IN THE ARRAY
C T). ANY VALUES OF T OUTSIDE THE
C (1-N1,1-N2,1-N3) ARRAY ARE TAKEN AS ZERO.
C Q IS THE ARRAY OF SOURCE TERMS DIMENSIONED
C Q(N1M,N2M,N3).
C
C T ARRAY STORING THE APPROXIMATE SOLUTION OF T
C IN THE ABOVE EQUATION WHICH IS PASSED TO THE
C ROUTINE. IF NO BETTER APPROXIMATION IS
C KNOWN, AN ARRAY OF ZEROS CAN BE USED.
C DIMENSIONED T(N1M,N2M,N3) CHANGED ON RETURN
C TO THE SOLUTION OBTAINED.
C
C APARAM IS AN ITERATION ACCELERATION PARAMETER FACTOR
C TYPICALLY SET TO 1.0. IF CONVERGENCE IS
C SLOW, IT CAN BE DECREASED. IF DIVERGENCE IS
C OBTAINED, IT SHOULD BE INCREASED. IN EITHER
C CASE IT MUST NOT GO OUTSIDE THE BOUNDS
C PRESCRIBED.
C
C ITMAX IS THE MAXIMUM NUMBER OF ITERATIONS TO BE USED
C
C ITCOUN IS AN INDICATOR SET EQUAL TO 1 ON THE FIRST
C CALL TO D03ECF FOR SUBSEQUENT CALLS FOR THE
C SAME PROBLEM, I.E. SAME N1 N2 N3, BUT
C POSSIBLY DIFFERENT COEFFICIENTS AND/OR
C SOURCE TERMS, AS OCCUR WITH NON-LINEAR
C SYSTEMS OR TIME-DEPENDENT SYSTEMS, ITCOUN
C SHOULD BE SET TO THE NUMBER OF ITERATIONS
C USED SO FAR. ON SUBSEQUENT CALLS TO ENSURE
C SUITABLE CYCLING OF THE ITERATION
C ACCELERATION PARAMETERS IN D03UBF. CHANGED ON
C EXIT TO THE NUMBER OF ITERATIONS USED SO FAR
C I.E. ITCOUN ON RETURN = ITCOUN ON INPUT
C + ITUSED ON RETURN
C NDIR IS AN INTEGER SET TO A NON-ZERO VALUE FOR
C SYSTEMS OF EQUATIONS WHICH HAVE A UNIQUE
C SOLUTION. FOR SYSTEMS DERIVED FROM, FOR
C EXAMPLE LAPACES EQUATION WITH ALL NEUMANN
C BOUNDARY CONDITIONS, PROBLEMS TO WHICH AN
C ARBITRARY CONSTANT CAN BE ADDED TO THE
C SOLUTION, NDIR SHOULD BE SET TO 0 (ZERO) AND
C THE VALUES OF THE NEXT THREE PARAMETERS
C MUST BE SPECIFIED. FOR SUCH PROBLEMS THE
C ROUTINE SUBTRACTS THE VALUE OF THE FUNCTION
C DERIVED AT THE NODE (IXN,IYN,IZN) FROM THE
C WHOLE SOLUTION AFTER EACH ITERATION TO
C REDUCE THE POSSIBILITY OF LARGE ROUNDING
C ERRORS. THE USER MUST ALSO ENSURE THAT FOR
C SUCH PROBLEMS THE APPROPRIATE CONSISTENCY
C CONDITION ON THE SOURCE TERMS IS SATISFIED.
C
C IXN,IYN,IZN NODAL INDECIES OF THE NODE AT WHICH THE
C SOLUTION IS TO BE REDUCED TO ZERO FOR
C PROBLEMS FOR WHICH NDIR IS SET TO 0 THE NODE
C SHOULD NOT CORRESPOND TO A CORNER NODE OF
C THE REGION OF INTEREST.
C
C CONVERGENCE CRITERIA
C
C CONRES CONVERGENCE CRITERION ON THE MAXIMUM ABSOLUTE
C VALUE OF THE NORMALIZED RESIDUAL, THE LATTER
C BEING DEFINED AS THE RESIDUAL OF THE
C EQUATION DIVIDED BY THE CENTRAL COEFFICIENT
C WHEN THE LATER IS NOT EQUAL TO 0.0, AND
C DEFINED AS THE RESIDUAL WHEN THE CENTRAL
C COEFFICIENT IS 0.0.
C
C CONCHN CONVERGENCE CRITERION ON THE MAXIMUM ABSOLUTE
C VALUE OF THE CHANGE MADE AT EACH ITERATION
C TO THE ARRAY T.
C
C CONVERGENCE IS ACHIEVED WHEN BOTH THE CRITERIA ARE
C SATISFIED.
C
C RESIDS ARRAY OF DIMENSION ITMAX ... SEE OUTPUT
C
C CHNGS ARRAY OF DIMENSION ITMAX ... SEE OUTPUT
C
C WRKSP1 IS A WORKSPACE ARRAY OF DIMENSION
C (N1M,N2M,N3) USED BY D03UBF
C
C WRKSP2 ... DITTO ...
C
C WRKSP3 ... DITTO ...
C
C WRKSP4 IS A WORKSPACE ARRAY OF DIMENSION
C (N1M,N2M,N3) USED TO PASS THE RESIDUALS TO
C D03UBF, AND IN WHICH THE UPDATE VALUES ARE
C RETURNED FROM D03UBF TO THIS ROUTINE.
C IFAIL IS AN ERROR PARAMETER INDICATOR SET BY THE USER TO
C INDICATE THE TYPE OF FAILURE IF AN ERROR IS
C ENCOUNTERED.
C
C ALL ABOVE PASSED AS ARGUMENTS
C
C PROCESS
C -------
C
C SET ERROR PARAMETER
C CHECK INPUT INTEGERS
C COMMENCEMENT OF CALCULATIONAL PROCEDURE
C START OF ITERATIVE SOLUTION PROCEDURE
C SET CONVERGENCE PARAMETER ACCUMULATORS TO ZERO
C CALCULATE THE RESIDUALS AND STORE IN ARRAY WRKSP4
C CALL SUBROUTINE D03UBF TO PERFORM ONE ITERATION OF S.I.P.
C AND WHICH RETURNS THE ARRAY OF UPDATES IN THE ARRAY WRKSP4
C COMBINE THE UPDATES WITH THE OLD SOLUTION
C CHECK FOR CONVERGENCE
C REPEAT TO CONVERGENCE OR UNTIL MAXIMUM NUMBER OF
C ITERATIONS HAVE BEEN USED.
C SET ERROR INDICATOR ACCORDINGLY.
C RETURN
C
C OUTPUTS
C -------
C
C T ARRAY STORING THE SOLUTION TO THE SYSTEM OF
C EQUATIONS PROVIDED.
C
C ITUSED NUMBER OF ITERATIONS ACTUALLY USED THIS CALL.
C
C ITCOUN ACCUMULATED NUMBER OF ITERATIONS USED
C ITCOUN ON OUTPUT = ITCOUN ON INPUT +
C ITUSED ON OUTPUT
C
C RESIDS VECTOR STORING THE MAXIMUM ABSOLUTE RESIDUALS
C OF EACH ITERATION, DIMENSION ITMAX. ONLY
C THE FIRST ITUSED VALUES ARE SET ON RETURN.
C
C CHNGS ARRAY STORING THE MAXIMUM ABSOLUTE CHANGES OF
C EACH ITERATION, DIMENSION ITMAX. ONLY THE
C FIRST ITUSED VALUES ARE SET ON RETURN.
C
C IFAIL ERROR INDICATOR
C = 0 CORRECT RETURN
C = 1 EITHER N1.LE.1 OR N2.LE.1 OR N3.LE.1
C = 2 N1M IS LESS THAN N1 OR N2M IS LESS THAN N2
C = 3 APARAM.LE.0.0
C = 4 APARAM.GT.((N1-1)**2+(N2-1)**2+(N3-1)**2)/3.
C = 5 CONVERGENCE NOT ACHIEVED. REFER TO VALUES
C IN RESIDS AND CHNGS TO DETERMINE IF THE
C ITERATION WAS CONVERGING. REFER TO GUIDE
C ON SETTING APARAM IF CONVERGENCE IS VERY
C SLOW.
C
C ALL PASSED AS ARGUMENTS
C
C ROUTINES USED
C -------------
C
C D03UBF SINGLE ITERATION OF S.I.P. FOR A SEVEN POINT
C MOLECULE SYSTEM.
C P01AAF ERROR HANDLING ROUTINE
C
C **************************************************************
C
C
C SET ERROR PARAMETER AND ITERATION COUNTER
C
SUBROUTINE D03EDF(NGX,NGY,LDA,A,RHS,UB,MAXIT,ACC,US,U,IOUT,NUMIT,
* IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C MARK 12B REVISED. IER-534 (FEB 1987).
C MARK 13 REVISED. IER-624 (APR 1988).
C MARK 14 REVISED. IER-811 (DEC 1989).
C
C MGD1 author -- P Wesseling, Delft University, Holland
C Modified by -- C P Thompson and G J McCarthy, CSSD, AERE Harwell
C Modified for NAG usage -- NAG Central Office
C
SUBROUTINE D03EEF(XMIN,XMAX,YMIN,YMAX,PDEF,BNDY,NGX,NGY,LDA,A,RHS,
* SCHEME,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE D03FAF(XS,XF,L,LBDCND,BDXS,BDXF,YS,YF,M,MBDCND,BDYS,
* BDYF,ZS,ZF,N,NBDCND,BDZS,BDZF,LAMBDA,LDIMF,
* MDIMF,F,PERTRB,W,LWRK,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
SUBROUTINE D03MAF(H,M,N,NB,NPTS,PLACES,INDEX,IDIM,IN,DIST,LD,
* IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C **************************************************************
C
C H IS THE LENGTH OF THE SIDES OF THE TRIANGLES IN THE ORIGINAL
C GRID.
C ON THE LINE Y=-H/2 THERE ARE M GRID POINTS AT X=0,H*SQRT(3.0),
C . . .,(M-1)*H*SQRT(3.0) IN THE ORIGINAL GRID.
C ON THE LINE X=0 THERE ARE N GRID POINTS AT Y=-H/2,H/2,...,
C (N-3/2)*H IN THE ORIGINAL GRID.
C WE WILL REFER TO A LINE OF POINTS OF THE ORIGINAL GRID WITH
C THE SAME COORDINATES AS A GRID COLUMN.
C THE REGION OVER WHICH THE EQUATION IS TO BE SOLVED MUST LIE
C WITHIN THE RECTANGLE WITH CORNERS AT (0,0),(0,(N-1)*H),
C ((M-1)*H*SQRT(3.0),0).
C NB IS THE NUMBER OF TIMES A TRIANGLE SIDE IS BISECTED
C TO FIND ITS POINT OF INTERSECTION WITH THE BOUNDARY.
C THE ACCURACY IS THEREFORE H/2**NB.
C NPTS IS THE NUMBER OF POINTS AT WHICH THE SOLUTION IS
C FOUND. THIS IS CALCULATED BY D03MAF. IN THE EVENT
C OF FAILURE BECAUSE NDIM IS TOO SMALL (IFAIL=1) OR
C BECAUSE AN INTERNAL POINT IS FOUND ON OR OUTSIDE THE
C BOUNDING RECTANGLE (IFAIL=2) OR BECAUSE LD IS TOO
C SMALL (IFAIL=3) THEN NPTS IS SET TO ZERO.
C PLACES IS AN ARRAY IN WHICH THE X AND Y COORDINATES OF THE
C GRID POINTS ARE STORED. ITS SECOND DIMENSION SHOULD
C BE AT LEAST IDIM WHICH MUST BE LESS THAN NPTS.
C D03MAF CHECKS THAT IDIM.GE.NPTS.
C FOR THE JTH POINT AND ITS THREE NEIGHBOURS AHEAD OF IT IN
C THE ORDERING, INDEX(I,J),I=1,4 HOLDS
C (THE POINT NUMBER) FOR INTERNAL POINTS
C - (THE POINT NUMBER) FOR BOUNDARY POINTS AND
C ZERO FOR EXTERNAL POINTS.
C IN IS THE NAME OF AN INTEGER VALUED FUNCTION TO BE
C SUPPLIED BY THE USER. IT HAS REAL ARGUMENTS X,Y AND
C SHOULD RETURN THE VALUE 1 IF (X,Y) LIES INSIDE THE
C REGION AND 0 IF OUTSIDE.
C DIST IS AN ARRAY USED FOR WORKSPACE. ITS SECOND DIMENSION
C SHOULD BE AT LEAST LD WHICH MUST BE AT LEAST 4*N.
C
C **************************************************************
C
SUBROUTINE D03PCF(NPDE,M,TS,TOUT,PDEDEF,BNDARY,U,NPTS,X,ACC,W,NW,
* IW,NIW,ITASK,ITRACE,IND,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C---------------------------------------------------------------------
C+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
C This code is the setup routine for PDE problems only.
C The remeshing option is not available and the linear algebra
C is restricted to using a banded matrices, with BDF method
C integrator.
C+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
C-------------------------------------------------------------------
SUBROUTINE D03PDF(NPDE,M,TS,TOUT,PDEDEF,BNDARY,U,NBKPTS,XBKPTS,
* NPOLY,NPTS,X,UINIT,ACC,W,NW,IW,NIW,ITASK,ITRACE,
* IND,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C MARK 16 REVISED. IER-1131 (JUL 1993).
C
C Modified to meet NAG standard by M.S. Derakhshan
C ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
C ----------------------------------------------------------------------
SUBROUTINE D03PEF(NPDE,TS,TOUT,PDEDEF,BNDARY,U,NPTS,X,NLEFT,ACC,W,
* NW,IW,NIW,ITASK,ITRACE,IND,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
C
C-----------------------------------------------------------------------
C+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
C This code is the setup routine for PDE problems only.
C The remeshing option is not available and the linear algebra
C is restricted to banded matrices, with BDF method integrator.
C+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
C-----------------------------------------------------------------------
SUBROUTINE D03PHF(NPDE,M,TS,TOUT,PDEDEF,BNDARY,U,NPTS,X,NCODE,
* ODEDEF,NXI,XI,NEQN,RTOL,ATOL,ITOL,NORM,LAOPT,
* ALGOPT,W,NW,IW,NIW,ITASK,ITRACE,IND,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C----------------------------------------------------------------------
C++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
SUBROUTINE D03PJF(NPDE,M,TS,TOUT,PDEDEF,BNDARY,U,NBKPTS,XBKPTS,
* NPOLY,NPTS,X,NCODE,ODEDEF,NXI,XI,NEQN,UVINIT,
* RTOL,ATOL,ITOL,NORM,LAOPT,ALGOPT,W,NW,IW,NIW,
* ITASK,ITRACE,IND,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C
C Modified to meet NAG standard by M.S. Derakhshan
C ----------------------------------------------------------------------
C ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
SUBROUTINE D03PKF(NPDE,TS,TOUT,PDEDEF,BNDARY,U,NPTS,X,NLEFT,NCODE,
* ODEDEF,NXI,XI,NEQN,RTOL,ATOL,ITOL,NORM,LAOPT,
* ALGOPT,W,NW,IW,NIW,ITASK,ITRACE,IND,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
C ----------------------------------------------------------------------
C ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
SUBROUTINE D03PPF(NPDE,M,TS,TOUT,PDEDEF,BNDARY,UVINIT,U,NPTS,X,
* NCODE,ODEDEF,NXI,XI,NEQN,RTOL,ATOL,ITOL,NORM,
* LAOPT,ALGOPT,REMESH,NXFIX,XFIX,NRMESH,DXMESH,
* TRMESH,IPMINF,XRATIO,CONST,MONFFD,W,NW,IW,NIW,
* ITASK,ITRACE,IND,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
C
C ----------------------------------------------------------------------
C N.B. Integer workspace is NXFIX+1 greater than in non-remeshing
C routines to allow allocation of memory for the array IXFIX
C with adjustable dimensions NXFIX+1 (the additional 1 being
C necessary since NXFIX can be zero). i.e. IW(NIW-NXFIX) is
C passed to D03PHZ as the start of the array(IXFIX(NXFIX+1).
C Local scalar NIWO is the size of the original integer
C workspace (passed to D03PHZ).
C ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
SUBROUTINE D03PRF(NPDE,TS,TOUT,PDEDEF,BNDARY,UVINIT,U,NPTS,X,
* NLEFT,NCODE,ODEDEF,NXI,XI,NEQN,RTOL,ATOL,ITOL,
* NORM,LAOPT,ALGOPT,REMESH,NXFIX,XFIX,NRMESH,
* DXMESH,TRMESH,IPMINF,XRATIO,CONST,MONFKB,W,NW,
* IW,NIW,ITASK,ITRACE,IND,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
C ----------------------------------------------------------------------
C SEE COMMENTS IN D03PPF ABOUT SIZE OF INTEGER WORKSPACE
C ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
SUBROUTINE D03PYF(NPDE,U,NBKPTS,XBKPTS,NPOLY,NPTS,XP,INTPTS,ITYPE,
* UOUT,W,NW,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C -------------------------------------------------------------------
C +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
C
C Routine to provide values of the solution and possibly the
C first derivative in space and the flux on the mesh
C XP(INTPTS).
C
C ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
C --------------------------------------------------------------------
SUBROUTINE D03PZF(NPDE,M,U,NPTS,X,XP,INTPTS,ITYPE,UOUT,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C -----------------------------------------------------
C +++++++++++++++++++++++++++++++++++++++++++++++++++++
C Routine to compute solution and possibly its first deriv
C at the interpolation points stored in the array XP.
C Linear interpolation is used.
C
C Parameter list :
C
C XP(INTPTS) ; the spatial interpolation points,
C XP(I) < XP(I+1) , I = 1, IPTS-1
C
C UOUT(NPDE,IPTS,ITYPE) ; array to hold the values found by
C linear interpolation
C
C UOUT(J,K,1) ; holds U(XP(K),T) for PDE J
C UOUT(J,K,2) ; holds DU/DX of UOUT(J,K,1)
C
C X(NPTS) ; is an array that contains the original mesh
C
C U(NEQN) ; holds the original solution; the first NPDE*NPTS
C components of this contains the PDE solution at the
C mesh points X(NPTS) and the last NV components are
C the additional coupled ode variables
C
C ITYPE ; has value 1 or 2 depending on how many components
C of the array UP are required
C
C IFAIL ; error flag ; set to I if XP(I) lies outside the range
C (X(1) , X(NPTS)) .
C ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
C ------------------------------------------------------------------
C
SUBROUTINE D03UAF(N1,N2,N1M,A,B,C,D,E,APARAM,IT,R,WRKSP1,WRKSP2,
* IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 10A REVISED. IER-386 (OCT 1982).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C **************************************************************
C D03UAF PERFORMS 1 ITERATION OF THE STRONGLY IMPLICIT PROCEDURE
C AT EACH CALL TO CALCULATE THE SUCCESSIVE APPROXIMATE
C CORRECTIONS TO THE SOLUTION OF A SYSTEM OF SIMULTANEOUS
C ALGEBRAIC EQUATIONS FOR WHICH THE ITERATIVE UPDATE MATRIX IS
C OF THE FIVE POINT MOLECULE FORM ON A TOPOLOGICALLY TWO-
C DIMENSIONAL RECTANGULAR MESH.
C
C STRONGLY IMPLICIT PROCEDURE ROUTINE FOR 2 DIMENSIONAL 05
C POINT MOLECULES.
C
C INPUTS
C
C N1 NUMBER OF NODES IN THE FIRST COORDINATE DIRECTION.
C N2 NUMBER OF NODES IN THE SECOND COORDINATE DIRECTION.
C N1M FIRST DIMENSION OF ALL THE TWO-DIMENSIONAL ARRAYS.
C A ARRAY OF DIMENSION (N1M,N2) STORING THE COEFFICIENT
C OF THE ITERATIVE UPDATE EQUATIONS AS SHOWN BELOW -
C B ... DIMENSION (N1M,N2) ...
C C ... DIMENSION (N1M,N2) ...
C D ... DIMENSION (N1M,N2) ...
C E ... DIMENSION (N1M,N2) ...
C
C A(I,J)*S(I,J-1)+B(I,J)*S(I-1,J)+C(I,J)*S(I,J)+D(I,J)*
C S(I+1,J)+E(I,J)*S(I,J+1)=R(I,J)
C
C WITH I=1,2,...,N1 AND J=1,2,...,N2 AND WHERE S(I,J)
C IS THE ARRAY WHOSE APPROXIMATE VALUES ARE SOUGHT
C (AND WHICH OVERWRITE THE RESIDUALS R(I,J) STORED ON
C INPUT IN THE ARRAY R). ANY VALUES OF S OUTSIDE THE
C (1-N1,1-N2) ARRAY WHICH DEFINES THE PROBLEM REGION
C ARE TAKEN AS ZERO.
C R IS INPUT AS THE ARRAY OF RESIDUALS, CORRESPONDING
C TO R ABOVE, CHANGED ON OUTPUT TO THE VALUES OF THE
C UPDATE ARRAY CORRESPONDING TO S ABOVE, DIMENSIONED
C (N1M,N2).
C APARAM IS AN ITERATION ACCELERATION PARAMETER FACTOR
C TYPICALLY SET TO 1.0. IF CONVERGENCE IS SLOW, IT
C CAN BE DECREASED. IF DIVERGENCE IS OBTAINED, IT
C SHOULD BE INCREASED. IN EITHER CASE IT MUST NOT GO
C OUTSIDE THE BOUNDS PRESCRIBED (SEE IFAIL PARAMETER
C FOR D03UAF).
C IT IS THE ITERATION COUNTER SET AND INCREMENTED BY
C THE CALLING ROUTINE, IT IS USED TO DETERMINE THE
C APPROPRIATE ITERATION PARAMETERS.
C WRKSP1 IS A WORKSPACE ARRAY OF DIMENSION (N1M,N2).
C WRKSP2 ... DITTO ...
C IFAIL IS AN ERROR PARAMETER INDICATOR SET BY THE USER TO
C INDICATE THE TYPE OF FAILURE IF AN ERROR IS
C ENCOUNTERED.
C
C PROCESS
C
C SET ERROR PARAMETER
C CHECK INPUT INTEGERS
C SET FREQUENTLY REQUIRED INTEGER VARIABLES
C COMMENCEMENT OF CALCULATIONAL PROCEDURE
C SET ODD/EVEN COUNTERS KS=1 FOR ODD ITERATIONS
C KS=2 FOR EVEN ITERATIONS
C DETERMINE THE NUMBER OF THE ACCELERATION PARAMETER TO BE USED
C (THE SAME PARAMETER IS USED FOR 2 ITERATIONS, THERE ARE 9
C PARAMETERS IN ALL).
C FIRST CALCULATE THE TERM, ALM, IN THE LARGEST PARAMETER
C CHECK THE VALUES OF ALM
C DETERMINE THE ITERATION PARAMETER
C START THE APPROXIMATE LU FACTORIZATION DETERMINING THE VALUES
C OF THE ELEMENTS SB,SC IN THE LOWER TRIANGULAR MATRIX L AND
C WRKSP1 AND WRKSP2 IN THE UPPER TRIANGULAR MATRIX U. PROGRESS
C FORWARDS FIRST AND INVERT THE LOWER TRIANGULAR MATRIX L AS
C ONE PROCEEDS SO THAT THE ELEMENTS SB,SC DO NOT HAVE TO BE
C STORED IN ARRAYS. THE ELEMENTS WRKSP1 AND WRKSP2 HAVE TO BE
C STORED FOR THE SUBSEQUENT INVERSION OF THE MATRIX U BY BACK
C SUBSTITUTION.
C DO JL = 1,N2
C (JL IS ONLY A COUNTER IN THE SECOND COORDINATE DIRECTION
C J IS THE INDEX OF THE SECOND COORDINATE AND ON
C ALTERNATE ITERATIONS SCANS FIRST INCREASING AND THEN
C DECREASING).
C DO I = 1,N1
C (I IS THE INDEX IN THE FIRST COORDINATE DIRECTION)
C STORE TH VALUE OF THE CENTRAL COEFFICIENT
C DETERMINE THE TYPE OF THE DIFFERENCE EQUATION
C FOR A FIVE POINT MOLECULE EQUATION -
C SET VALUES OF ADJACENT ARRAY ELEMENTS DEPENDENT
C ON NODAL POSITION
C CALCULATE THE OFF-DIAGONAL ELEMENTS OF L,
C NAMELY SB AND SC
C CALCULATE THE OFTEN USED EXPRESSIONS
C CALCULATE THE VALUE OF THE DIAGONAL ELEMENT OF L
C CALCULATE AND STORE THE ELEMENTS OF U, NAMELY
C WRKSP1 AND WRKSP2
C CALCULATE THE ELEMENTS OF L**(-1) * RESIDUAL
C ARRAY
C FOR THE EXPLICIT EQUATION DO THE SAME
C END OF SCAN FOR FORWARD ELIMINATION
C BACK SUBSTITUTION TO MULTIPLY BY U**(-1)
C DO JL = 1,N2
C J RUNS BACKWARDS AND FORWARDS ALTERNATELY
C DO I = N1,N1-1,....,2,1
C R IS INITIALLY L**(-1) * RESIDUAL, IT BECOMES
C (U**(-1) * (L**(-1) *RESIDUAL) = CHANGE
C END OF SCAN FOR BACK SUBSTITUTION
C RETURN
C
C OUTPUTS
C
C R ARRAY STORING THE APPROXIMATE SOLUTION TO THE SYSTEM
C OF EQUATIONS PROVIDED, AFTER ONE ITERATION. NOTE
C THAT IT HAS OVERWRITTEN THE INPUT RESIDUAL.
C IFAIL ERROR INDICATOR
C =0 CORRECT RETURN
C =1 EITHER N1.LE.1 OR N2.LE.1
C =2 N1M IS LESS THAN N1
C =3 APARAM.LE.0.0
C =4 APARAM.GT.((N1-1)**2+N2-1)**2)/2.
C
C ROUTINES USED
C
C P01AAF ERROR HANDLING ROUTINE
C
C **************************************************************
C
SUBROUTINE D03UBF(N1,N2,N3,N1M,N2M,A,B,C,D,E,F,G,APARAM,IT,R,
* WRKSP1,WRKSP2,WRKSP3,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 10A REVISED. IER-387 (OCT 1982).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C **************************************************************
C
C DAVID A. H. JACOBS 22/5/79
C D03UBF PERFORMS ONE ITERATION OF THE STRONGLY IMPLICIT
C PROCEDURE AT EACH CALL TO CALCULATE THE SUCCESSIVE
C APPROXIMATE CORRECTIONS TO THE SOLUTION OF A SYSTEM
C OF SIMULTANEOUS ALGEBRAIC EQUATIONS FOR WHICH THE
C ITERATIVE UPDATE MATRIX IS OF THE SEVEN POINT
C MOLECULE FORM ON A TOPOLOGICALLY THREE-DIMENSIONAL
C RECTANGULAR MESH.
C
C INPUTS
C ------
C
C N1 NUMBER OF NODES IN THE FIRST COORDINATE DIRECTION.
C
C N2 NUMBER OF NODES IN THE SECOND COORDINATE DIRECTION.
C
C N3 NUMBER OF NODES IN THE THIRD COORDINATE DIRECTION.
C
C N1M FIRST DIMENSION OF ALL THREE DIMENSIONAL ARRAYS.
C
C N2M SECOND DIMENSION OF ALL THREE DIMENSIONAL ARRAYS.
C
C A,B,C, ARRAYS ALL OF DIMENSION (N1M,N2M,N3) STORING
C D,E,F,G THE COEFFICIENTS OF THE ITERATIVE UPDATE
C EQUATIONS -
C
C A(I,J,K)*S(I,J,K-1)+B(I,J,K)*S(I,J-1,K)+C(I,J,K)*S(I-1,J,K)+
C D(I,J,K)*S(I,J,K)+E(I,J,K)*S(I+1,J,K)+F(I,J,K)*S(I,J+1,K)+
C G(I,J,K)*S(I,J,K+1)=R(I,J,K)
C
C WITH I=1,2,...,N1 , J=1,2,...,N2 AND K=1,2,...,N3
C AND WHERE S(I,J,K) IS THE ARRAY WHOSE
C APPROXIMATE VALUES ARE SOUGHT (AND WHICH
C OVERWRITE THE RESIDUALS R(I,J,K) STORED ON
C INPUT IN THE ARRAY R). ANY VALUES OF S
C OUTSIDE THE (1-N1,1-N2,1-N3) ARRAY ARE
C TAKEN AS ZERO.
C
C R IS INPUT AS THE ARRAY OF RESIDUALS,
C CORRESPONDING TO R ABOVE, CHANGED ON OUTPUT
C TO THE VALUES OF THE UPDATE ARRAY
C CORRESPONDING TO S ABOVE, DIMENSIONED
C R(N1M,N2M,N3)
C
C APARAM IS AN ITERATION ACCELERATION PARAMETER FACTOR
C TYPICALLY SET TO 1.0. IF CONVERGENCE IS
C SLOW, IT CAN BE DECREASED. IF DIVERGENCE IS
C OBTAINED, IT SHOULD BE INCREASED. IN EITHER
C CASE IT MUST NOT GO OUTSIDE THE BOUNDS
C PRESCRIBED. (SEE IFAIL PARAMETER DETAILS.)
C
C IT IS THE ITERATION COUNTER SET AND
C INCREMENTED BY THE CALLING ROUTINE, IT IS
C USED TO DETERMINE THE APPROPRIATE ITERATION
C PARAMETERS.
C
C WRKSP1 IS A WORKSPACE ARRAY OF DIMENSION (N1M,N2M,N3)
C
C WRKSP2 ... DITTO ...
C
C WRKSP3 ... DITTO ...
C
C IFAIL IS AN ERROR PARAMETER INDICATOR SET BY THE USER TO
C INDICATE THE TYPE OF FAILURE IF AN ERROR IS
C ENCOUNTERED.
C
C ALL ABOVE PASSED AS ARGUMENTS
C
C PROCESS
C -------
C
C SET ERROR PARAMETER
C CHECK INPUT INTEGERS
C SET FREQUENTLY REQUIRED INTEGER VARIABLES
C COMMENCEMENT OF CALCULATIONAL PROCEDURE
C SET ODD/EVEN COUNTERS KS=1 FOR ODD ITERATIONS
C KS=2 FOR EVEN ITERATIONS
C DETERMINE THE NUMBER OF THE ACCELERATION PARAMETER TO
C BE USED (THE SAME PARAMETER IS USED FOR 2 ITERATIONS,
C THERE ARE 9 PARAMETERS IN ALL).
C FIRST CALCULATE THE TERM, ALM, IN THE LARGEST PARAMETER
C CHECK THE VALUES OF ALM
C DETERMINE THE ITERATION PARAMETER
C START OF APPROXIMATE LU FACTORIZATION DETERMINING THE
C VALUES OF THE ELEMENTS SA,SB,SC IN THE LOWER
C TRIANGULAR MATRIX L AND WRKSP1, WRKSP2 AND WRKSP3 IN
C THE UPPER TRIANGULAR MATRIX U. PROGRESS FORWARDS
C FIRST AND INVERT THE LOWER TRIANGULAR MATRIX L AS
C ONE PROCEEDS SO THAT THE ELEMENTS SA,SB,SC DO NOT
C HAVE TO BE STORED IN ARRAYS.THE ELEMENTS WRKSP1,
C WRKSP2 AND WRKSP3 HAVVE TO BE STORED FOR THE
C SUBSEQUENT INVERSION OF THE MATRIX U BY BACK
C SUBSTITUTION.
C DO KL=1,N3
C (KL IS ONLY A COUNTER IN THE THIRD COORDINATE DIRECTION
C K IS THE INDEX OF THE THIRD COORDINATE AND ON ALTERNATE
C ITERATIONS SCANS FIRST INCREASING AND THEN DECREASING).
C DO JL=1,N2
C (JL IS ONLY A COUNTER IN THE SECOND COORDINATE DIRECTION
C J IS THE INDEX OF THE SECOND COORDINATE AND ON ALTERNATE
C ITERATIONS SCANS FIRST INCREASING AND THEN DECREASING).
C DO I=1,N1
C (I IS THE INDEX IN THE FIRST COORDINATE DIRECTION)
C STORE THE VALUE OF THE CENTRAL COEFFICIENT
C DETERMINE THE TYPE OF THE DIFFERENCE EQUATION
C FOR A SEVEN POINT MOLECULE EQUATION -
C SET VALUES OF ADJACENT ARRAY ELEMENTS
C DEPENDENT ON NODAL POSITION
C
C CALCULATE THE OFF-DIAGONAL ELEMENTS OF
C L, NAMELY SA,SB,SC
C CALCULATE OFTEN USED EXPRESIONS
C CALCULATE THE VALUE OF THE DIAGONAL ELEMENT OF L
C CALCULATE AND STORE THE ELEMENTS OF U,
C NAMELY WRKSP1, WRKSP2 AND WRKSP3
C
C CALCULATE THE ELEMENTS OF L**(-1) *
C RESIDUAL ARRAY
C FOR THE EXPLICIT EQUATION DO THE SAME
C END OF SCAN FOR THE FORWARD ELIMINATION
C BACK SUBSTITUTION TO MULTIPLY BY U**(-1)
C DO KL=1,N3
C K RUNS BACKWARDS AND FORWARDS ALTERNATELY
C DO JL=1,N2
C J RUNS BACKWARDS AND FORWARDS ALTERNATELY
C DO I=N1,N1-1,...,2,1
C R IS INITIALLY L**(-1) * RESIDUAL, IT BECOMES
C (U**(-1) * (L**(-1) * RESIDUAL) = CHANGE
C END OF SCAN FOR THE BACK SUBSTITUTION
C RETURN
C
C OUTPUTS
C -------
C
C R ARRAY STORING THE APPROXIMATE SOLUTION TO
C THE SYSTEM OF EQUATIONS PROVIDED, AFTER
C THE ONE ITERATION. NOTE THAT IT HAS
C OVERWRITTEN THE INPUT RESIDUAL.
C
C IFAIL ERROR INDICATOR
C = 0 CORRECT RETURN
C = 1 EITHER N1.LE.1 OR N2.LE.1 OR N3.LE.1
C = 2 N1M IS LESS THAN N1 OR N2M IS LESS THAN N2
C = 3 APARAM.LE.0.0
C = 4 APARAM.GT.((N1-1)**2+(N2-1)**2+(N3-1)**2)/3.
C
C BOTH PASSED AS ARGUMENTS
C
C ROUTINES USED
C -------------
C
C P01AAF ERROR HANDLING ROUTINE
C
C **************************************************************
C
SUBROUTINE D04AAF(XVAL,NDER,HBASE,DER,EREST,FUN,IFAIL)
C
C *** PURPOSE ***
C
C A SUBROUTINE FOR NUMERICAL DIFFERENTIATION AT A POINT. IT
C RETURNS A SET OF APPROXIMATIONS TO THE J-TH ORDER DERIVATIVE
C (J=1,2,...14) OF FUN(X) EVALUATED AT X = XVAL AND, FOR EACH
C DERIVATIVE, AN ERROR ESTIMATE ( WHICH INCLUDES THE EFFECT OF
C AMPLIFICATION OF ROUND- OFF ERRORS).
C
C
C *** INPUT PARAMETERS ***
C
C (1) XVAL REAL. THE ABSCISSA AT WHICH THE SET OF
C DERIVATIVES IS REQUIRED.
C (2) NDER INTEGER. THE HIGHEST ORDER DERIVATIVE REQUIRED.
C IF(NDER.GT.0) ALL DERIVATIVES UP TO MIN(NDER,14) ARE
C CALCULATED.
C IF(NDER.LT.0 AND NDER EVEN) EVEN ORDER DERIVATIVES
C UP TO MIN(-NDER,14) ARE CALCULATED.
C IF(NDER.LT.0 AND NDER ODD ) ODD ORDER DERIVATIVES
C UP TO MIN(-NDER,13) ARE CALCULATED.
C (3) HBASE REAL. A STEP LENGTH.
C (6) FUN THE NAME OF A REAL FUNCTION SUBPROGRAMME,
C WHICH IS REQUIRED BY THE ROUTINE AS A SUBPROGRAMME
C AND WHICH REPRESENTS THE FUNCTION BEING DIFFERENTIATED
C THE ROUTINE REQUIRES 21 FUNCTION EVALUATIONS FUN(X)
C LOCATED AT X = XVAL AND AT
C X = XVAL + (2*J-1)*HBASE, J=-9,-8, .... +9,+10.
C THE FUNCTION VALUE AT X = XVAL IS DISREGARDED WHEN
C ONLY ODD ORDER DERIVATIVES ARE REQUIRED.
C
C *** OUTPUT PARAMETERS ***
C
C (4) DER(J) J=1,2,...14. REAL. A VECTOR OF
C APPROXIMATIONS TO THE J-TH DERIVATIVE OF FUN(X)
C EVALUATED AT X = XVAL.
C (5) EREST(J) J=1,2,...14. REAL. A VECTOR OF
C ESTIMATES OF THE DABSOLUTE ACCURACY OF DER(J). THESE
C ARE NEGATIVE WHEN EREST(J).GT.ABS(DER(J)), OR WHEN,
C FOR SOME OTHER REASON THE ROUTINE IS DOUBTFUL ABOUT
C THE VALIDITY OF THE RESULT.
C
C *** IMPORTANT WARNING ***
C
C EREST IS AN ESTIMATE OF THE OVERALL ERROR. IT IS ESSENTIAL
C FOR
C PROPER USE THAT THE USER CHECKS EACH VALUE OF DER
C SUBSEQUENTLY
C USED TO SEE THAT IT IS ACCURATE ENOUGH FOR HIS PURPOSES.
C FAILURE TO DO THIS MAY RESULT IN THE CONTAMINATION OF ALL
C SUBSEQUENT RESULTS.
C IT IS TO BE EXPECTED THAT IN NEARLY ALL CASES DER(14) WILL
C BE
C UNUSABLE.( 14 REPRESENTS A LIMIT IN WHICH FOR THE EASIEST
C FUNCTION LIKELY TO BE ENCOUNTERED THIS ROUTINE MIGHT JUST
C OBTAIN
C AN APPROXIMATION OF THE CORRECT SIGN.)
C
C *** NOTE ON CALCULATION ***
C
C THE CALCULATION IS BASED ON THE EXTENDED T-TABLE (T SUB
C K,P,S)
C DESCRIBED IN LYNESS AND MOLER, NUM. MATH., 8, 458-464,(1966).
C REFERENCES IN COMMENT CARDS TO TTAB(NK,NP,NS) REFER TO
C T-TABLE
C WITH NK=K+1, NP=P+1, NS=S+1. SINCE ONLY PART OF THE
C EXTENDED
C T-TABLE IS NEEDED AT ONE TIME, THAT PART IS STORED IN
C RTAB(10,7)
C AND IS SUBSEQUENTLY OVERWRITTEN. HERE
C RTAB(NK,NP-NS+1) = TTAB(NK,NP,NS).
C
C NAG COPYRIGHT 1976
C MARK 5 RELEASE
C MARK 6B REVISED IER-116 (MAR 1978)
C MARK 7 REVISED IER-139 (DEC 1978)
C MARK 8 REVISED. IER-219 (MAR 1980)
C MARK 8D REVISED. IER-271 (DEC 1980).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
SUBROUTINE D05AAF(LAMBDA,A,B,K1,K2,G,F,C,N,IND,W1,W2,WD,NMAX,MN,
* IFAIL)
C MARK 5 RELEASE NAG COPYRIGHT 1976
C MARK 6 REVISED
C MARK 9 REVISED. IER-304 (SEP 1981).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C BASED UPON NPL DNAC LIBRARY SUBROUTINE F2926
C THIS SUBROUTINE SOLVES THE FREDHOLM INTEGRAL EQUATION OF THE
C SECOND KIND
C F(X) - LAMBDA * (INTEGRAL FROM A TO B OF K(X,S) * F(S) DS) =
C G(X)
C FOR A.LE.X.LE.B, WHEN THE KERNEL IS DEFINED IN TWO PARTS
C K = K1 FOR A.LE.S.LE.X AND K = K2 FOR X.LT.S.LE.B.
C THE METHOD USED IS THAT OF EL-GENDI, WHICH REQUIRES
C THAT EACH OF THE FUNCTIONS K1 AND K2 SHOULD BE SMOOTH
C AND NON SINGULAR FOR ALL X AND S IN THE CLOSED INTERVAL
C (A,B).
C THE SUBROUTINE USES AUXILIARY SUBROUTINE F04AAF
C AND FUNCTIONS P01AAF AND C06DBF.
C NAG COPYRIGHT 1976
C MARK 5 RELEASE
SUBROUTINE D05ABF(K,G,LAMBDA,A,B,ODOREV,EV,N,CM,F1,WK,NMAX,NT2P1,
* F,C,IFAIL)
C MARK 6 RELEASE. NAG COPYRIGHT 1977.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C NPL DNAC LIBRARY SUBROUTINE F2925.
C
C THIS SUBROUTINE SOLVES THE NON-SINGULAR FREDHOLM INTEGRAL
C EQUATION OF THE SECOND KIND-
C F(X) - LAMBDA*(INTEGRAL FROM A TO B OF K(X,S)*F(S) DS) = G(X)
C FOR A .LE. X .LE. B, BY THE METHOD OF S.E. EL-GENDI
C (COMPUTER J., 1969, VOL.12, PP.282-287).
C
C THE SUBROUTINE USES AUXILIARY SUBROUTINES F04AAF
C AND FUNCTIONS X01AAF,P01AAF AND C06DBF.
C
SUBROUTINE D05BAF(CK,CG,CF,METHOD,IORDER,ALIM,TLIM,YN,ERREST,NOUT,
* TOL,THRESH,WORK,IWK,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C MARK 14B REVISED. IER-845 (MAR 1990).
C
C <<<<<<<<<<<<<<<<<<<<<<<<<<< >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
C
C A code for the solution of convolution Volterra
C equation of the second kind
C
C by
C
C M.S Derakhshan, January 1989.
C ---------------------------------------------------------------
C +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
C This subroutine computes the numerical solution of the
C convolution Volterra integeral equation of the second kind
C
C t
C (1) y(t) = f(t) + I k(t - s) g(s,y(s))ds, (t >= a).
C a
C
C The underlying methods for the numerical solution of (1)
C are the reducible linear multistep formulae of the Adam's
C and Backward Differentiation (BD) type.
C +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
C ---------------------------------------------------------------
SUBROUTINE D05BDF(CK,CF,CG,INITWT,IORDER,TLIM,TOLNL,NMESH,YN,WORK,
* LWK,NCT,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
C
C <<<<<<<<<<<<<<<<<<<<<<<<<<<< >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
C
C A code for the solution of nonlinear convolution
C weakly singular Abel-Volterra equation
C of the second kind
C
C M.S Derakhshan,
C Mark 16. Nag Copyright, 1991.
C --------------------------------------------------------------
C ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
C This routine computes the numerical solution of a weakly
C singular Volterra-Abel integral equation of the 2nd kind of
C the form :
C t -1/2
C y(t) = f(t) + (1/sqrt(pi)) I (t-s) k(t-s) g(s, y(s)) ds,
C 0
C (t >= 0).
C The solution YN(i) approximates y(t) at t = (i-1)*H,
C i = 1, ..., NMESH. The code is based on
C the BDF fractional linear multistep method of orders
C 4, 5 and 6 and FFT techniques can be used for the
C computation of the lag term.
C For a complete description of the parameters, see the
C routine specification.
C ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
C --------------------------------------------------------------
SUBROUTINE D05BEF(CK,CF,CG,INITWT,IORDER,TLIM,TOLNL,NMESH,YN,WORK,
* LWK,NCT,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
C
C <<<<<<<<<<<<<<<<<<<<<<<<<<<< >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
C
C A code for the solution of nonlinear convolution
C weakly singular Abel-Volterra equation
C of the first kind
C
C M.S Derakhshan,
C Mark 16. Nag Copyright, 1991.
C ---------------------------------------------------------------
C +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
C This routine computes the numerical solution of a weakly
C singular Volterra-Abel integral equation of the 1st kind of
C the form :
C t -1/2
C 0 = f(t) + (1/sqrt(pi)) I (t-s) k(t-s) g(s, y(s)) ds,
C 0
C (t >= 0).
C The solution YN(i) approximates y(t) at t = (i-1)*H,
C i = 1, ..., NMESH. The code is based on
C the BDF fractional linear multistep method of orders
C 4, 5 and 6 and FFT techniques can be used for the
C computation of the lag term. The solution at zero should be
C available.
C For a complete description of the parameters, see the
C routine specification.
C +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
C -------------------------------------------------------------
SUBROUTINE D05BWF(METHOD,IORDER,OMEGA,NOMG,LENSW,SW,LDSW,NWT,
* IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
C
C <<<<<<<<<<<<<<<<<<<<<<<<<<< >>>>>>>>>>>>>>>>>>>>>>>>>>>
C
C A code for generating the quadrature weights
C associated with a reducible linear multistep
C method for solving Volterra
C equations
C
C Mark 16 Release. NAG Copyright 1991.
C M.S Derakhshan, May 1991.
C ------------------------------------------------------
C ++++++++++++++++++++++++++++++++++++++++++++++++++++++
C This subroutine generates the weights associated with
C the Adams formulae of the orders 3 to 6, and the
C Backward Differentiation Formulae of the orders
C 2 to 5.
C ++++++++++++++++++++++++++++++++++++++++++++++++++++++
C ------------------------------------------------------
C
C
C
SUBROUTINE D05BYF(IORDER,IQ,LENFW,WT,SW,LDSW,WORK,LWK,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
C
C <<<<<<<<<<<<<<<<<<<<<<<<< >>>>>>>>>>>>>>>>>>>>>>>>>>
C
C A routine for computing Fractional weights
C of BDF formulae of orders 4, 5 and 6.
C
C M.S. Derakhshan.
C --------------------------------------------------------------
C ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
C This routine computes the square root convolution weights
C of BDF reducible rules of orders 4 to 6. It also computes
C the values of the starting fractional weights associated with
C these BDF formulae.
C ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
C --------------------------------------------------------------
SUBROUTINE E01AAF(A,B,C,N1,N2,N,X)
C MARK 1 RELEASE. NAG COPYRIGHT 1971
C MARK 4.5 REVISED
C MARK 11.5(F77) REVISED. (SEPT 1985.)
SUBROUTINE E01ABF(N,P,A,G,N1,N2,IFAIL)
C MARK 2 RELEASE. NAG COPYRIGHT 1972
C MARK 3 REVISED.
C MARK 4.5 REVISED
C MARK 7A REVISED IER-158 (MAR 1979)
C MARK 9 REVISED. IER-351 (SEP 1981)
C MARK 11.5(F77) REVISED. (SEPT 1985.)
SUBROUTINE E01AEF(M,XMIN,XMAX,X,Y,IP,N,ITMIN,ITMAX,A,WRK,LWRK,
* IWRK,LIWRK,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C *******************************************************
C
C NPL ALGORITHMS LIBRARY ROUTINE PINTRP
C
C CREATED 17 07 79. UPDATED 14 05 80. RELEASE 00/42
C
C AUTHORS ... GERALD T. ANTHONY, MAURICE G. COX,
C J. GEOFFREY HAYES AND MICHAEL A. SINGER.
C NATIONAL PHYSICAL LABORATORY, TEDDINGTON,
C MIDDLESEX TW11 OLW, ENGLAND
C
C *******************************************************
C
C E01AEF. A ROUTINE, WITH CHECKS, WHICH DETERMINES AND
C REFINES A POLYNOMIAL INTERPOLANT Q(X) TO DATA WHICH
C MAY CONTAIN DERIVATIVES.
C
C INPUT PARAMETERS
C M NUMBER OF DISTINCT X-VALUES
C XMIN,
C XMAX LOWER AND UPPER ENDPOINTS OF INTERVAL
C X INDEPENDENT VARIABLE VALUES (DISTINCT)
C Y VALUES AND DERIVATIVES OF
C DEPENDENT VARIABLE.
C IP HIGHEST ORDER OF DERIVATIVE AT EACH X-VALUE.
C N NUMBER OF INTERPOLATING CONDITIONS.
C N = M + IP(1) + IP(2) + ... + IP(M).
C ITMIN,
C ITMAX MINIMUM AND MAXIMUM NUMBER OF ITERATIONS TO BE
C PERFORMED.
C
C OUTPUT PARAMETERS
C A CHEBYSHEV COEFFICIENTS OF Q(X)
C
C WORKSPACE (AND ASSOCIATED DIMENSION) PARAMETERS
C WRK REAL WORKSPACE ARRAY. THE FIRST IMAX ELEMENTS
C CONTAIN, ON EXIT, PERFORMANCE INDICES FOR
C THE INTERPOLATING POLYNOMIAL, AND THE NEXT
C N ELEMENTS THE COMPUTED RESIDUALS
C LWRK DIMENSION OF WRK. LWRK MUST BE AT LEAST
C 7*N + 5*IMAX + M + 2, WHERE
C IMAX IS ONE MORE THAN THE LARGEST ELEMENT
C OF THE ARRAY IP.
C IWRK INTEGER WORKSPACE ARRAY. ON EXIT, IWRK(1)
C CONTAINS THE NUMBER OF ITERATIONS TAKEN
C LIWRK DIMENSION OF IWRK. AT LEAST 2*M + 2.
C
C FAILURE INDICATOR PARAMETER
C IFAIL FAILURE INDICATOR.
C 0 - SUCCESSFUL TERMINATION.
C 1 - AT LEAST ONE OF THE FOLLOWING CONDITIONS
C HAS BEEN VIOLATED -
C M AT LEAST 1,
C N = M + IP(1) + IP(2) + ... + IP(M),
C LWRK AT LEAST 7*N + 5*IMAX + M + 2,
C LIWRK AT LEAST 2*M + 2.
C 2 - FOR SOME I, IP(I) IS LESS THAN 0.
C 3 - AT LEAST ONE OF THE FOLLOWING CONDITIONS
C HAS BEEN VIOLATED -
C XMIN STRICTLY LESS THAN XMAX,
C FOR EACH I, X(I) MUST LIE IN THE
C INTERVAL XMIN TO XMAX,
C THE X-VALUES MUST ALL BE DISTINCT
C 4 - NOT ALL PERFORMANCE INDICES LESS THAN
C ONE, BUT ITMAX ITERATIONS PERFORMED,
C 5 - COMPUTATION TERMINATED BECAUSE
C ITERATIONS DIVERGING.
C
C
C CHECK AND SET ITERATION LIMITS
C
SUBROUTINE E01BAF(M,X,Y,K,C,LCK,WRK,LWRK,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C ******************************************************
C
C NPL ALGORITHMS LIBRARY ROUTINE SP3INT
C
C CREATED 16/5/79. RELEASE 00/00
C
C AUTHORS ... GERALD T. ANTHONY, MAURICE G.COX
C J.GEOFFREY HAYES AND MICHAEL A. SINGER.
C NATIONAL PHYSICAL LABORATORY, TEDDINGTON,
C MIDDLESEX TW11 OLW, ENGLAND
C
C ******************************************************
C
C E01BAF. AN ALGORITHM, WITH CHECKS, TO DETERMINE THE
C COEFFICIENTS IN THE B-SPLINE REPRESENTATION OF A CUBIC
C SPLINE WHICH INTERPOLATES (PASSES EXACTLY THROUGH) A
C GIVEN SET OF POINTS.
C
C INPUT PARAMETERS
C M THE NUMBER OF DISTINCT POINTS WHICH THE
C SPLINE IS TO INTERPOLATE.
C (M MUST BE AT LEAST 4.)
C X ARRAY CONTAINING THE DISTINCT VALUES OF THE
C INDEPENDENT VARIABLE. NB X(I) MUST BE
C STRICTLY GREATER THAN X(J) WHENEVER I IS
C STRICTLY GREATER THAN J.
C Y ARRAY CONTAINING THE VALUES OF THE DEPENDENT
C VARIABLE.
C LCK THE SMALLER OF THE ACTUALLY DECLARED DIMENSIONS
C OF K AND C. MUST BE AT LEAST M + 4.
C
C OUTPUT PARAMETERS
C K ON SUCCESSFUL EXIT, K CONTAINS THE KNOTS
C SET UP BY THE ROUTINE. IF THE SPLINE IS
C TO BE EVALUATED (BY NPL ROUTINE E02BEF,
C FOR EXAMPLE) THE ARRAY K MUST NOT BE
C ALTERED BEFORE CALLING THAT ROUTINE.
C C ON SUCCESSFUL EXIT, C CONTAINS THE B-SPLINE
C COEFFICIENTS OF THE INTERPOLATING SPLINE.
C THESE ARE ALSO REQUIRED BY THE EVALUATING
C ROUTINE E02BEF.
C IFAIL FAILURE INDICATOR
C 0 - SUCCESSFUL TERMINATION.
C 1 - ONE OF THE FOLLOWING CONDITIONS HAS
C BEEN VIOLATED -
C M AT LEAST 4
C LK AT LEAST M + 4
C LWORK AT LEAST 6 * M + 16
C 2 - THE VALUES OF THE INDEPENDENT VARIABLE
C ARE DISORDERED. IN OTHER WORDS, THE
C CONDITION MENTIONED UNDER X IS NOT
C SATISFIED.
C
C WORKSPACE (AND ASSOCIATED DIMENSION) PARAMETERS
C WRK WORKSPACE ARRAY, OF LENGTH LWRK.
C LWRK ACTUAL DECLARED DIMENSION OF WRK.
C MUST BE AT LEAST 6 * M + 16.
C
SUBROUTINE E01BEF(N,X,F,D,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C
C Computes a monotonicity-preserving piecewise cubic Hermite
C interpolant to a set of data points.
C
C E01BEF is a driver for E01BEZ (derived from PCHIP routine PCHIM),
C specialiSed for the case INCFD = 1.
C
SUBROUTINE E01BFF(N,X,F,D,M,PX,PF,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C
C Evaluates a piecewise cubic Hermite interpolant at a set of
C points.
C
C E01BFF is a driver for E01BFZ (derived from PCHIP routine PCHFE),
C specialised for the case INCFD = 1.
C
SUBROUTINE E01BGF(N,X,F,D,M,PX,PF,PD,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C
C Evaluates a piecewise cubic Hermite interpolant and its first
C derivative at a set of data points.
C
C E01BGF is a driver for E01BGZ (derived from PCHIP routine PCHFD),
C specialised for the case INCFD = 1.
C
SUBROUTINE E01BHF(N,X,F,D,A,B,PINT,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C
C Evaluates the definite integral of a piecewise cubic Hermite
C interpolant over the interval (a,b).
C
C E01BHF is a driver for E01BHZ (derived from PCHIP routine PCHIA),
C specialised for the case INCFD = 1.
C
SUBROUTINE E01DAF(MX,MY,X,Y,F,PX,PY,LAMDA,MU,C,WRK,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C Derived from DASL routine B2IRE.
SUBROUTINE E01RAF(N,X,F,M,A,U,IW,IFAIL)
C MARK 9 RELEASE. NAG COPYRIGHT 1981
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C
C E01RAF PRODUCES, FROM A SET OF FUNCTION VALUES
C AND CORRESPONDING ABSCISSAE, THE COEFFICIENTS OF
C AN INTERPOLATING RATIONAL FUNCTION EXPRESSED IN
C CONTINUED FRACTION FORM
C
C ON ENTRY
C N -INTEGER- NUMBER OF INTERPOLATION POINTS
C X -REAL ARRAY- ABSCISSA POINTS
C F -REAL ARRAY- FUNCTION VALUES
C ON EXIT
C M -INTEGER- NUMBER OF THIELE COEFFICIENTS
C U -REAL ARRAY- REORDERED ABSCISSA POINTS
C A -REAL ARRAY- THIELE COEFFICIENTS
C IW -INTEGER ARRAY- WORKSPACE
C IFAIL -INTEGER- ERROR INDICATOR
C
C VERSION U.K.C. 21/11/79
C BY P.R. GRAVES-MORRIS, T.R. HOPKINS AND D.J. WINSTANLEY
C BASED ON A PROCEDURE BY A.R. CURTIS.
C
C EPS IS A MACHINE DEPENDENT VARIABLE.
SUBROUTINE E01RBF(M,A,U,X,F,IFAIL)
C MARK 9 RELEASE. NAG COPYRIGHT 1981
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C
C E01RBF EVALUATES CONTINUED FRACTIONS OF THE FORM
C PRODUCED BY E01RAF
C
C ON ENTRY
C X -REAL- POINT AT WHICH THIELE INTERPOLANT IS TO BE
C EVALUATED
C U -REAL ARRAY- ABSCISSA POINTS (REARRANGED)
C A -REAL ARRAY- THIELE COEFFICIENTS
C M -INTEGER- NUMBER OF COEFFICIENTS
C ON EXIT
C F -REAL- CALCULATED VALUE OF THE THIELE INTERPOLANT
C IFAIL -INTEGER- ERROR INDICATOR
C
C EPS IS A MACHINE DEPENDENT VARIABLE.
SUBROUTINE E01SAF(M,X,Y,F,TRIANG,GRADS,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C
C Creates a Thiessen triangulation of (M,X,Y) and computes
C derivatives required for interpolation.
C
C Input arguments:
C M is the number of data points (nodes).
C X, Y, F are the scattered data to be interpolated, F = F(X,Y).
C
C Output arguments:
C TRIANG contains the data structure defining the triangulation;
C used as parameters IADJ and IEND to subroutine E01SAZ.
C GRADS contains the estimated partial derivatives at the nodes;
C first row contains derivatives with respect to X, second row
C with respect to Y. Used as parameter ZXZY to subroutine E01SAZ.
C
C Parameters:
C TOL is a convergence criterion for estimating gradients at nodes;
C TOL .ge. 0.0; TOL = 0.01 is sufficient.
C MAXIT is the maximum number of iterations allowed for computation
C of estimated gradients; MAXIT .ge. 0.
C
C M, X, Y, F, TRIANG and GRADS should be used as input to NAG
C library routine E01SBF to compute interpolated values of F(X,Y).
C
SUBROUTINE E01SBF(M,X,Y,F,TRIANG,GRADS,PX,PY,PF,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C
C Takes a Thiessen triangulation of a set of points in the plane,
C and estimated partial derivatives with respect to X and Y, as
C returned by NAG Fortran Library routine E01SAF. Returns the
C value of a C1 function F(X,Y) interpolating the points and their
C partial derivatives, evaluated at the point (PX,PY).
C If (PX,PY) lies outside the triangulation boundary, extrapolation
C is performed. Interpolation is exact for quadratic data.
C
SUBROUTINE E01SEF(M,X,Y,F,RNW,RNQ,NW,NQ,FNODES,MINNQ,WRK,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C MARK 14C REVISED. IER-876 (NOV 1990).
C
C This subroutine serves to construct a Shepard's method type of
C surface through a set of scattered data points, using a least
C squares quadratic nodal function through each point.
C
C Routine created - December 1986
C Author - Richard Franke
C Naval Postgraduate School Monterey,
C California 93940
C Adapted for Nag by H.Scullion (Leic Univ.)
C and I. Gladwell (Nag Ltd.)
C
C Input Parameters:
C
C M - The number of data points.
C
C X,Y,F - The data points, (X(I),Y(I),F(I),I=1,M)
C
C RNW - The radius for the weight functions.
C
C RNQ - The radius for the nodal functions. If either of
C RNW or RNQ is .le. zero on entry, then their values
C will be computed, using the values of NW and NQ,
C and will be returned on exit.
C
C NW - The approximate number of neighbouring nodes to
C affect each weight function.
C
C NQ - The approximate number of neighbouring nodes to
C affect each nodal function. If either of
C NW or NQ is .le. zero on entry, then a default
C value will be used.
C
C Output Parameters:
C
C FNODES - Real array of dimension at least (5*M).
C This array is used to store the coefficients for the
C nodal functions.
C
C WRK - Real work array of dimension at least (6*M),
C used in E01SEZ.
C
C On exit, if IFAIL = 0, normal return.
C = 1, input error. M.lt.3
C = 2, input error. RNQ.lt.RNW
C = 3, input error. NQ.lt.NW
C = 4, input error. The (X(I),Y(I)) points are
C not unique for I = 1,2,...,M.
C
SUBROUTINE E01SFF(M,X,Y,F,RNW,FNODES,PX,PY,PF,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C
C This function serves as the evaluation module in a quadratic
C Shepard interpolation process. This module should be called
C after a call to subroutine E01SEF.
C
C Routine created - December 1986
C Author - Richard Franke
C Naval Postgraduate School Monterey,
C California 93940
C Adapted for Nag by H.Scullion (Leic Univ.)
C and I. Gladwell (Nag Ltd.)
C
C Input Parameters:
C
C M - The number of data points.
C Unchanged on exit.
C
C X,Y,F - The data points, (X(I),Y(I),F(I),I=1,M).
C Unchanged on exit.
C
C RNW - The radius for the weight functions. As
C defined in E01SEF.
C Unchanged on exit.
C
C FNODES - Real array of dimension at least (5*M).
C This array is used to store the coefficients for
C the nodal functions.
C Unchanged on exit.
C
C PX,PY - The point (PX,PY) where the function is to be
C evaluated.
C Unchanged on exit.
C
C Output Parameter:
C
C PF - The value of the interpolant at point (PX,PY).
C
C On exit, if IFAIL = 0, normal return.
C = 1, M is .lt. 3.
C = 2, evaluation point is outside support region.
C
SUBROUTINE E02ACF(X,Y,N,AA,M1,REF)
C MARK 1 RELEASE. NAG COPYRIGHT 1971
C MARK 4.5 REVISED
C MARK 5C REVISED
C MARK 9B REVISED. IER-361 (JAN 1982)
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 14C REVISED. IER-877 (NOV 1990).
C CALCULATES A MINIMAX POLYNOMIAL FIT TO A SET OF DATA POINTS
C AS A
C SERIES OF CHEBYSHEV POLYNOMIALS.
SUBROUTINE E02ADF(M,KPLUS1,NROWS,X,Y,W,WORK1,WORK2,A,S,IFAIL)
C
C NAG LIBRARY SUBROUTINE E02ADF
C
C E02ADF COMPUTES WEIGHTED LEAST-SQUARES POLYNOMIAL
C APPROXIMATIONS TO AN ARBITRARY SET OF DATA POINTS.
C
C FORSYTHE-CLENSHAW METHOD WITH MODIFICATIONS DUE TO
C REINSCH AND GENTLEMAN.
C
C USES NAG LIBRARY ROUTINE P01AAF.
C USES BASIC EXTERNAL FUNCTION SQRT.
C
C STARTED - 1973.
C COMPLETED - 1976.
C AUTHOR - MGC AND JGH.
C
C WORK1 AND WORK2 ARE WORKSPACE AREAS.
C WORK1(1, R) CONTAINS THE VALUE OF THE R TH WEIGHTED
C RESIDUAL FOR THE CURRENT DEGREE I.
C WORK1(2, R) CONTAINS THE VALUE OF X(R) TRANSFORMED
C TO THE RANGE -1 TO +1.
C WORK1(3, R) CONTAINS THE WEIGHTED VALUE OF THE CURRENT
C ORTHOGONAL POLYNOMIAL (OF DEGREE I) AT THE R TH
C DATA POINT.
C WORK2(1, J) CONTAINS THE COEFFICIENT OF THE CHEBYSHEV
C POLYNOMIAL OF DEGREE J - 1 IN THE CHEBYSHEV-SERIES
C REPRESENTATION OF THE CURRENT ORTHOGONAL POLYNOMIAL
C (OF DEGREE I).
C WORK2(2, J) CONTAINS THE COEFFICIENT OF THE CHEBYSHEV
C POLYNOMIAL OF DEGREE J - 1 IN THE CHEBYSHEV-SERIES
C REPRESENTATION OF THE PREVIOUS ORTHOGONAL POLYNOMIAL
C (OF DEGREE I - 1).
C
C NAG COPYRIGHT 1975
C MARK 5 RELEASE
C MARK 6 REVISED IER-84
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C CHECK THAT THE VALUES OF M AND KPLUS1 ARE REASONABLE
C
SUBROUTINE E02AEF(NPLUS1,A,XCAP,P,IFAIL)
C NAG LIBRARY SUBROUTINE E02AEF
C
C E02AEF EVALUATES A POLYNOMIAL FROM ITS CHEBYSHEV-
C SERIES REPRESENTATION.
C
C CLENSHAW METHOD WITH MODIFICATIONS DUE TO REINSCH
C AND GENTLEMAN.
C
C USES NAG LIBRARY ROUTINES P01ABF AND X02AJF.
C USES INTRINSIC FUNCTION ABS.
C
C STARTED - 1973.
C COMPLETED - 1976.
C AUTHOR - MGC AND JGH.
C
C NAG COPYRIGHT 1975
C MARK 5 RELEASE
C MARK 7 REVISED IER-140 (DEC 1978)
C MARK 9 REVISED. IER-352 (SEP 1981)
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C
SUBROUTINE E02AFF(NPLUS1,F,A,IFAIL)
C NAG LIBRARY SUBROUTINE E02AFF
C
C E02AFF COMPUTES THE COEFFICIENTS OF A POLYNOMIAL,
C IN ITS CHEBYSHEV-SERIES FORM, WHICH INTERPOLATES
C (PASSES EXACTLY THROUGH) DATA AT A SPECIAL SET OF
C POINTS. LEAST-SQUARES POLYNOMIAL APPROXIMATIONS
C CAN ALSO BE OBTAINED.
C
C CLENSHAW METHOD WITH MODIFICATIONS DUE TO REINSCH
C AND GENTLEMAN.
C
C USES NAG LIBRARY ROUTINES P01AAF AND X01AAF.
C USES BASIC EXTERNAL FUNCTION SIN.
C
C STARTED - 1973.
C COMPLETED - 1976.
C AUTHOR - MGC AND JGH.
C
C NAG COPYRIGHT 1975
C MARK 5 RELEASE
C MARK 5B REVISED IER-73
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C
SUBROUTINE E02AGF(M,KPLUS1,NROWS,XMIN,XMAX,X,Y,W,MF,XF,YF,LYF,IP,
* A,S,NP1,WRK,LWRK,IWRK,LIWRK,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 9 REVISED. IER-314 (SEP 1981).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 14C REVISED. IER-878 (NOV 1990).
C
C **************************************************************
C
C NPL ALGORITHMS LIBRARY ROUTINE CONFIT
C
C CREATED 20/8/1979 UPDATED 16/5/80 RELEASE NO. 00/05
C
C AUTHOR... GERALD T ANTHONY.
C NATIONAL PHYSICAL LABORATORY,
C TEDDINGTON, MIDDLESEX TW11 0LW,
C ENGLAND.
C
C **************************************************************
C
C E02AGF CALLS E01AEW TO DETERMINE A POLYNOMIAL MU(X) WHICH
C INTERPOLATES THE GIVEN CONSTRAINTS AND A POLYNOMIAL NU(X)
C WHICH HAS VALUE ZERO WHERE A CONSTRAINED VALUE IS SPECIFIED.
C IT THEN CALLS E02ADZ TO FIT Y-MU(X) AS A POLYNOMIAL IN X
C WITH FACTOR NU(X). FINALLY THE COEFFICIENTS OF MU ARE ADDED
C TO THESE FITS TO GIVE THE COEFFICIENTS OF THE CONSTRAINED
C FITS TO Y. ALL POLYNOMIALS ARE EXPRESSED IN CHEBYSHEV
C SERIES FORM
C
C INPUT PARAMETERS
C M THE NUMBER OF DATA POINTS TO BE FITTED
C KPLUS1 FITS WITH UP TO KPLUS1 COEFFICIENTS ARE REQUIRED
C NROWS FIRST DIMENSION OF ARRAY A WHERE COEFFICIENTS ARE
C TO BE STORED
C XMIN, END POINTS OF THE RANGE OF THE
C XMAX INDEPENDENT VARIABLE
C X, Y, W ARRAYS OF DATA VALUES OF THE INDEPENDENT VARIABLE,
C DEPENDENT VARIABLE AND WEIGHT, RESPECTIVELY
C MF NUMBER OF X VALUES AT WHICH A CONSTRAINT
C IS SPECIFIED
C XF ARRAY OF VALUES OF THE INDEPENDENT
C VARIABLE AT WHICH CONSTRAINTS ARE
C SPECIFIED
C YF ARRAY OF SPECIFIED VALUES AND DERIVATIVES OF THE
C DEPENDENT VARIABLE IN THE ORDER
C Y1, Y1 DERIV, Y1 2ND DERIV,...., Y2,....
C LYF DIMENSION OF ARRAY YF
C IP INTEGER ARRAY OF DEGREES OF DERIVATIVES
C SPECIFIED AT EACH POINT XF
C
C OUTPUT PARAMETERS
C A ON EXIT, 2 PARAMETER ARRAY CONTAINING THE
C COEFFICIENTS OF THE CHEBYSHEV SERIES
C REPRESENTATION OF THE FITS, A(I+1, J+1)
C CONTAINS THE COEFFICIENT OF TJ IN THE FIT
C OF DEGREE I, I = N,N+1,...,K, J =
C 0,1,...,I WHERE N = NP1 - 1
C S ON EXIT, ARRAY CONTAINING THE R.M.S. RESIDUAL FOR
C EACH DEGREE OF FIT FROM N TO K
C NP1 ON EXIT, CONTAINS N + 1, WHERE N IS THE
C TOTAL NUMBER OF INTERPOLATION CONDITIONS
C
C IFAIL FAILURE INDICATOR
C 0 - SUCCESSFUL TERMINATION
C 1 - AT LEAST ONE OF THE FOLLOWING CONDITIONS
C HAS BEEN VIOLATED
C LYF AT LEAST N
C LWRK AT LEAST 2*N + 2 + THE LARGER OF
C 4*M + 3*KPLUS1 AND 8*NP1 +
C 5*IMAX + MF - 3 WHERE IMAX =
C 1 + MAX(IP(I))
C LIWRK AT LEAST 2*MF + 2
C KPLUS1 AT LEAST NP1
C M AT LEAST 1
C NROWS AT LEAST KPLUS1
C MF AT LEAST 1
C 2 - FOR SOME I, IP(I) IS LESS THAN 0
C 3 - XMIN IS NOT STRICTLY LESS THAN XMAX
C OR FOR SOME I, XF(I) IS NOT IN RANGE
C XMIN TO XMAX OR THE XF(I) ARE NOT
C DISTINCT
C 4 - FOR SOME I, X(I) IS NOT IN RANGE XMIN TO XMAX
C 5 - THE X(I) ARE NOT NON-DECREASING
C 6 - THE NUMBER OF DISTINCT VALUES OF X(I) WITH
C NON-ZERO WEIGHT IS LESS THAN KPLUS1 - NP1
C 7 - E01AEW HAS FAILED TO CONVERGE, IE
C THE CONSTRAINT CANNOT BE SATISFIED
C WITH SUFFICIENT ACCURACY
C
C WORKSPACE PARAMETERS
C WRK REAL WORKSPACE ARRAY
C LWRK DIMENSION OF WRK. LWRK MUST BE AT LEAST
C 2*N + 2 + THE LARGER OF
C 4*M + 3*KPLUS1 AND 8*NP1 + 5*IMAX + MF - 3
C WHERE IMAX = 1 + MAX(IP(I))
C IWRK INTEGER WORKSPACE ARRAY
C LIWRK DIMENSION OF IWRK. LIWRK MUST BE AT LEAST
C 2*MF + 2
C
SUBROUTINE E02AHF(NP1,XMIN,XMAX,A,IA1,LA,PATM1,ADIF,IADIF1,LADIF,
* IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C
C NPL DATA FITTING LIBRARY ROUTINE CHBDIF
C
C CREATED 1/5/79 UPDATED 23/1/80 RELEASE NO. 00/03
C
C AUTHORS.. GERALD T ANTHONY, MAURICE G COX, J GEOFFREY HAYES.
C NATIONAL PHYSICAL LABORATORY
C TEDDINGTON, MIDDLESEX, ENGLAND.
C
C * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C
C INPUT PARAMETERS
C NP1 = N+1 WHERE N IS DEGREE OF GIVEN POLYNOMIAL
C XMIN LOWER LIMIT OF RANGE OF X
C XMAX UPPER LIMIT OF RANGE OF X
C A COEFFICIENTS A0, A1,...AN OF THE GIVEN POLYNOMIAL
C IA1 ARE STORED IN ARRAY A IN POSITIONS 1, 1+IA1,...
C 1+N*IA1, RESPECTIVELY
C LA THE DECLARED DIMENSION OF ARRAY A
C
C OUTPUT PARAMETERS
C PATM1 THE VALUE OF THE GIVEN POLYNOMIAL AT XMIN
C ADIF THE COEFFICIENTS OF THE DERIVATIVE POLYNOMIAL
C IADIF1 ARE RETURNED IN ARRAY ADIF IN POSITIONS
C 1, 1+IADIF1,...1+(N-1)*IADIF1
C LADIF THE DECLARED DIMENSION OF ARRAY ADIF
C IFAIL ERROR INDICATOR
C
C DIFFERENTIATE THE SERIES WITH COEFFICIENTS A OF DEGREE N
C (I.E. NP1 COEFFICIENTS) TO OBTAIN THE SERIES WITH COEFFICIENTS
C ADIF OF DEGREE N-1. ALSO SET NEXT HIGHER COEFFICIENT TO ZERO.
C
SUBROUTINE E02AJF(NP1,XMIN,XMAX,A,IA1,LA,QATM1,AIN,IAINT1,LAINT,
* IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 9 REVISED. IER-315 (SEP 1981).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C
C NPL DATA FITTING LIBRARY ROUTINE CHBINT
C
C CREATED 1/5/79 UPDATED 23/1/80 RELEASE NO. 00/03
C
C AUTHORS.. GERALD T ANTHONY, MAURICE G COX, J GEOFFREY HAYES.
C NATIONAL PHYSICAL LABORATORY
C TEDDINGTON, MIDDLESEX, ENGLAND.
C
C * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C
C INPUT PARAMETERS
C NP1 = N+1 WHERE N IS DEGREE OF GIVEN POLYNOMIAL
C XMIN LOWER LIMIT OF RANGE OF X
C XMAX UPPER LIMIT OF RANGE OF X
C A COEFFICIENTS A0, A1,...AN OF THE GIVEN POLYNOMIAL
C IA1 ARE STORED IN ARRAY A IN POSITIONS 1, 1+IA1,...
C 1+N*IA1, RESPECTIVELY
C LA THE DECLARED DIMENSION OF ARRAY A
C QATM1 THE VALUE OF THE INTEGRATED POLYNOMIAL AT XMIN
C
C OUTPUT PARAMETERS
C AIN THE COEFFICIENTS OF THE INTEGRATED POLYNOMIAL
C IAINT1 ARE RETURNED IN ARRAY AIN IN POSITIONS
C 1, 1+IAINT1,...1+NP1*IAINT1
C LAINT THE DECLARED DIMENSION OF ARRAY AIN
C IFAIL ERROR INDICATOR
C
C INTEGRATE THE SERIES WITH COEFFICIENTS A OF DEGREE N
C (I.E. NP1 COEFFICIENTS) TO OBTAIN THE SERIES WITH COEFFICIENTS
C AIN OF DEGREE N + 1. THE SUM OF THE INTEGRATED SERIES IS
C QATM1 AT THE LEFT HAND END OF THE INTERVAL OF DEFINITION, XMIN
C
SUBROUTINE E02AKF(NP1,XMIN,XMAX,A,IA1,LA,X,RESULT,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C * * * * * * * * * * * * * * * * * * * * * * * * * * *
C
C NPL DATA FITTING LIBRARY ROUTINE TVAL1C
C
C CREATED 19/3/79 UPDATED 6/7/79 RELEASE NO. 00/04.
C
C AUTHORS.. GERALD T ANTHONY, MAURICE G COX, BETTY CURTIS
C AND J GEOFFREY HAYES.
C NATIONAL PHYSICAL LABORATORY
C TEDDINGTON, MIDDLESEX, ENGLAND.
C
C * * * * * * * * * * * * * * * * * * * * * * * * * * *
C
C INPUT PARAMETERS
C NP1 NP1 = N + 1. N IS THE DEGREE OF THE
C CHEBYSHEV SERIES
C XMIN MINIMUM VALUE OF X
C XMAX MAXIMUM VALUE OF X
C A THE ARRAY WHERE THE COEFFICIENTS ARE STORED
C IA1 THE ADDRESS INCREMENT OF A
C LA DIMENSION OF A
C X UNNORMALIZED ARGUMENT IN THE RANGE (XMIN, XMAX)
C
C OUTPUT PARAMETERS
C RESULT VALUE OF THE SUMMATION
C IFAIL ERROR INDICATOR
C
C NP1 CHEBYSHEV COEFFICIENTS A0, A1, ..., AN, ARE
C STORED IN THE ARRAY A IN POSITIONS 1, 1+IA1, 1+2*IA1, ...,
C 1+N*IA1, WHERE N = NP1 - 1.
C IA1 MUST NOT BE NEGATIVE.
C LA MUST BE AT LEAST EQUAL TO 1 + N*IA1.
C THE VALUE OF THE POLYNOMIAL OF DEGREE N
C A0T0(XCAP)/2 + A1T1(XCAP) + A2T2(XCAP) + + ... + ANTN(XCAP),
C IS CALCULATED FOR THE ARGUMENT XCAP, WHERE XCAP IS
C THE NORMALIZED VALUE OF X IN THE RANGE (XMIN, XMAX),
C STORING IT IN RESULT.
C UNLESS THE ROUTINE DETECTS AN ERROR, IFAIL CONTAINS
C ZERO ON EXIT.
C IFAIL = 1 INDICATES AT LEAST ONE OF THE RESTRICTIONS ON
C INPUT PARAMETERS IS VIOLATED - IE
C NP1 .GT. 0
C IA1 .GE. 0
C LA .GE. 1 + N * IA1
C XMIN .LT. XMAX
C IFAIL = 2 INDICATES THAT
C X DOES NOT SATISFY THE RESTRICTION XMIN .LE. X .LE. XMAX.
C THE RECURRENCE RELATION BY CLENSHAW, MODIFIED BY REINSCH
C AND GENTLEMAN, IS USED.
C
SUBROUTINE E02BAF(M,NCAP7,X,Y,W,K,WORK1,WORK2,C,SS,IFAIL)
C NAG COPYRIGHT 1975
C MARK 5 RELEASE
C MARK 6 REVISED IER-84
C MARK 8 RE-ISSUE. IER-224 (APR 1980).
C MARK 9A REVISED. IER-356 (NOV 1981)
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C NAG LIBRARY SUBROUTINE E02BAF
C
C E02BAF COMPUTES A WEIGHTED LEAST-SQUARES APPROXIMATION
C TO AN ARBITRARY SET OF DATA POINTS BY A CUBIC SPLINE
C WITH KNOTS PRESCRIBED BY THE USER. CUBIC SPLINE
C INTERPOLATION CAN ALSO BE CARRIED OUT.
C
C COX-DE BOOR METHOD FOR EVALUATING B-SPLINES WITH
C ADAPTATION OF GENTLEMAN*S PLANE ROTATION SCHEME FOR
C SOLVING OVER-DETERMINED LINEAR SYSTEMS.
C
C USES NAG LIBRARY ROUTINE P01AAF.
C
C STARTED - 1973.
C COMPLETED - 1976.
C AUTHOR - MGC AND JGH.
C
C REDESIGNED TO USE CLASSICAL GIVENS ROTATIONS IN
C ORDER TO AVOID THE OCCASIONAL UNDERFLOW (AND HENCE
C OVERFLOW) PROBLEMS EXPERIENCED BY GENTLEMAN*S 3-
C MULTIPLICATION PLANE ROTATION SCHEME
C
C WORK1 AND WORK2 ARE WORKSPACE AREAS.
C WORK1(R) CONTAINS THE VALUE OF THE R TH DISTINCT DATA
C ABSCISSA AND, SUBSEQUENTLY, FOR R = 1, 2, 3, 4, THE
C VALUES OF THE NON-ZERO B-SPLINES FOR EACH SUCCESSIVE
C ABSCISSA VALUE.
C WORK2(L, J) CONTAINS, FOR L = 1, 2, 3, 4, THE VALUE OF
C THE J TH ELEMENT IN THE L TH DIAGONAL OF THE
C UPPER TRIANGULAR MATRIX OF BANDWIDTH 4 IN THE
C TRIANGULAR SYSTEM DEFINING THE B-SPLINE COEFFICIENTS.
C
SUBROUTINE E02BBF(NCAP7,K,C,X,S,IFAIL)
C NAG LIBRARY SUBROUTINE E02BBF
C
C E02BBF EVALUATES A CUBIC SPLINE FROM ITS
C B-SPLINE REPRESENTATION.
C
C DE BOOR*S METHOD OF CONVEX COMBINATIONS.
C
C USES NAG LIBRARY ROUTINE P01AAF.
C
C STARTED - 1973.
C COMPLETED - 1976.
C AUTHOR - MGC AND JGH.
C
C NAG COPYRIGHT 1975
C MARK 5 RELEASE
C MARK 7 REVISED IER-141 (DEC 1978)
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
SUBROUTINE E02BCF(NCAP7,K,C,X,LEFT,S,IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C **************************************************
C * *
C * NAG LIBRARY SUBROUTINE E02BCF *
C * *
C * EVALUATION OF CUBIC SPLINE AND ITS *
C * DERIVATIVES FROM ITS B-SPLINE REPRESENTATION *
C * *
C * ROUTINE CREATED ... 17 NOV 1977 *
C * LATEST UPDATE .... 24 APR 1978 *
C * RELEASE NUMBER ... 01 *
C * AUTHORS ... MAURICE G. COX AND *
C * J. GEOFFREY HAYES, N.P.L. *
C * *
C **************************************************
C
SUBROUTINE E02BDF(NCAP7,K,C,DEFINT,IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C **************************************************
C * *
C * NAG LIBRARY SUBROUTINE E02BDF *
C * *
C * DEFINITE INTEGRAL OF CUBIC SPLINE FROM ITS *
C * B-SPLINE REPRESENTATION *
C * *
C * ROUTINE CREATED ... 17 NOV 1977 *
C * LATEST UPDATE .... 24 APR 1978 *
C * RELEASE NUMBER ... 01 *
C * AUTHORS ... MAURICE G. COX AND *
C * J. GEOFFREY HAYES, N.P.L. *
C * *
C **************************************************
C
SUBROUTINE E02BEF(START,M,X,Y,W,S,NEST,N,K,C,FP,WRK,LWRK,IWRK,
* IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE E02CAF(M,N,K,L,X,Y,F,W,NX,A,NA,XMIN,XMAX,NUX,INUXP1,
* NUY,INUYP1,WORK,NWORK,IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C E02CAF IS A CALLING ROUTINE WHICH SETS UP WORK SPACE ARRAYS
C FOR THE SUBROUTINE E02CAZ AND THEN CALLS IT TO OBTAIN AN
C APPROXIMATION TO THE LEAST SQUARES POLYNOMIAL SURFACE FIT
C TO DATA ARBITRARILY DISTRIBUTED ON LINES PARALLEL TO ONE
C INDEPENDENT COORDINATE AXIS.
C
C STARTED - 1978.
C COMPLETED - 1978.
C AUTHOR - GTA.
C
C
C FIND MAXIMUM OF N AND THE ELEMENTS OF M, AND LARGER OF K AND
C L, AND THE SUM OF THE ELEMENTS OF M
C
SUBROUTINE E02CBF(MFIRST,MLAST,K,L,X,XMIN,XMAX,Y,YMIN,YMAX,FF,A,
* NA,WORK,NWORK,IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C THIS SUBROUTINE EVALUATES A POLYNOMIAL OF DEGREE K AND L
C RESPECTIVELY IN THE INDEPENDENT VARIABLES X AND Y. THE
C POLYNOMIAL IS GIVEN IN DOUBLE CHEBYSHEV SERIES FORM
C A(I,J) * TI(XCAP) * TJ(YCAP),
C SUMMED OVER I = 0,1,...K AND J = 0,1,...L WITH THE CONVENTION
C THAT TERMS WITH EITHER I OR J ZERO ARE HALVED AND THE TERM
C WITH BOTH I AND J ZERO IS MULTIPLIED BY 0.25. HERE TI(XCAP)
C IS THE CHEBYSHEV POLYNOMIAL OF THE FIRST KIND OF DEGREE I
C WITH ARGUMENT XCAP=((X - XMIN) - (XMAX - X))/(XMAX - XMIN).
C TJ(YCAP) IS DEFINED SIMILARLY. THE COEFFICIENT A(I,J)
C SHOULD BE STORED IN ELEMENT (L + 1)*I + J + 1 OF THE SINGLE
C DIMENSION ARRAY A. THE EVALUATION IS PERFORMED FOR A SINGLE
C GIVEN VALUE OF Y WITH EACH X VALUE GIVEN IN X(R), FOR R =
C MFIRST, MFIRST+1,....,MLAST.
C
C STARTED - 1978.
C COMPLETED - 1978.
C AUTHOR - GTA.
C
SUBROUTINE E02DAF(M,PX,PY,X,Y,F,W,LAMDA,MU,POINT,NPOINT,DL,C,NC,
* WS,NWS,EPS,SIGMA,RANK,IFAIL)
C MARK 6 RELEASE. NAG COPYRIGHT 1977
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C THIS IS A CALLING ROUTINE WHICH SETS UP WORK SPACE ARRAYS
C FOR THE SUBROUTINE E02DAZ, AND THEN CALLS IT.
C
SUBROUTINE E02DCF(START,MX,X,MY,Y,F,S,NXEST,NYEST,NX,LAMDA,NY,MU,
* C,FP,WRK,LWRK,IWRK,LIWRK,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE E02DDF(START,M,X,Y,F,W,S,NXEST,NYEST,NX,LAMDA,NY,MU,C,
* FP,RANK,WRK,LWRK,IWRK,LIWRK,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE E02DEF(M,PX,PY,X,Y,LAMDA,MU,C,FF,WRK,IWRK,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C Derived from DASL routine B2VRE.
C
C E02DEF. An algorithm for evaluating a bicubic polynomial
C spline S(X,Y) from its B-spline representation at the M
C points (X(I),Y(I)), I = 1, 2, ..., M.
C
C Input Parameters:
C M The number of evaluation points.
C PX NXKNTS + 8, where NXKNTS is number
C of interior X-knots.
C PY NYKNTS + 8, where NYKNTS is number
C of interior Y-knots.
C X X-values.
C Y Y-values.
C LAMDA The X-knots.
C MU The Y-knots.
C C B-spline coefficients of S. First
C subscript relates to X.
C
C Output parameter:
C FF Values of spline.
C On exit, FF(I) contains the value of
C the spline evaluated at point (X(I),Y(I)),
C for I = 1,..,M.
C
C Workspace parameters:
C WRK Real workspace of dimension at least (PY-4).
C IWRK Integer workspace of dimension at least (PY-4).
C
C Failure indicator parameter:
C IFAIL Failure indicator:
C 1 - PX .LT. 8, or
C PY .LT. 8, or
C M .LT. 1.
C 2 - E02DFW failure for X or Y.
C 3 - At least one point (X(K),Y(K)) lies
C outside the rectangle defined by
C LAMDA(4), LAMDA(PX-3), MU(4) and MU(PY-3).
C
SUBROUTINE E02DFF(MX,MY,PX,PY,X,Y,LAMDA,MU,C,FF,WRK,LWRK,IWRK,
* LIWRK,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C Derived from DASL routine B2VRE.
C
C E02DFF. An algorithm for evaluating a bivariate
C polynomial spline S(X,Y) from its B-spline
C representation at all vertices of a rectangular mesh.
C
C The routine is designed to be fast, but this is at the
C expense of some workspace. If speed is less important
C than space, E02DFF may be called several times with
C one meshline of values, for example, at each call.
C
C Input Parameters:
C PX NXKNTS + 8, where NXKNTS is number
C of interior X-knots.
C LAMDA The X-knots.
C PY NYKNTS + 8, where NYKNTS is number
C of interior Y-knots.
C MU The Y-knots.
C C B-spline coefficients of S. First
C subscript relates to X.
C MX The number of X-meshlines.
C X X-meshline values.
C MY The number of Y-meshlines.
C Y The Y-meshline values.
C
C Output parameters:
C FF Values of spline.
C On exit, FF((I-1)*MY+J) contains the value of
C the spline evaluated at point (X(I),Y(J)),
C for I = 1,..,MX, J = 1,..,MY.
C
C Workspace (and associated dimension) parameters:
C WRK Real workspace.
C LWRK Dimension of WRK. .ge. NWRK =
C min(NWRK1, NWRK2), where
C NWRK1 = MX*4 + PX,
C NWRK2 = MY*4 + PY.
C IWRK Integer workspace.
C LIWRK Dimension of IWRK. .ge. NIWRK =
C MX + PX - 4 (if NWRK1 .le. NWRK2),
C MY + PY - 4 (otherwise).
C
C Failure indicator parameter:
C IFAIL Failure indicator:
C 1 - PX .LT. 8, or
C PY .LT. 8, or
C MX .LT. 0, or
C MY .LT. 0, or
C LWRK .LT. NWRK or
C LIWRK .LT. NIWRK
C 2 - E02DFW failure for X or Y.
C 3 - LAMDA(4) .le. X(1) .lt. X(2) .lt. ...
C .lt. X(MX) .le. LAMDA(PX-3) or
C MU(4) .le. Y(1) .lt. Y(2) .lt. ...
C .lt. Y(MY) .le. MU(PY-3)
C violated.
C
SUBROUTINE E02GAF(M,A,LA,B,NPLUS2,TOL,X,RESID,IRANK,ITER,IWORK,
* IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 10A REVISED. IER-396 (OCT 1982).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C
C THIS SUBROUTINE USES A MODIFICATION OF THE SIMPLEX METHOD
C OF LINEAR PROGRAMMING TO CALCULATE AN L1 SOLUTION TO AN
C OVER-DETERMINED SYSTEM OF LINEAR EQUATIONS.
C
SUBROUTINE E02GBF(M,N,MPL1,E,IER,F,X,MXS1,MONIT,IPRINT,K,EL1N,
* INDX,W,IW,IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 8A REVISED. IER-256 (AUG 1980).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C MARK 14 REVISED. IER-812 (DEC 1989).
C MONIT
SUBROUTINE E02GCF(M,N,MDIM,NDIM,A,B,TOL1,RELER,X,RESMAX,IRANK,
* ITER,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1980.
C MARK 9 REVISED. IER-316 (SEP 1981).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C
C THIS SUBROUTINE USES A MODIFICATION OF THE SIMPLEX METHOD
C OF LINEAR PROGRAMMING TO CALCULATE A CHEBYSHEV SOLUTION TO
C AN OVER-DETERMINED SYSTEM OF LINEAR EQUATIONS.
C
C DERIVED FROM ACM ALGORITHM 495 BY I. BARRODALE AND C. PHILLIPS
C
SUBROUTINE E02RAF(IA,IB,C,IC,A,B,W,JW,IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C GIVEN A POWER SERIES E02RAF CALCULATES THE COEFFICIENTS OF
C THE (L/M) PADE APPROXIMANT.
C
C ARGUMENT LIST.
C --------------
C
C IA (INTEGER) ON ENTRY IA=L+1. THIS PARAMETER IMPLICITLY
C SPECIFIES THE ORDER OF THE NUMERATOR OF THE
C APPROXIMANT.
C UNCHANGED ON EXIT.
C IB (INTEGER) ON ENTRY IB=M+1. THIS PARAMETER IMPLICITLY
C SPECIFIES THE ORDER OF THE DENOMINATOR OF
C THE APPROXIMANT. UNCHANGED ON EXIT.
C C (REAL ARRAY) ON ENTRY CONTAINS THE POWER SERIES
C COEFFICIENTS. UNCHANGED ON EXIT.
C IC (INTEGER) LENGTH OF ARRAY C. AT LEAST L+M+1
C A (REAL ARRAY) ON EXIT CONTAINS THE NUMERATOR COEFFICIENTS
C A(1)= COEFF OF X**0 ETC.
C B (REAL ARRAY) ON EXIT CONTAINS THE DENOMINATOR
C COEFFICIENTS. B(1)=COEFF OF X**0 ETC.
C W (REAL ARRAY) ARRAY USED AS WORKSPACE.
C JW (INTEGER) DIMENSION OF W AT LEAST IB*(2*IB+3).
C IFAIL (INTEGER) USED FOR ERROR EXITS.
C
SUBROUTINE E02RBF(A,IA,B,IB,X,ANS,IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C
C E02RBF EVALUATES AN (L/M) PADE APPROXIMANT, WHOSE
C NUMERATOR AND DENOMINATOR COEFFICIENTS ARE STORED IN
C ARRAYS A AND B RESPECTIVELY, AT A GIVEN POINT X.
C
C ARGUMENT LIST
C -------------
C
C A,B (REAL ARRAYS) ON ENTRY CONTAINS THE NUMERATOR
C (DENOMINATOR) COEFFICIENTS OF THE PADE
C APPROXIMANT. UNCHANGED ON EXIT.
C IA,IB (INTEGERS) ON ENTRY DEFINE IMPLICITLY THE ORDERS
C OF THE NUMERATOR IA=L+1, AND DENOMINATOR
C IB=M+1. UNCHANGED ON EXIT.
C X (REAL) ON ENTRY DEFINES THE POINT OF EVALUATION
C OF THE PADE APPROXIMANT. UNCHANGED ON
C EXIT.
C ANS (REAL) ON EXIT CONTAINS THE VALUE OF THE PADE
C APPROXIMANT AT X.
C IFAIL (INTEGER) USED FOR ERROR EXITS.
C
SUBROUTINE E02ZAF(PX,PY,LAMDA,MU,M,X,Y,POINT,NPOINT,ADRES,NADRES,
* IFAIL)
C MARK 6 RELEASE. NAG COPYRIGHT 1977
C MARK 10 REVISED. IER-385 (JUN 1982).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C THE REAL ARRAYS LAMDA AND MU OF DIMENSION AT LEAST
C PX-4 AND PY-4 RESPECTIVELY EACH CONTAIN A SEQUENCE
C OF NON-DECREASING VALUES. THE POINTS WITH COORDINATES
C GIVEN IN X(1) TO X(M) AND Y(1) TO Y(M) ARE SORTED INTO
C THE RECTANGULAR PANELS DETERMINED BY THE LAMDA AND MU
C VALUES. ARRAYS X AND Y ARE UNCHANGED, THE ORDERING
C BEING INDICATED BY NADRES LINKED LISTS STORED IN ARRAY
C POINT(1) TO POINT(NPOINT), WHERE
C NADRES = (PX-7)*(PY-7) IS THE NUMBER OF PANELS, AND
C NPOINT .GE. NADRES + M
C
SUBROUTINE E04ABF(FUN,EPS,T,A,B,MAXCAL,X,F,IFAIL)
C
C MARK 6 RELEASE NAG COPYRIGHT 1977
C MARK 8 REVISED. IER-231 (MAR 1980).
C MARK 8D REVISED. IER-272 (DEC 1980).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C
C **************************************************************
C
C E04ABF ATTEMPTS TO FIND A MINIMUM IN AN INTERVAL A .LE. X .LE.
C B OF A FUNCTION F(X) OF THE SCALAR X, USING FUNCTION VALUES
C ONLY.
C
C IT IS BASED ON THE SUBROUTINE UNIFUN IN THE NPL ALGORITHMS
C LIBRARY (REF. NO. E4/13/F). THE FUNCTION F(X) IS DEFINED BY
C THE USER-SUPPLIED SUBROUTINE FUN. T AND EPS DEFINE A TOLERANCE
C TOL = EPS * ABS(X) + T, AND FUN IS NEVER EVALUATED AT TWO
C POINTS CLOSER THAN TOL. IF FUN IS DELTA-UNIMODAL, FOR SOME
C DELTA LESS THAN TOL, THEN X APPROXIMATES THE GLOBAL MINIMUM OF
C FUN WITH AN ERROR LESS THAN 3*TOL. IF FUN IS NOT DELTA-
C UNIMODAL ON (A, B), THEN X MAY APPROXIMATE A LOCAL, BUT NON
C GLOBAL, MINIMUM. EPS SHOULD BE NO SMALLER THAN 2*EPSMCH, AND
C PREFERABLY NOT MUCH LESS THAN SQRT(EPSMCH), WHERE EPSMCH IS
C THE RELATIVE MACHINE PRECISION. T SHOULD BE POSITIVE. NOTE
C THAT, FOR CONSISTENCY WITH OTHER E04 DOCUMENTATION, THE NAME
C FUNCT IS USED INSTEAD OF FUN IN THE WRITE-UP.
C
C PHILIP E. GILL, WALTER MURRAY, SUSAN M. PICKEN, HAZEL M.
C BARBER AND MARGARET H. WRIGHT, D.N.A.C., NATIONAL PHYSICAL
C LABORATORY, ENGLAND
C
C **************************************************************
C
C FUN
SUBROUTINE E04BBF(FUN,EPS,T,A,B,MAXCAL,X,F,G,IFAIL)
C
C MARK 6 RELEASE NAG COPYRIGHT 1977
C MARK 8D REVISED. IER-272 (DEC 1980).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C
C **************************************************************
C
C E04BBF ATTEMPTS TO FIND A MINIMUM IN AN INTERVAL A .LE. X .LE.
C B OF A FUNCTION F(X) OF THE SCALAR X, USING FUNCTION AND
C GRADIENT VALUES.
C
C IT IS BASED ON THE SUBROUTINE UNIGRD IN THE NPL ALGORITHMS
C LIBRARY (REF. NO. E4/14/F). THE FUNCTION F(X) IS DEFINED BY
C THE USER-SUPPLIED SUBROUTINE FUN. T AND EPS DEFINE A TOLERANCE
C TOL = EPS * ABS(X) + T, AND FUN IS NEVER EVALUATED AT TWO
C POINTS CLOSER THAN TOL. IF FUN IS DELTA-UNIMODAL, FOR SOME
C DELTA LESS THAN TOL, THEN X APPROXIMATES THE GLOBAL MINIMUM OF
C FUN WITH AN ERROR LESS THAN 3*TOL. IF FUN IS NOT DELTA-
C UNIMODAL ON (A, B), THEN X MAY APPROXIMATE A LOCAL, BUT NON
C GLOBAL, MINIMUM. EPS SHOULD BE NO SMALLER THAN 2*EPSMCH, AND
C PREFERABLY NOT MUCH LESS THAN SQRT(EPSMCH), WHERE EPSMCH IS
C THE RELATIVE MACHINE PRECISION. T SHOULD BE POSITIVE. NOTE
C THAT, FOR CONSISTENCY WITH OTHER E04 DOCUMENTATION, THE NAME
C FUNCT IS USED INSTEAD OF FUN IN THE WRITE-UP.
C
C PHILIP E. GILL, WALTER MURRAY, SUSAN M. PICKEN, HAZEL M.
C BARBER AND MARGARET H. WRIGHT, D.N.A.C., NATIONAL PHYSICAL
C LABORATORY, ENGLAND
C
C **************************************************************
C
C FUN
SUBROUTINE E04CCF(N,X,FMIN,EPS,N1,PDSTAR,PSTAR,PBAR,STEP,Y,P,
* FUNCT,MONIT,MAXIT,IFAIL)
C MARK 1 RELEASE. NAG COPYRIGHT 1971
C MARK 3 REVISED.
C MARK 4.5 REISSUE. LER-F7
C MARK 8 REVISED. IER-220 (MAR 1980)
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C MARK 14 REVISED. IER-716 (DEC 1989).
C MARK 16A REVISED. IER-984 (JUN 1993).
SUBROUTINE E04DGF(N,FUNGRD,ITER,OBJF,OBJGRD,X,IWORK,WORK,IUSER,
* USER,IFAIL)
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
C MARK 16 REVISED. IER-1053 (JUL 1993).
C
C ==================================================================
C E04DGF is a pre-conditioned, limited memory quasi-Newton method,
C based on method PLMA of "Conjugate gradient methods for
C large-scale nonlinear optimization", Technical Report SOL 79-15.
C ==================================================================
C
C -----------
C Parameters
C -----------
C
C N - INTEGER.
C On entry, N must contain the number of variables in the
C problem.
C
C FUNGRD - SUBROUTINE, supplied by the user.
C FUNGRD must be declared as EXTERNAL in the routine that
C calls E04DGF.
C If MODE = 0 then FUNGRD must calculate the objective
C function, if MODE = 2 it must also calculate the gradient
C vector.
C
C It's specification is:
C SUBROUTINE FUNGRD( MODE, N, X, OBJF, G, NSTATE, IUSER, USER )
C INTEGER MODE, N, NSTATE, IUSER(*)
C REAL X(N), OBJF, G(N), USER(*)
C
C MODE - INTEGER.
C MODE indicates which parameter values within FUNGRD need to
C be set.
C If MODE = 0, FUNGRD must only return the value of the
C objective function (OBJF) at X.
C If MODE = 2, FUNGRD must also return the gradient ( G(1)...
C ...G(N) ).
C If MODE is negative on exit from FUNGRD, the execution of
C E04DGF is terminated with IFAIL set to MODE.
C
C N - INTEGER.
C On entry, N specifies the number of variables as input to
C E04DGF.
C N must not be altered by FUNGRD.
C
C X - REAL array of dimension (N).
C On entry, X contains the point at which the objective
C function is to be evaluated.
C X must not be altered by FUNGRD.
C
C OBJF - REAL.
C On exit, OBJF must contain the value of the objective
C function.
C
C G - REAL array of dimension (N).
C On exit, if MODE = 2 G(1) ... G(N) must contain the gradient
C vector of the objective function. The j-th component of G
C must contain the partial derivative of the function with
C respect to the j-th variable.
C
C NSTATE - INTEGER.
C On entry, NSTATE will be set to 1 on the first call of FUNGRD
C by E04DGF, and is set to 0 for all subsequent calls. Thus, if
C the user wishes, NSTATE may be tested within FUNGRD in order
C to perform cerain calculations once only. For example, the
C user may read data or initialise COMMON blocks when NSTATE
C = 1.
C
C IUSER - INTEGER array of dimension as least (1).
C USER - REAL array of dimension at least (1).
C FUNGRD is called from E04DGF with the parameters IUSER and
C USER as supplied to E04DGF. The user is free to use arrays
C IUSER and USER to supply information to FUNGRD as an
C alternative to using COMMON.
C
C ITER - INTEGER.
C On exit, ITER is set to the number of iterations performed
C by E04DGF.
C
C OBJF - REAL.
C On exit, contains the value of the objective function at the
C final iterate.
C
C OBJGRD - REAL array of length at least (N).
C On exit, OBJGRD contains the gradient vector at the final
C iterate.
C
C X - REAL array of length at least (N).
C On entry, X must contain the initial estimate of the
C solution.
C On exit, X contains the final estimate of the solution.
C
C IWORK - INTEGER array of dimension at least (N+1).
C Work array.
C
C WORK - REAL array of dimension at least (13*N).
C Work array.
C
C IFAIL - INTEGER.
C On entry, IFAIL must be set to 0,-1 or 1. The recommended
C value for IFAIL is -1.
C On exit, IFAIL represents a diagnostic indicator. The values
C of IFAIL on exit are as follows:-
C
C IFAIL < 0 the user requested termination by setting MODE
C negative within routine FUNGRD.
C
C IFAIL = 0 indicates successful termination.
C
C IFAIL = 3 maximum number of function evaluations have
C been performed.
C
C IFAIL = 4 the computed upper bound on the step length
C taken during the linesearch was too small.
C A re-run with a larger value assigned to the
C maximum step length ( i.e. BIGDX ) may be
C successful. If BIGDX is already large ( .ge.
C the default value ) then current point cannot
C be improved upon.
C
C IFAIL = 6 the current point cannot be improved upon.
C A sufficient decrease in the function value
C could not be attained during the final
C linesearch. If the subroutine OBJFUN computes
C the function and gradients correctly, then
C this may occur because an overly stringent
C accuracy has been requested i.e. Optimality
C tolerance (FTOL) is too small or if the
C minimum lies close to a step length of zero.
C
C IFAIL = 7 large errors were found in the derivatives of
C the objective function. This value of IFAIL
C will occur if the verification process
C indicated that at least one gradient component
C had no correct figures. The user should refer
C to the printed output to determine which
C elements are suspect to be in error.
C
C IFAIL = 8 indicates that the gradient (g) at the starting
C point is too small. The value g(trans)*g is
C less than machine precision. The problem
C should be rerun at a different starting point.
C
C IFAIL = 9 an input parameter is invalid.
C
C
C -------------------
C Optional Parameters
C -------------------
C
C ITMAX an INTEGER, the maximum number of iterations to be
C performed. If ITMAX .lt. 0, then a default value of
C max(50,5*N) is taken.
C
C BIGDX a REAL variable, the maximum allowable step length. The
C default value is 1.0E+20.
C
C FGUESS a REAL variable, user supplied guess of optimum objective
C function value. FGUESS is F(est) of CG paper (p.10 ).
C
C EPSRF a REAL variable which specifies the relative precision
C of the objective function at the starting point.
C
C ETA is a REAL variable used to specify the accuracy of the
C linesearch.
C
C FTOL is a REAL variable which indicates the correct number of
C figures the user desires in the optimum function value.
C For this purpose, leading zeros after the decimal are
C considered as candidates for correct figures.
C
C MSGCG is an INTEGER variable which indicates the amount of
C output required by the user.
C The values of MSGCG are as follows:-
C MSGCG = 0 No printout.
C MSGCG = 1 Final solution only.
C MSGCG = 5 One line of output for each iteration.
C MSGCG = 10 Final solution and brief line of output for each
C iteration.
C
C LDBGCG is an INTEGER variable controlling debug output. If LDBGCG
C > 0, then debug printout will be produced.
C
C IDBGCG is an INTEGER variable which indicates the number of the
C iteration from which debug print should start.
C
C ******************************************************************
C
C -- Written on 4th June 1986.
C Sven Hammarling and Janet Welding, NAG Central Office.
C
SUBROUTINE E04DJF(IOPTNS,INFORM)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C E04DJF reads the options file from unit IOPTNS and loads the
C options into the relevant elements of IPRMCG and RPRMCG.
C
C If IOPTNS .lt. 0 or IOPTNS .gt. 99 then no file is read,
C otherwise the file associated with unit IOPTNS is read.
C
C Output:
C
C INFORM = 0 if a complete OPTIONS file was found
C (starting with BEGIN and ending with END);
C 1 if IOPTNS .lt. 0 or IOPTNS .gt. 99;
C 2 if BEGIN was found, but end-of-file
C occurred before END was found;
C 3 if end-of-file occurred before BEGIN or
C ENDRUN were found;
C 4 if ENDRUN was found before BEGIN.
C
C -- Written on 5-June-1986.
C Sven Hammarling and Janet Welding, NAG Central Office.
C
SUBROUTINE E04DKF(STRING)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C E04DKF loads the option supplied in STRING into the relevant
C element of IPRMLS or RPRMLS.
C
C -- Written on 5-June-1986.
C Janet Welding and Sven Hammarling, NAG Central Office.
C
SUBROUTINE E04FCF(M,N,LSFN1,LSMON,IPRINT,MAXCAL,ETA,XTOL,STEPMX,X,
* FSUMSQ,FVEC,FJAC,LJ,S,VT,LVT,NITER,NFTOTL,IW,
* LIW,W,LW,IFAIL)
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
C
C **************************************************************
C
C E04FCF IS A COMPREHENSIVE QUASI-NEWTON ALGORITHM FOR FINDING
C AN UNCONSTRAINED MINIMUM OF A SUM OF SQUARES OF M NONLINEAR
C FUNCTIONS IN N VARIABLES (M .GE. N). NO DERIVATIVES ARE
C REQUIRED.
C
C THE ROUTINE IS ESSENTIALLY IDENTICAL TO THE SUBROUTINE LSNDN
C IN THE NPL ALGORITHMS LIBRARY (REF. NO. E4/22/F).
C
C PHILIP E. GILL, ENID M. R. LONG, WALTER MURRAY,
C SUSAN M. PICKEN AND BRIAN T. HINDE
C D.N.A.C., NATIONAL PHYSICAL LABORATORY, ENGLAND
C
C **************************************************************
C
C Modified to output explanatory messages.
C Peter Mayes, NAG Central Office, December 1987.
C
SUBROUTINE E04FDF(M,N,X,FSUMSQ,IW,LIW,W,LW,IFAIL)
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
C
C **************************************************************
C
C E04FDF IS AN EASY-TO-USE ALGORITHM FOR FINDING
C AN UNCONSTRAINED MINIMUM OF A SUM OF SQUARES OF M NONLINEAR
C FUNCTIONS IN N VARIABLES (M .GE. N). NO DERIVATIVES ARE
C REQUIRED.
C
C THE ROUTINE IS ESSENTIALLY IDENTICAL TO THE SUBROUTINE LSNDN1
C IN THE NPL ALGORITHMS LIBRARY (REF. NO. E4/26/F) AND CALLS
C E04FCZ WITH SUITABLE DEFAULT SETTINGS FOR PARAMETERS. IT CALLS
C THE USER-SUPPLIED ROUTINE LSFUN1.
C
C N.B. LSFUN1 IS A DESIGNATED NAME.
C --------------------------------
C
C GIVEN AN INITIAL APPROXIMATION TO THE MINIMUM, E04FDF COMPUTES
C THE POSITION OF THE MINIMUM AND THE CORRESPONDING FUNCTION
C VALUE. THE REAL ARRAY W AND THE INTEGER ARRAY IW ARE USED
C AS WORKSPACE. W MUST BE DIMENSIONED AT LEAST
C 7*N + N*N + 2*M*N + 3*M + N*(N - 1)/2 OR 9 + 5*M IF N = 1
C AND IW MUST HAVE AT LEAST ONE ELEMENT.
C
C PHILIP E. GILL, WALTER MURRAY, SUSAN M. PICKEN,
C NICHOLAS I. M. GOULD AND ENID M. R. LONG
C D.N.A.C., NATIONAL PHYSICAL LABORATORY, ENGLAND
C
C **************************************************************
C
C Modified to call BLAS.
C Peter Mayes, NAG Central Office, October 1987.
C
C Modified to output explanatory messages.
C Peter Mayes, NAG Central Office, December 1987.
C
SUBROUTINE E04GBF(M,N,LSQLIN,LSFJC,LSMON,IPRINT,MAXCAL,ETA,XTOL,
* STEPMX,X,FSUMSQ,FVEC,FJAC,LJ,S,VT,LVT,NITER,
* NFTOTL,IW,LIW,W,LW,IFAIL)
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
C MARK 16 REVISED. IER-989 (JUN 1993).
C
C **************************************************************
C
C E04GBF IS A COMPREHENSIVE QUASI-NEWTON ALGORITHM FOR FINDING
C AN UNCONSTRAINED MINIMUM OF A SUM OF SQUARES OF M NONLINEAR
C FUNCTIONS IN N VARIABLES (M .GE. N). FIRST DERIVATIVES ARE
C REQUIRED.
C
C THE ROUTINE IS ESSENTIALLY IDENTICAL TO THE SUBROUTINE LSFDQ
C IN THE NPL ALGORITHMS LIBRARY (REF. NO. E4/21/F). E04GBF
C WILL NORMALLY BE CALLED WITH E04FCV (WHICH CALLS E04ABZ)
C OR E04HEV (WHICH CALLS E04BBZ) AS THE PARAMETER LSQLIN.
C
C PHILIP E. GILL, SUSAN M. PICKEN, WALTER MURRAY
C BRIAN T. HINDE AND NICHOLAS I. M. GOULD
C D.N.A.C., NATIONAL PHYSICAL LABORATORY, ENGLAND.
C
C **************************************************************
C
C Modified to output explanatory messages.
C Peter Mayes, NAG Central Office, December 1987.
C
SUBROUTINE E04GCF(M,N,X,FSUMSQ,IW,LIW,W,LW,IFAIL)
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
C
C **************************************************************
C
C E04GCF IS AN EASY-TO-USE QUASI-NEWTON ALGORITHM FOR FINDING
C AN UNCONSTRAINED MINIMUM OF A SUM OF SQUARES OF M NONLINEAR
C FUNCTIONS IN N VARIABLES (M .GE. N). FIRST DERIVATIVES ARE
C REQUIRED.
C
C THE ROUTINE IS ESSENTIALLY IDENTICAL TO THE SUBROUTINE LSFDQ2
C IN THE NPL ALGORITHMS LIBRARY (REF. NO. E4/21/F) AND CALLS
C E04GBF WITH SUITABLE DEFAULT SETTINGS FOR PARAMETERS. IT
C CALLS THE USER-SUPPLIED ROUTINE LSFUN2.
C
C N.B. LSFUN2 IS A DESIGNATED NAME.
C --------------------------------
C
C GIVEN AN INITIAL APPROXIMATION TO THE MINIMUM, E04GCF COMPUTES
C THE POSITION OF THE MINIMUM AND THE CORRESPONDING FUNCTION
C VALUE. THE REAL ARRAY W AND THE INTEGER ARRAY IW ARE USED
C AS WORKSPACE. W MUST BE DIMENSIONED AT LEAST
C 8*N + N*N + 2*M*N + 3*M + N*(N - 1)/2 + N*(N + 1)/2 OR
C 11 + 5*M IF N = 1 AND IW MUST BE HAVE AT LEAST ONE ELEMENT.
C
C PHILIP E. GILL, WALTER MURRAY, SUSAN M. PICKEN
C AND NICHOLAS I. M. GOULD.
C D.N.A.C., NATIONAL PHYSICAL LABORATORY, ENGLAND.
C
C **************************************************************
C
C Modified to call BLAS.
C Peter Mayes, NAG Central Office, October 1987.
C
C Modified to output explanatory messages.
C Peter Mayes, NAG Central Office, December 1987.
C
SUBROUTINE E04GDF(M,N,LSFJC,LSMON,IPRINT,MAXFUN,ETA,XTOL,STEPMX,X,
* FSUMSQ,FVEC,FJAC,LJ,S,VT,LVT,NITER,NFTOTL,IW,
* LIW,W,LW,IFAIL)
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
C
C **************************************************************
C
C E04GDF IS A COMPREHENSIVE MODIFIED GAUSS-NEWTON ALGORITHM
C FOR FINDING AN UNCONSTRAINED MINIMUM OF A SUM OF SQUARES OF
C M NONLINEAR FUNCTIONS IN N VARIABLES (M .GE. N). FIRST
C DERIVATIVES ARE REQUIRED.
C
C THE ROUTINE IS ESSENTIALLY IDENTICAL TO THE SUBROUTINE LSFDN
C IN THE NPL ALGORITHMS LIBRARY (REF. NO. E4/18/F).
C
C PHILIP E. GILL, SUSAN M. PICKEN, WALTER MURRAY,
C BRIAN T. HINDE AND ENID M. R. LONG
C D.N.A.C., NATIONAL PHYSICAL LABORATORY, ENGLAND.
C
C **************************************************************
C
C Modified to output explanatory messages.
C Peter Mayes, NAG Central Office, December 1987.
C
SUBROUTINE E04GEF(M,N,X,FSUMSQ,IW,LIW,W,LW,IFAIL)
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
C
C **************************************************************
C
C E04GEF IS AN EASY-TO-USE MODIFIED GAUSS-NEWTON ALGORITHM
C FOR FINDING AN UNCONSTRAINED MINIMUM OF A SUM OF SQUARES OF
C M NONLINEAR FUNCTIONS IN N VARIABLES (M .GE. N). FIRST
C DERIVATIVES ARE REQUIRED.
C
C THE ROUTINE IS ESSENTIALLY IDENTICAL TO THE SUBROUTINE LSFDN2
C IN THE NPL ALGORITHMS LIBRARY (REF. NO. E4/24/F) AND CALLS
C E04HEX WITH SUITABLE DEFAULT SETTINGS FOR PARAMETERS. IT
C CALLS THE USER-SUPPLIED ROUTINE LSFUN2.
C
C N.B. LSFUN2 IS A DESIGNATED NAME.
C --------------------------------
C
C GIVEN AN INITIAL APPROXIMATION TO THE MINIMUM, E04GEF COMPUTES
C THE POSITION OF THE MINIMUM AND THE CORRESPONDING FUNCTION
C VALUE. THE REAL ARRAY W AND THE INTEGER ARRAY IW ARE USED
C AS WORKSPACE. W MUST BE DIMENSIONED AT LEAST
C 8*N + N*N + 2*M*N + 3*M + N*(N - 1)/2 + N*(N + 1)/2 OR
C 11 + 5*M IF N = 1 AND IW MUST HAVE AT LEAST ONE ELEMENT.
C
C PHILIP E. GILL, WALTER MURRAY, SUSAN M. PICKEN
C AND NICHOLAS I. M. GOULD.
C D.N.A.C., NATIONAL PHYSICAL LABORATORY, ENGLAND.
C
C **************************************************************
C
C Modified to call BLAS.
C Peter Mayes, NAG Central Office, October 1987.
C
C Modified to output explanatory messages
C Peter Mayes, NAG Central Office, December 1987.
C
SUBROUTINE E04HCF(N,SFUN,X,F,G,IW,LIW,W,LW,IFAIL)
C
C MARK 6 RELEASE NAG COPYRIGHT 1977
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C MARK 14 REVISED. IER-799 (DEC 1989).
C
C **************************************************************
C
C E04HCF CHECKS THAT A USER-SUPPLIED ROUTINE FOR EVALUATING AN
C OBJECTIVE FUNCTION AND ITS FIRST DERIVATIVES PRODUCES
C DERIVATIVE VALUES WHICH ARE CONSISTENT WITH THE FUNCTION
C VALUES CALCULATED.
C
C THE ROUTINE IS ESSENTIALLY IDENTICAL TO THE SUBROUTINE CHKGRD
C IN THE NPL ALGORITHMS LIBRARY (REF. NO. E4/05/F). W(I), I = 1,
C 2, . . . , 3*N ARE USED AS WORKSPACE. (NOTE THAT, FOR
C CONSISTENCY WITH OTHER E04 DOCUMENTATION, THE NAME FUNCT IS
C USED INSTEAD OF SFUN IN THE WRITE-UP.)
C
C PHILIP E. GILL, ENID M. R. LONG, WALTER MURRAY, SUSAN M.
C PICKEN, D.N.A.C., NATIONAL PHYSICAL LABORATORY, ENGLAND.
C
C **************************************************************
C
C SFUN
C
C A MACHINE DEPENDENT CONSTANT IS SET HERE. EPSMCH IS THE
C SMALLEST POSITIVE REAL NUMBER SUCH THAT 1 + EPSMCH .GT. 1.
C
SUBROUTINE E04HDF(N,SFUN,SHESS,X,G,HESL,LH,HESD,IW,LIW,W,LW,IFAIL)
C
C MARK 6 RELEASE NAG COPYRIGHT 1977
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C MARK 13 REVISED. IER-645 (APR 1988).
C MARK 14 REVISED. IER-800 (DEC 1989).
C
C **************************************************************
C
C E04HDF CHECKS THAT A USER-SUPPLIED ROUTINE FOR CALCULATING
C SECOND DERIVATIVES OF AN OBJECTIVE FUNCTION IS CONSISTENT WITH
C A USER-SUPPLIED ROUTINE FOR CALCULATING THE CORRESPONDING
C FIRST DERIVATIVES.
C
C THE ROUTINE IS ESSENTIALLY IDENTICAL TO THE SUBROUTINE CHKHES
C IN THE NPL ALGORITHMS LIBRARY (REF. NO. E4/04/F). W(I), I = 1,
C 2, . . . , 5*N ARE USED AS WORKSPACE. (NOTE THAT, FOR
C CONSISTENCY WITH OTHER E04 DOCUMENTATION, THE NAMES FUNCT AND
C HESS ARE USED INSTEAD OF SFUN AND SHESS IN THE WRITE-UP.)
C
C PHILIP E. GILL, ENID M. R. LONG, WALTER MURRAY, SUSAN M.
C PICKEN, D.N.A.C., NATIONAL PHYSICAL LABORATORY, ENGLAND.
C
C **************************************************************
C
C SFUN, SHESS
C
C A MACHINE DEPENDENT CONSTANT IS SET HERE. EPSMCH IS THE
C SMALLEST POSITIVE REAL NUMBER SUCH THAT 1.0 + EPSMCH .GT. 1.0.
C
SUBROUTINE E04HEF(M,N,LSFJC,LSHES,LSMON,IPRINT,MAXCAL,ETA,XTOL,
* STEPMX,X,FSUMSQ,FVEC,FJAC,LJ,S,VT,LVT,NITER,
* NFTOTL,IW,LIW,W,LW,IFAIL)
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
C
C **************************************************************
C
C E04HEF IS A COMPREHENSIVE MODIFIED GAUSS-NEWTON ALGORITHM
C FOR FINDING AN UNCONSTRAINED MINIMUM OF A SUM OF SQUARES OF
C M NONLINEAR FUNCTIONS IN N VARIABLES (M .GE. N). FIRST AND
C SECOND DERIVATIVES ARE REQUIRED.
C
C THE ROUTINE IS ESSENTIALLY IDENTICAL TO THE SUBROUTINE LSSDN
C IN THE NPL ALGORITHMS LIBRARY (REF. NO. E4/17/F).
C
C PHILIP E. GILL, SUSAN M. PICKEN, WALTER MURRAY AND
C NICHOLAS I. M. GOULD.
C D.N.A.C., NATIONAL PHYSICAL LABORATORY, ENGLAND.
C
C **************************************************************
C
C Modified to output explanatory messages.
C Peter Mayes, NAG Central Office, December 1987.
C
SUBROUTINE E04HFF(M,N,X,FSUMSQ,IW,LIW,W,LW,IFAIL)
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
C
C **************************************************************
C
C E04HFF IS AN EASY-TO-USE MODIFIED GAUSS-NEWTON ALGORITHM
C FOR FINDING AN UNCONSTRAINED MINIMUM OF A SUM OF SQUARES OF
C M NONLINEAR FUNCTIONS IN N VARIABLES (M .GE. N). FIRST AND
C SECOND DERIVATIVES ARE REQUIRED.
C
C THE ROUTINE IS ESSENTIALLY IDENTICAL TO THE SUBROUTINE LSSDN2
C IN THE NPL ALGORITHMS LIBRARY (REF. NO. E4/23/F) AND CALLS
C E04HEX WITH SUITABLE DEFAULT SETTINGS FOR PARAMETERS. IT
C CALLS THE USER-SUPPLIED ROUTINES LSFUN2 AND LSHES2.
C
C N.B. LSFUN2 AND LSHES2 ARE DESIGNATED NAMES.
C -------------------------------------------
C
C GIVEN AN INITIAL APPROXIMATION TO THE MINIMUM, E04HFF COMPUTES
C THE POSITION OF THE MINIMUM AND THE CORRESPONDING FUNCTION
C VALUE. THE REAL ARRAY W AND THE INTEGER ARRAY IW ARE USED
C AS WORKSPACE. W MUST BE DIMENSIONED AT LEAST
C 8*N + N*N + 2*M*N + 3*M + N*(N - 1)/2 + N*(N + 1)/2 OR
C 11 + 5*M IF N = 1 AND IW MUST HAVE AT LEAST ONE ELEMENT.
C
C PHILIP E. GILL, WALTER MURRAY, SUSAN M. PICKEN
C AND NICHOLAS I. M. GOULD.
C D.N.A.C., NATIONAL PHYSICAL LABORATORY, ENGLAND.
C
C **************************************************************
C
C Modified to call BLAS.
C Peter Mayes, NAG Central Office, October 1987.
C
C Modified to output explanatory messages.
C Peter Mayes, NAG Central Office, December 1987.
C
SUBROUTINE E04JAF(N,IBOUND,BL,BU,X,F,IW,LIW,W,LW,IFAIL)
C
C MARK 6 RELEASE NAG COPYRIGHT 1977
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C MARK 13A REVISED. IER-632 (APR 1988).
C MARK 14 REVISED. IER-815 (DEC 1989).
C MARK 16 REVISED. IER-1109 (JUL 1993).
C
C **************************************************************
C
C E04JAF IS AN EASY-TO-USE QUASI-NEWTON ALGORITHM FOR FINDING A
C MINIMUM OF A FUNCTION F(X1, X2, . . . XN), SUBJECT TO FIXED
C UPPER AND LOWER BOUNDS ON THE INDEPENDENT VARIABLES X1, X2, .
C . . , XN , USING FUNCTION VALUES ONLY.
C
C GIVEN AN INITIAL ESTIMATE OF THE POSITION OF A CONSTRAINED
C MINIMUM, THE ROUTINE ATTEMPTS TO COMPUTE THE POSITION OF THE
C MINIMUM AND THE CORRESPONDING FUNCTION VALUE. IT IS ASSUMED
C THAT F IS TWICE CONTINUOUSLY-DIFFERENTIABLE.
C
C THE ROUTINE IS ESSENTIALLY IDENTICAL TO THE SUBROUTINE BCNDQ1
C IN THE NPL ALGORITHMS LIBRARY (REF. NO. E4/49/F) AND CALLS
C E04JBL WITH SUITABLE DEFAULT SETTINGS FOR PARAMETERS.
C
C N.B. FUNCT1 IS A DESIGNATED NAME.
C --------------------------------
C
C THE REAL ARRAY W, USED AS WORKSPACE BY E04JAF, MUST BE
C DIMENSIONED AT LEAST 12*N + N*(N - 1)/2, OR 13 IF N = 1. THE
C INTEGER ARRAY IW, ALSO USED AS WORKSPACE BY E04JAF, MUST BE
C DIMENSIONED AT LEAST (N + 2).
C
C PHILIP E. GILL, WALTER MURRAY, SUSAN M. PICKEN, MARGARET H.
C WRIGHT AND ENID M. R. LONG, D.N.A.C., NATIONAL PHYSICAL
C LABORATORY, ENGLAND
C
C **************************************************************
C
C
C A MACHINE-DEPENDENT CONSTANT IS SET HERE. EPSMCH IS THE
C SMALLEST POSITIVE REAL NUMBER SUCH THAT 1 + EPSMCH .GT. 1.
C
SUBROUTINE E04KAF(N,IBOUND,BL,BU,X,F,G,IW,LIW,W,LW,IFAIL)
C
C MARK 6 RELEASE NAG COPYRIGHT 1977
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C MARK 13A REVISED. IER-632 (APR 1988).
C MARK 14 REVISED. IER-816 (DEC 1989).
C MARK 16 REVISED. IER-1110 (JUL 1993).
C
C **************************************************************
C
C E04KAF IS AN EASY-TO-USE QUASI-NEWTON ALGORITHM FOR FINDING A
C MINIMUM OF A FUNCTION F(X1, X2, . . . , XN), SUBJECT TO FIXED
C UPPER AND LOWER BOUNDS ON THE INDEPENDENT VARIABLES X1, X2, .
C . . , XN , WHEN FIRST DERIVATIVES OF F ARE AVAILABLE.
C
C GIVEN AN INITIAL ESTIMATE OF THE POSITION OF A CONSTRAINED
C MINIMUM, THE ROUTINE ATTEMPTS TO COMPUTE THE POSITION OF THE
C MINIMUM AND THE CORRESPONDING FUNCTION AND GRADIENT VALUES. IT
C IS ASSUMED THAT F IS TWICE CONTINUOUSLY-DIFFERENTIABLE.
C
C THE ROUTINE IS ESSENTIALLY IDENTICAL TO THE SUBROUTINE BCFDQ2
C IN THE NPL ALGORITHMS LIBRARY (REF. NO. E4/48/F) AND CALLS
C E04KBN WITH SUITABLE DEFAULT SETTINGS FOR PARAMETERS. (IN
C PARTICULAR, E04LBS IS USED AS MINLIN.)
C
C N.B. FUNCT2 IS A DESIGNATED NAME.
C --------------------------------
C
C THE REAL ARRAY W, USED AS WORKSPACE BY E04KAF, MUST BE
C DIMENSIONED AT LEAST 10*N + N*(N - 1)/2, OR 11 IF N = 1. THE
C INTEGER ARRAY IW, ALSO USED AS WORKSPACE BY E04KAF, MUST BE
C DIMENSIONED AT LEAST (N + 2).
C
C PHILIP E. GILL, WALTER MURRAY, SUSAN M. PICKEN, MARGARET H.
C WRIGHT AND ENID M. R. LONG, D.N.A.C., NATIONAL PHYSICAL
C LABORATORY, ENGLAND
C
C **************************************************************
C
C
C A MACHINE-DEPENDENT CONSTANT IS SET HERE. EPSMCH IS THE
C SMALLEST POSITIVE REAL NUMBER SUCH THAT 1 + EPSMCH .GT.
C EPSMCH.
C
SUBROUTINE E04KCF(N,IBOUND,BL,BU,X,F,G,IW,LIW,W,LW,IFAIL)
C
C MARK 6 RELEASE NAG COPYRIGHT 1977
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C MARK 13A REVISED. IER-632 (APR 1988).
C MARK 14 REVISED. IER-817 (DEC 1989).
C
C **************************************************************
C
C E04KCF IS AN EASY-TO-USE MODIFIED NEWTON ALGORITHM FOR FINDING
C A MINIMUM OF A FUNCTION F(X1, X2, . . . , XN), SUBJECT TO
C FIXED UPPER AND LOWER BOUNDS ON THE INDEPENDENT VARIABLES X1,
C X2, . . . , XN , WHEN FIRST DERIVATIVES OF F ARE AVAILABLE.
C
C GIVEN AN INITIAL ESTIMATE OF THE POSITION OF A CONSTRAINED
C MINIMUM, THE ROUTINE ATTEMPTS TO COMPUTE THE POSITION OF THE
C MINIMUM AND THE CORRESPONDING FUNCTION AND GRADIENT VALUES. IT
C IS ASSUMED THAT F IS TWICE CONTINUOUSLY-DIFFERENTIABLE.
C
C THE ROUTINE IS ESSENTIALLY IDENTICAL TO THE SUBROUTINE BCFDN2
C IN THE NPL ALGORITHMS LIBRARY (REF. E4/47/F) AND CALLS E04LBR
C WITH SUITABLE DEFAULT SETTINGS FOR PARAMETERS.
C
C N.B. FUNCT2 IS A DESIGNATED NAME.
C --------------------------------
C
C THE REAL ARRAY W, USED AS WORKSPACE BY E04KCF, MUST BE
C DIMENSIONED AT LEAST N*(N + 7), OR 10 IF N = 1. THE INTEGER
C ARRAY IW, ALSO USED AS WORKSPACE BY E04KCF, MUST BE
C DIMENSIONED AT LEAST (N + 2).
C
C PHILIP E. GILL, WALTER MURRAY, SUSAN M. PICKEN, MARGARET H.
C WRIGHT AND ENID M. R. LONG, D.N.A.C., NATIONAL PHYSICAL
C LABORATORY, ENGLAND
C
C **************************************************************
C
C
C A MACHINE-DEPENDENT CONSTANT IS SET HERE. EPSMCH IS THE
C SMALLEST POSITIVE REAL NUMBER SUCH THAT 1 + EPSMCH .GT. 1.
C
SUBROUTINE E04KDF(N,SFUN,MONIT,IPRINT,MAXFUN,ETA,XTOL,DELTA,
* STEPMX,IBOUND,BL,BU,X,HESL,LH,HESD,ISTATE,F,G,
* IW,LIW,W,LW,IFAIL)
C
C MARK 6 RELEASE NAG COPYRIGHT 1977
C MARK 6A REVISED IER-100
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C **************************************************************
C
C E04KDF IS A COMPREHENSIVE MODIFIED NEWTON ALGORITHM FOR
C FINDING
C - AN UNCONSTRAINED MINIMUM OF A FUNCTION OF SEVERAL VARIABLES
C - A MINIMUM OF A FUNCTION OF SEVERAL VARIABLES SUBJECT TO
C FIXED UPPER AND/OR LOWER BOUNDS ON THE VARIABLES.
C FIRST DERIVATIVES ARE REQUIRED.
C
C THE ROUTINE IS ESSENTIALLY IDENTICAL TO THE SUBROUTINE BCMNAF
C IN THE NPL ALGORITHMS LIBRARY (REF. NO. E4/32/F). IT CALLS
C E04LBR WITH E04KDZ AS THE PARAMETER SHESS AND E04LBS (WHICH
C CALLS E04BBZ) AS THE PARAMETER MINLIN.
C
C THE USER MUST SUPPLY AN INITIAL APPROXIMATION TO THE MINIMUM
C AND A SUBROUTINE SFUN TO COMPUTE THE FUNCTION VALUE AND/OR
C GRADIENT VECTOR. (NOTE THAT FOR CONSISTENCY WITH OTHER E04
C DOCUMENTATION, THE NAME FUNCT IS USED INSTEAD OF SFUN IN THE
C WRITE-UP.) THE ARRAYS W AND IW ARE USED AS WORK-SPACE BY
C E04LBR AND MUST BE DIMENSIONED AT LEAST MAX(8, 7*N + N(N-1)/2
C ) AND 2, RESPECTIVELY. ON EXIT, E04KDF GIVES THE ESTIMATED
C POSITION OF THE CONSTRAINED MINIMUM IN X AND THE LOWEST
C FUNCTION VALUE IN F.
C
C THE MAJORITY OF THE ARITHMETIC IS PERFORMED BY F01BQZ, F01DEF
C AND F04AQZ, WHICH ARE CALLED MANY TIMES DURING THE SOLUTION OF
C A PROBLEM. IMPLEMENTORS CAN REDUCE THE TIME TAKEN BY E04KDF
C (PARTICULARLY WHEN N IS LARGE) BY PUTTING THE BODIES OF THESE
C ROUTINES INTO MACHINE CODE.
C
C PHILIP E. GILL, WALTER MURRAY, SUSAN M. PICKEN, MARGARET H.
C WRIGHT AND ENID M. R. LONG, D.N.A.C., NATIONAL PHYSICAL
C LABORATORY, ENGLAND
C
C **************************************************************
C
SUBROUTINE E04LAF(N,IBOUND,BL,BU,X,F,G,IW,LIW,W,LW,IFAIL)
C
C MARK 6 RELEASE NAG COPYRIGHT 1977
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C MARK 13A REVISED. IER-632 (APR 1988).
C MARK 14 REVISED. IER-818 (DEC 1989).
C
C **************************************************************
C
C E04LAF IS AN EASY-TO-USE MODIFIED NEWTON ALGORITHM FOR FINDING
C A MINIMUM OF A FUNCTION F(X1, X2, . . . , XN), SUBJECT TO
C FIXED UPPER AND LOWER BOUNDS ON THE INDEPENDENT VARIABLES X1,
C X2, . . . , XN , WHEN FIRST AND SECOND DERIVATIVES ARE
C AVAILABLE.
C
C GIVEN AN INITIAL ESTIMATE OF THE POSITION OF A CONSTRAINED
C MINIMUM, THE ROUTINE ATTEMPTS TO COMPUTE THE POSITION OF THE
C MINIMUM AND THE CORRESPONDING FUNCTION AND GRADIENT VALUES. IT
C IS ASSUMED THAT F IS TWICE CONTINUOUSLY-DIFFERENTIABLE.
C
C THE ROUTINE IS ESSENTIALLY IDENTICAL TO THE SUBROUTINE BCSDN2
C IN THE NPL ALGORITHMS LIBRARY (REF. E4/46/F) AND CALLS E04LBR
C WITH SUITABLE DEFAULT SETTINGS FOR PARAMETERS.
C
C N.B. FUNCT2 AND HESS2 ARE DESIGNATED NAMES.
C ------------------------------------------
C
C THE REAL ARRAY W, USED AS WORKSPACE BY E04LAF, MUST BE
C DIMENSIONED AT LEAST N*(N + 7), OR 10 IF N = 1. THE INTEGER
C ARRAY IW, ALSO USED AS WORKSPACE BY E04LAF, MUST BE
C DIMENSIONED AT LEAST (N + 2).
C
C PHILIP E. GILL, WALTER MURRAY, SUSAN M. PICKEN, MARGARET H.
C WRIGHT AND ENID M. R. LONG, D.N.A.C., NATIONAL PHYSICAL
C LABORATORY, ENGLAND
C
C **************************************************************
C
C
C A MACHINE-DEPENDENT CONSTANT IS SET HERE. EPSMCH IS THE
C SMALLEST POSITIVE REAL NUMBER SUCH THAT 1 + EPSMCH .GT. 1.
C
SUBROUTINE E04LBF(N,SFUN,SHESS,MONIT,IPRINT,MAXFUN,ETA,XTOL,
* STEPMX,IBOUND,BL,BU,X,HESL,LH,HESD,ISTATE,F,G,
* IW,LIW,W,LW,IFAIL)
C
C MARK 6 RELEASE NAG COPYRIGHT 1977
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C **************************************************************
C
C E04LBF IS A COMPREHENSIVE MODIFIED NEWTON ALGORITHM FOR
C FINDING
C - AN UNCONSTRAINED MINIMUM OF A FUNCTION OF SEVERAL VARIABLES
C - A MINIMUM OF A FUNCTION OF SEVERAL VARIABLES SUBJECT TO
C FIXED UPPER AND/OR LOWER BOUNDS ON THE VARIABLES.
C FIRST AND SECOND DERIVATIVES ARE REQUIRED.
C
C THE ROUTINE IS ESSENTIALLY IDENTICAL TO THE SUBROUTINE BCMNA
C IN THE NPL ALGORITHMS LIBRARY (REF. NO. E4/31/F). IT CALLS
C E04LBR WITH E04LBS (WHICH CALLS E04BBZ) AS THE PARAMETER
C MINLIN.
C
C THE USER MUST SUPPLY AN INITIAL APPROXIMATION TO THE MINIMUM,
C A SUBROUTINE SFUN TO COMPUTE THE FUNCTION VALUE AND/OR
C GRADIENT VECTOR, AND A SUBROUTINE SHESS TO CALCULATE THE
C HESSIAN MATRIX. (NOTE THAT, FOR CONSISTENCY WITH OTHER E04
C DOCUMENTATION, THE NAMES FUNCT AND HESS ARE USED INSTEAD OF
C SFUN AND SHESS IN THE WRITE-UP.) THE ARRAYS W AND IW ARE USED
C AS WORK-SPACE BY E04LBR AND MUST BE DIMENSIONED AT LEAST
C MAX(8, 7*N + N(N-1)/2 ) AND 2, RESPECTIVELY. ON EXIT, E04LBF
C GIVES THE ESTIMATED POSITION OF THE CONSTRAINED MINIMUM IN X
C AND THE LOWEST FUNCTION VALUE IN F.
C
C THE MAJORITY OF THE ARITHMETIC IS PERFORMED BY F01BQZ, F01DEF
C AND F04AQZ, WHICH ARE CALLED MANY TIMES DURING THE SOLUTION OF
C A PROBLEM. IMPLEMENTORS CAN REDUCE THE TIME TAKEN BY E04LBF
C (PARTICULARLY WHEN N IS LARGE) BY PUTTING THE BODIES OF THESE
C ROUTINES INTO MACHINE CODE.
C
C PHILIP E. GILL, WALTER MURRAY, SUSAN M. PICKEN, MARGARET H.
C WRIGHT AND ENID M. R. LONG, D.N.A.C., NATIONAL PHYSICAL
C LABORATORY, ENGLAND
C
C **************************************************************
C
SUBROUTINE E04MFF(N,NCLIN,A,LDA,BL,BU,CVEC,ISTATE,X,ITER,OBJ,AX,
* CLAMDA,IW,LENIW,W,LENW,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
C
C ******************************************************************
C E04MFF solves the linear programming problem
C
C minimize c' x
C x
C ( x )
C subject to bl .le.( ).ge. bu,
C ( Ax )
C
C where A is a constant nclin by n matrix.
C The feasible region is defined by a mixture of linear equality or
C inequality constraints on x.
C
C n is the number of variables (dimension of x).
C (n must be positive.)
C
C nclin is the number of general linear constraints (rows of A).
C (nclin may be zero.)
C
C The first n elements of bl and bu are lower and upper
C bounds on the variables. The next nclin elements are
C lower and upper bounds on the general linear constraints.
C
C The matrix A of coefficients in the general linear constraints
C is entered as the two-dimensional array A (of dimension
C LDA by n). If nclin = 0, A is not referenced.
C
C The vector x must contain an initial estimate of the solution,
C and will contain the computed solution on output.
C
C Documentation for E04MFF is coming real soon now.
C Wait for the release of users guide for LPOPT (Version 1.00-6),
C by P. E. Gill, W. Murray and M. A. Saunders,
C
C Version 1.0-6 Jun 30, 1991. (Nag Mk 16 version.)
C
C Copyright 1989 Optimates.
C ******************************************************************
C
SUBROUTINE E04MGF(IOPTNS,INFORM)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
C
C ******************************************************************
C E04MGF reads the options file from unit IOPTNS and loads the
C options into the relevant elements of IPRMLC and RPRMLC.
C
C If IOPTNS .lt. 0 or IOPTNS .gt. 99 then no file is read,
C otherwise the file associated with unit IOPTNS is read.
C
C Output:
C
C INFORM = 0 if a complete OPTIONS file was found
C (starting with BEGIN and ending with END);
C 1 if IOPTNS .lt. 0 or IOPTNS .gt. 99;
C 2 if BEGIN was found, but end-of-file
C occurred before END was found;
C 3 if end-of-file occurred before BEGIN or
C ENDRUN were found;
C 4 if ENDRUN was found before BEGIN.
C ******************************************************************
C
SUBROUTINE E04MHF(STRING)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
C
C ******************************************************************
C E04MHF loads the option supplied in STRING into the relevant
C element of IPRMLC or RPRMLC.
C ******************************************************************
C
SUBROUTINE E04NCF(MM,N,NCLIN,LDA,LDR,A,BL,BU,CVEC,ISTATE,KX,X,R,B,
* ITER,OBJ,CLAMDA,IW,LENIW,W,LENW,IFAIL)
C MARK 14 RE-ISSUE. NAG COPYRIGHT 1989.
C MARK 16 REVISED. IER-1060 (JUL 1993).
C
C ******************************************************************
C E04NCF solves problems of the form
C
C Minimize F(x)
C x
C ( x )
C subject to bl .le.( ).ge. bu,
C ( Ax )
C
C where ' denotes the transpose of a column vector, x denotes
C the n-vector of parameters and F(x) is one of the following
C functions..
C
C FP = None (find a feasible point).
C LP = c'x
C QP1= 1/2 x'Rx R n times n, symmetric pos. def.
C QP2= c'x + 1/2 x'Rx . . .. .. .. ..
C QP3= 1/2 x'R'Rx R m times n, upper triangular.
C QP4= c'x + 1/2 x'R'Rx . . .. . .. ...
C LS1= 1/2 (b - Rx)'(b - Rx) R m times n, rectangular.
C LS2= c'x + 1/2 (b - Rx)'(b - Rx) . . .. . ...
C LS3= 1/2 (b - Rx)'(b - Rx) R m times n, upper triangular.
C LS4= c'x + 1/2 (b - Rx)'(b - Rx) . . .. . .. ...
C
C The matrix R is entered as the two-dimensional array R (of row
C dimension LDR). If LDR = 0, R is not accessed.
C
C The vector c is entered in the one-dimensional array CVEC.
C
C NCLIN is the number of general linear constraints (rows of A).
C (NCLIN may be zero.)
C
C The first N elements of BL and BU are lower and upper
C bounds on the variables. The next NCLIN elements are
C lower and upper bounds on the general linear constraints.
C
C The matrix A of coefficients in the general linear constraints
C is entered as the two-dimensional array A (of dimension
C LDA by N). If NCLIN = 0, A is not accessed.
C
C The vector x must contain an initial estimate of the solution,
C and will contain the computed solution on output.
C
C
C Complete documentation for E04NCF is contained in
C Report SOL 86-1,
C Users Guide for LSSOL (Version 1.0), by P.E. Gill,
C S. J. Hammarling, W. Murray, M.A. Saunders and M.H. Wright,
C Department of Operations Research, Stanford University, Stanford,
C California 94305.
C
C Systems Optimization Laboratory, Stanford University.
C Version 1.0 Dated 24-January-1986.
C Version 1.01 Dated 30-June-1986. Level-2 BLAS added.
C Version 1.02 Dated 13-May-1988. Level-2 matrix routines added.
C Version 1.05 Dated 20-Sep-1992. Output modified.
C 20-Oct-1992. Summary file included.
C
C Copyright 1984 Stanford University.
C
C This material may be reproduced by or for the U.S. Government
C pursuant to the copyright license under DAR clause 7-104.9(a)
C (1979 Mar).
C
C This material is based upon work partially supported by the
C National Science Foundation under Grants MCS-7926009 and
C ECS-8312142; the Department of Energy Contract AM03-76SF00326,
C PA No. DE-AT03-76ER72018; the Army Research Office Contract DAA29-
C 84-K-0156; and the Office of Naval Research Grant N00014-75-C-0267
C ******************************************************************
C
SUBROUTINE E04NDF(IOPTNS,INFORM)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C***********************************************************************
C E04NDF reads the options file from unit IOPTNS and loads the
C options into the relevant elements of IPRMLS and RPRMLS.
C
C If IOPTNS .lt. 0 or IOPTNS .gt. 99 then no file is read,
C otherwise the file associated with unit IOPTNS is read.
C
C Output:
C
C INFORM = 0 if a complete OPTIONS file was found
C (starting with BEGIN and ending with END);
C 1 if IOPTNS .lt. 0 or IOPTNS .gt. 99;
C 2 if BEGIN was found, but end-of-file
C occurred before END was found;
C 3 if end-of-file occurred before BEGIN or
C ENDRUN were found;
C 4 if ENDRUN was found before BEGIN.
C***********************************************************************
C
SUBROUTINE E04NEF(STRING)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C***********************************************************************
C E04NEF loads the option supplied in STRING into the relevant
C element of IPRMLS or RPRMLS.
C***********************************************************************
C
SUBROUTINE E04NFF(N,NCLIN,A,LDA,BL,BU,CVEC,H,LDH,QPHESS,ISTATE,X,
* ITER,OBJ,AX,CLAMDA,IW,LENIW,W,LENW,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
C
C ******************************************************************
C E04NFF solves problems of the form
C
C minimize f(x)
C x
C ( x )
C subject to bl .le.( ).ge. bu,
C ( Ax )
C
C where ' denotes the transpose of a column vector, x denotes
C the n-vector of parameters and f(x) is one of the following...
C
C FP = none (find a feasible point)
C LP = c'x
C QP1 = 1/2 x'Hx H n x n symmetric
C QP2 (default) = c'x + 1/2 x'Hx H n x n symmetric
C QP3 = 1/2 x'H'Hx H m x n upper trapezoidal
C QP4 = c'x + 1/2 x'H'Hx H m x n upper trapezoidal
C
C The matrix H is stored in the two-dimensional array H of
C row dimension LDH. H can be entered explicitly as the matrix
C H, or implicitly via a user-supplied version of the
C subroutine QPHESS. If LDH = 0, H is not touched.
C
C The vector c is entered in the one-dimensional array CVEC.
C
C nclin is the number of general linear constraints (rows of A).
C (nclin may be zero.)
C
C The first n elements of bl and bu are lower and upper
C bounds on the variables. The next nclin elements are
C lower and upper bounds on the general linear constraints.
C
C The matrix A of coefficients in the general linear constraints
C is entered as the two-dimensional array A (of dimension
C LDA by n). If nclin = 0, A is not referenced.
C
C The vector x must contain an initial estimate of the solution,
C and will contain the computed solution on output.
C
C Documentation for QPOPT is coming real soon now.
C Wait for the release of users guide for QPOPT (Version 1.0), by
C P. E. Gill, W. Murray and M. A. Saunders.
C
C Version 1.0-6 Jun 30, 1991. (Nag Mk 16 version).
C
C This version of E04NFF dated 11-Nov-92.
C Copyright 1988 Optimates.
C ******************************************************************
C
SUBROUTINE E04NGF(IOPTNS,INFORM)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
C
C ******************************************************************
C E04NGF reads the options file from unit IOPTNS and loads the
C options into the relevant elements of IPRMLC and RPRMLC.
C
C If IOPTNS .lt. 0 or IOPTNS .gt. 99 then no file is read,
C otherwise the file associated with unit IOPTNS is read.
C
C Output:
C
C INFORM = 0 if a complete OPTIONS file was found
C (starting with BEGIN and ending with END);
C 1 if IOPTNS .lt. 0 or IOPTNS .gt. 99;
C 2 if BEGIN was found, but end-of-file
C occurred before END was found;
C 3 if end-of-file occurred before BEGIN or
C ENDRUN were found;
C 4 if ENDRUN was found before BEGIN.
C ******************************************************************
C
SUBROUTINE E04NHF(STRING)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
C
C ******************************************************************
C E04NHF loads the option supplied in STRING into the relevant
C element of IPRMLC or RPRMLC.
C ******************************************************************
C
SUBROUTINE E04UCF(N,NCLIN,NCNLN,LDA,LDCJU,LDR,A,BL,BU,CONFUN,
* OBJFUN,ITER,ISTATE,C,CJACU,CLAMDA,OBJF,GRADU,R,
* X,IW,LENIW,W,LENW,IUSER,USER,IFAIL)
C MARK 14 RE-ISSUE. NAG COPYRIGHT 1989.
C MARK 15B REVISED. IER-951 (NOV 1991).
C MARK 16 REVISED. IER-1077 (JUL 1993).
C
C ==================================================================
C E04UCF solves the nonlinear program
C
C minimize F(x)
C
C ( x )
C subject to bl .le. ( A*x ) .le. bu
C ( c(x) )
C
C where F(x) is a smooth scalar function, A is a constant matrix
C and c(x) is a vector of smooth nonlinear functions. The
C feasible region is defined by a mixture of linear and nonlinear
C equality or inequality constraints on x.
C
C The dimensions of the problem are...
C
C N the number of variables (dimension of x),
C
C NCLIN the number of linear constraints (rows of the matrix A),
C
C NCNLN the number of nonlinear constraints (dimension of c(x)),
C
C
C E04UCF uses a sequential quadratic programming algorithm, with a
C positive-definite quasi-Newton approximation to the transformed
C Hessian Q'HQ of the Lagrangian function (which will be stored in
C the array R).
C
C
C Complete documentation for E04UCF is contained in Report
C SOL 86-2, Users guide for E04UCF (Version 4.0), by P.E. Gill,
C W. Murray, M.A. Saunders and M.H. Wright, Department of Operations
C Research, Stanford University, Stanford, California 94305.
C
C Systems Optimization Laboratory, Stanford University.
C Version 1.1, April 12, 1983. The less said about this one...
C Version 2.0, April 30, 1984.
C Version 3.0, March 20, 1985. First Fortran 77 version
C Version 3.2, August 20, 1985.
C Version 4.0, April 16, 1986. First version with differences
C Version 4.01, June 30, 1986. Level 2 BLAS + F77 linesearch
C Version 4.02, August 5, 1986. Reset SSBFGS. One call to E04XAY
C Version 4.03, June 14, 1987. Step limit
C Version 4.04, June 28, 1989. Vectorizable BLAS
C Version 4.05, November 28, 1989. Load and save files added
C (but not to NAG version!)
C Version 4.06, November 5, 1991. E04UCJ and E04UCK updated
C October 17, 1992. Summary/print file option.
C
C Copyright 1983 Stanford University.
C
C This material may be reproduced by or for the U.S. Government
C pursuant to the copyright license under DAR Clause 7-104.9(a)
C (1979 Mar).
C
C This material is based upon work partially supported by the
C National Science Foundation under Grants MCS-7926009 and
C ECS-8312142; the Department of Energy Contract AM03-76SF00326,
C PA No. DE-AT03-76ER72018; the Army Research Office Contract
C DAA29-84-K-0156; and the Office of Naval Research Grant
C N00014-75-C-0267.
C ==================================================================
C
SUBROUTINE E04UDF(IOPTNS,INFORM)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C***********************************************************************
C E04UDF reads the options file from unit IOPTNS and loads the
C options into the relevant elements of IPRMNP and RPRMNP.
C
C If IOPTNS .lt. 0 or IOPTNS .gt. 99 then no file is read,
C otherwise the file associated with unit IOPTNS is read.
C
C Output:
C
C INFORM = 0 if a complete OPTIONS file was found
C (starting with BEGIN and ending with END);
C 1 if IOPTNS .lt. 0 or IOPTNS .gt. 99;
C 2 if BEGIN was found, but end-of-file
C occurred before END was found;
C 3 if end-of-file occurred before BEGIN or
C ENDRUN were found;
C 4 if ENDRUN was found before BEGIN.
C***********************************************************************
C
SUBROUTINE E04UEF(STRING)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C***********************************************************************
C E04UEF loads the option supplied in STRING into the relevant
C element of IPRMLS, RPRMLS, IPRMNP or RPRMNP.
C***********************************************************************
C
SUBROUTINE E04UPF(M,N,NCLIN,NCNLN,LDA,LDCJU,LDFJU,LDR,A,BL,BU,
* CONFUN,OBJFUN,ITER,ISTATE,C,CJACU,F,FJACU,
* CLAMDA,OBJF,R,X,IW,LENIW,W,LENW,IUSER,USER,
* IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C MARK 16 REVISED. IER-1097 (JUL 1993).
C
C ------------------------------------------------------------------
C
C E04UPF solves the constrained least-squares problem
C
C minimize F(x) = 0.5 * f(x)'f(x)
C
C
C ( x )
C subject to bl .le. ( A*x ) .le. bu
C ( c(x) )
C
C where f(x) is a vector of smooth functions, A is a constant
C matrix and c(x) is a vector of smooth nonlinear functions.
C The feasible region is defined by a mixture of linear and
C nonlinear equality or inequality constraints on x.
C
C The dimensions of the problem are...
C
C M the number of elements of the vector f(x),
C
C N the number of variables (dimension of x),
C
C NCLIN the number of linear constraints (rows of the matrix A),
C
C NCNLN the number of nonlinear constraints (dimension of c(x)).
C
C
C E04UPF uses a sequential quadratic programming algorithm, with a
C positive-definite quasi-Newton approximation to the transformed
C Hessian Q'HQ of the Lagrangian function (which will be stored in
C the array R).
C
C Version 4.04, June 9, 1989.
C Version 4.06, November 5, 1991. (Matches E04UCF).
C
C Copyright 1989 Optimates.
C ------------------------------------------------------------------
C
SUBROUTINE E04UQF(IOPTNS,INFORM)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C ******************************************************************
C E04UQF reads the options file from unit IOPTNS and loads the
C options into the relevant elements of IPRMNP, RPRMNP, IPRMNL
C and RPRMNL.
C
C If IOPTNS .lt. 0 or IOPTNS .gt. 99 then no file is read,
C otherwise the file associated with unit IOPTNS is read.
C
C Output:
C
C INFORM = 0 if a complete OPTIONS file was found
C (starting with BEGIN and ending with END);
C 1 if IOPTNS .lt. 0 or IOPTNS .gt. 99;
C 2 if BEGIN was found, but end-of-file
C occurred before END was found;
C 3 if end-of-file occurred before BEGIN or
C ENDRUN were found;
C 4 if ENDRUN was found before BEGIN.
C ******************************************************************
C
SUBROUTINE E04URF(STRING)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C ******************************************************************
C E04URF loads the option supplied in STRING into the relevant
C element of IPRMLS, RPRMLS, IPRMNP, RPRMNP, IPRMNL or RPRMNL.
C ******************************************************************
C
SUBROUTINE E04XAF(MSGLVL,N,EPSRF,X,MODE,OBJFUN,LHES,HFORW,FX,GRAD,
* HCNTRL,HESIAN,IWARN,WORK,IUSER,USER,INFO,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C MARK 13B REVISED. IER-659 (AUG 1988).
C MARK 14 REVISED. IER-725 (DEC 1989).
C MARK 16 REVISED. IER-1106 (JUL 1993).
C***********************************************************************
C E04XAF (FDCALC) computes a set of forward-difference intervals
C for the n-variable function objfun at the point x by repeatedly
C calling subroutine E04XAZ (CHCORE).
C
C Full documentation of fdcalc is contained in report SOL 83-6,
C Documentation of FDCORE and FDCALC, by P.E. Gill, W. Murray,
C M.A. Saunders and M.H. Wright, Department of Operations Research,
C Stanford University, Stanford, California 94305, June 1983.
C
C The input parameters of E04XAF are ...
C
C msglvl integer
C -Controls the level of printout.
C msglvl = 0 No printout.
C msglvl = 1 Summary printout for each variable and
C warning messages.
C msglvl = 2 Full debug printout from E04XAF and CHCORE.
C
C n integer
C -The number of variables.
C
C epsrf real
C -A good bound on the relative error in computing fx at x.
C If epsrf is negative on entry or if the users value of
C epsrf is too small or too large then the default value
C of e**0.9 is used, where e is the machine precision.
C
C x real array of dimension at least (n).
C -The point at which the intervals are to be computed.
C
C mode integer
C -If mode = 0 E04XAF returns the gradient vector and
C hessian diagonal given function only.
C If mode = 1 E04XAF returns the hessian matrix given
C function and gradients.
C If mode = 2 E04XAF returns the gradient vector and
C hessian matrix given function only.
C
C objfun subroutine provided by the user.
C -Objfun must calculate the objective function ( and
C gradients if mode = 1 ). Objfun must be declared as
C external in the routine that calls E04XAF.
C Its specification is:
C subroutine objfun( mode, n, x, fx grad, nstate)
C
C lhes integer
C -The first dimension of array hesian as declared in the
C calling program.
C lhes .ge. n
C
C the input/output parameters of E04XAF are ...
C
C hforw real array of dimension at least (n).
C -Before entry, hforw should contain the initial trial
C intervals for:-
C variable x(i) if mode = 0 or mode = 2
C gradient grad(i) if mode = 1
C If hforw(i) .le. 0.0 then the initial trial interval is
C computed by E04XAF.
C On exit hforw contains the forward-difference intervals.
C
C the output parameters of E04XAF are ...
C
C fx real
C -The value of the function at x.
C
C grad real array of dimension at least (n).
C -The approximate gradient vector at x (computed by
C forward-differences with hforw if mode = 0 or 2).
C
C hcntrl real array of dimension at least (n).
C -The central-difference intervals.
C
C hesian real array of dimension (lhes,n) if mode = 1 or mode = 2.
C '' '' '' '' (lhes,1) if mode = 0.
C -If mode = 1 or mode = 2 then the leading n by n part of
C this array will contain the full hessian matrix.
C If mode = 0 then the first n elements of this array will
C contain the hessian diagonals.
C
C iwarn integer
C -On successful exit iwarn = 0.
C If the value of epsrf on entry is too small or too large
C then iwarn is set to 1 or 2 respectively on exit.
C
C the workspace parameters of E04XAF are ...
C
C work real array of dimension at least (n**2 + n) if mode = 1
C mode = 2.
C real array of dimension at least (n) if mode = 0.
C -Used as workspace.
C
C iuser integer array of length at least (1)
C -This array is not used by E04XAF, but is passed directly
C to user supplied routine objfun and may be used to
C supply information to objfun.
C
C user real array of length at least (1)
C -This array is not used by E04XAF, but is passed directly
C to user supplied routine objfun and may be used to
C supply information to objfun.
C
C the diagnostic parameters of E04XAF are ...
C
C info integer array of dimension at least (n)
C -The i-th component indicates the status of the interval
C for variable i (if mode = 0 or 2) or gradient i (if
C mode = 1). info(i) = 0 means that the interval was
C satisfactory.
C
C ifail integer
C -Before entry ifail must be set to 0, -1 or 1 (see
C Chapter p01). The recommended value is -1.
C On successful exit ifail = 0.
C
C Original Fortran 66 Version 2.1 of June 1983. (SOL)
C Fortran 77 version dated 17-May-1985. (SOL)
C Edited to output full hessian matrix. Mark 12. (NAG)
C
C***********************************************************************
SUBROUTINE E04YAF(M,N,LSQFUN,X,FVEC,FJAC,LJ,IW,LIW,W,LW,IFAIL)
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
C
C **************************************************************
C
C E04YAF CHECKS THAT A USER-SUPPLIED ROUTINE FOR EVALUATING A
C VECTOR OF FUNCTIONS AND THE MATRIX OF THEIR FIRST DERIVATIVES
C PRODUCES DERIVATIVE VALUES WHICH ARE CONSISTENT WITH THE
C FUNCTION VALUES CALCULATED.
C
C THIS SUBROUTINE MAY BE USED WHEN THE PARAMETER LIST OF LSQFUN
C IS (IFLAG, M, N, X, FVEC, FJAC, LJ, IW, LIW, W, LW)
C
C PHILIP E. GILL, ENID M. R. LONG, WALTER MURRAY,
C SUSAN M. PICKEN
C D.N.A.C., NATIONAL PHYSICAL LABORATORY, ENGLAND.
C
C **************************************************************
C
C Revised to output explanatory messages.
C Peter Mayes, NAG Central Office, December 1987.
C
SUBROUTINE E04YBF(M,N,LSQFUN,LSQHES,X,FVEC,FJAC,LJ,B,LB,IW,LIW,W,
* LW,IFAIL)
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
C MARK 14A REVISED. IER-683 (DEC 1989).
C
C **************************************************************
C
C E04YBF CHECKS THAT A USER-SUPPLIED ROUTINE FOR EVALUATING
C THE SECOND DERIVATIVE TERM OF THE HESSIAN MATRIX OF A SUM OF
C SQUARES IS CONSISTENT WITH A USER-SUPPLIED ROUTINE FOR
C CALCULATING THE CORRESPONDING FIRST DERIVATIVES.
C
C THIS SUBROUTINE MAY BE USED WHEN THE PARAMETER LIST OF LSQHES
C IS (IFLAG, M, N, LB, FVEC, X, B, IW, LIW, W, LW)
C AND THE PARAMETER LIST OF LSQFUN IS
C (IFLAG, M, N, X, FVEC, FJAC, LJ, IW, LIW, W, LW)
C
C PHILIP E. GILL, ENID M. R. LONG, WALTER MURRAY,
C SUSAN M. PICKEN
C D.N.A.C., NATIONAL PHYSICAL LABORATORY, ENGLAND.
C
C **************************************************************
C
C Revised to output explanatory messages.
C Peter Mayes, NAG Central Office, December 1987.
C
SUBROUTINE E04YCF(JOB,M,N,FSUMSQ,S,V,LV,CJ,WORK,IFAIL)
C MARK 11 RELEASE. NAG COPYRIGHT 1983.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C E04YCF RETURNS ESTIMATES OF ELEMENTS OF THE VARIANCE-COVARIANCE
C MATRIX FOR THE ESTIMATED REGRESSION COEFFICIENTS OF A NONLINEAR
C LEAST SQUARES PROBLEMS.
C
C FOR A DESCRIPTION OF THE PARAMETERS AND USE OF THIS ROUTINE SEE
C THE NAG LIBRARY MANUAL.
C
C -- WRITTEN ON 10-JANUARY-1984. S.J. HAMMARLING.
C
C NAG FORTRAN 66 ROUTINE.
C
SUBROUTINE E04ZCF(N,M,LA,CON,FUN,C,A,F,G,X,W,LW,IFAIL)
C MARK 11 RELEASE. NAG COPYRIGHT 1983.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C MARK 14 REVISED. IER-810 (DEC 1989).
C MARK 15B REVISED. IER-952 (NOV 1991).
C
C THIS VERSION MODIFIED (FROM E04ZAF) FOR USE WITH ROUTINES
C E04VCF AND E04VDF. S.J.HAMMARLING - 25 JULY 1983.
C
C **************************************************************
C
C E04ZCF CHECKS THAT USER-SUPPLIED ROUTINES FOR EVALUATING AN
C OBJECTIVE FUNCTION, CONSTRAINT FUNCTIONS AND THEIR FIRST
C DERIVATIVES PRODUCE DERIVATIVE VALUES WHICH ARE CONSISTENT
C WITH THE FUNCTION AND CONSTRAINT VALUES CALCULATED.
C
C THE ROUTINE IS ESSENTIALLY IDENTICAL TO THE SUBROUTINE CHKNCD
C IN THE NPL ALGORITHMS LIBRARY (REF. NO. E4/66/F). W(I), I = 1,
C 2, . . . , 4*N + M + N*M ARE USED AS WORKSPACE. (NOTE THAT,
C FOR CONSISTENCY WITH OTHER E04 DOCUMENTATION, THE NAME FUNCT
C IS USED INSTEAD OF SFUN IN THE WRITE-UP.)
C
C PHILIP E. GILL, ENID M. R. LONG, WALTER MURRAY AND SUSAN M.
C PICKEN, D.N.A.C., NATIONAL PHYSICAL LABORATORY, ENGLAND.
C
C **************************************************************
C
C CON, FUN
C
C A MACHINE DEPENDENT CONSTANT IS SET HERE. EPSMCH IS THE
C SMALLEST POSITIVE REAL NUMBER SUCH THAT 1.0 + EPSMCH .GT. 1.0.
C
SUBROUTINE F01ABF(A,IA,N,B,IB,Z,IFAIL)
C MARK 2 RELEASE. NAG COPYRIGHT 1972
C MARK 4 REVISED.
C MARK 4.5 REVISED
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C MARK 16 REVISED. IER-1001 (JUN 1993).
C
C ACCURATE INVERSE OF A REAL SYMMETRIC POSITIVE DEFINITE
C MATRIX USING CHOLESKYS METHOD, SIMPLIFIED PARAMETERS.
C 1ST AUGUST 1971
C
SUBROUTINE F01ADF(N,A,IA,IFAIL)
C MARK 15 RE-ISSUE. NAG COPYRIGHT 1991.
C
C CHOLINVERSION1
C The upper triangle of a positive definite symmetric matrix,
C A, is stored in the upper triangle of an (N+1)*N array
C A(I,J), I=1,N+1, J=1,N. The Cholesky decomposition A=LU,
C where U is the transpose of L , is performed and L is stored
C in the remainder of the array A. The reciprocals of the
C diagonal elements are stored instead of the elements them-
C selves. L is then replaced by its inverse and this in turn
C is replaced by the lower triangle of the inverse of A. A is
C retained so that the inverse can be subsequently improved.
C The subroutine will fail if A, modified by the rounding
C errors, is not positive definite.
C 1st December 1971
C
C Rewritten to call LAPACK routines SPOTRF/F07FDF and
C SPOTRI/F07FJF; new IFAIL exit for illegal input
C parameters inserted; error messages inserted (February 1991).
C
SUBROUTINE F01BLF(M,N,T,A,IA,AIJMAX,IRANK,INC,D,U,IU,DU,IFAIL)
C NAG COPYRIGHT 1976. MARK 5 RELEASE.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C PDK, DNAC, NPL, TEDDINGTON. NOVEMBER 1975.
C NPL DNAC LIBRARY SUBROUTINE QUINV.
C
C DETERMINE THE RANK AND THE PSEUDO-INVERSE OF AN M*N
C MATRIX A, WHERE M.GE.N. HOUSEHOLDERS FACTORISATION
C INVOLVING ORTHOGONAL MATRICES IS USED IN THE
C DECOMPOSITION A=QU. A CRITERION T IS USED TO
C DECIDE WHEN ELEMENTS CAN BE REGARDED AS ZERO IN
C THE DETERMINATION OF RANK. THE ELEMENT OF LARGEST
C MODULUS IN THE CURRENT MATRIX AT EACH STAGE OF THE
C REDUCTION IS RECORDED IN THE VECTOR AIJMAX. THE
C TRANSPOSE OF THE PSEUDO-INVERSE IS OVERWRITTEN ON
C A. A FAILURE EXIT WILL BE DUE TO AN INCORRECT
C CHOICE OF T OR M LESS THAN N. USES AUXILIARY SUBROUTINES
C F01BLZ AND F01BLY.
C
SUBROUTINE F01BRF(N,NZ,A,LICN,IRN,LIRN,ICN,U,IKEEP,IW,W,LBLOCK,
* GROW,ABORT,IDISP,IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C DERIVED FROM HARWELL LIBRARY ROUTINE MA28A
C
C THE PARAMETERS ARE AS FOLLOWS.....
C N INTEGER ORDER OF MATRIX NOT ALTERED BY SUBROUTINE.
C NZ INTEGER NUMBER OF NON-ZEROS IN INPUT MATRIX NOT
C . ALTERED BY SUBROUTINE.
C A REAL ARRAY LENGTH LICN. HOLDS NON-ZEROS OF
C . MATRIX ON ENTRY AND NON-ZEROS OF FACTORS ON EXIT.
C . REORDERED BY F01BRZ AND F01BRY AND ALTERED BY F01BRT.
C LICN INTEGER LENGTH OF ARRAYS A AND ICN. NOT ALTERED BY
C . SUBROUTINE.
C IRN INTEGER ARRAY LENGTH LIRN. HOLDS ROW INDICES ON INPUT
C . AND USED AS WORKSPACE BY F01BRT TO HOLD COLUMN
C . ORIENTATION OF MATRIX.
C LIRN INTEGER LENGTH OF ARRAY IRN.
C ICN INTEGER ARRAY LENGTH LICN. HOLDS COLUMN INDICES ON
C . ENTRY AND COLUMN INDICES OF DECOMPOSED MATRIX ON EXIT.
C . REORDERED BY F01BRZ AND F01BRY AND ALTERED BY F01BRT.
C U REAL VARIABLE SET BY USER TO CONTROL BIAS TOWARDS
C . NUMERIC OR SPARSITY PIVOTING. U=1.0 GIVES PARTIAL
C . PIVOTING WHILE U=0. DOES NOT CHECK MULTIPLIERS AT ALL.
C . VALUES OF U GREATER THAN ONE ARE TREATED AS ONE WHILE
C . NEGATIVE VALUES ARE TREATED AS ZERO. NOT ALTERED BY
C . SUBROUTINE.
C IKEEP INTEGER ARRAY LENGTH 5*N USED AS WORKSPACE BY F01BRF
C . (SEE LATER COMMENTS). IT IS NOT REQUIRED TO BE SET ON
C . ENTRY AND, ON EXIT, IT CONTAINS INFORMATION ABOUT THE
C . DECOMPOSITION. IT SHOULD BE PRESERVED BETWEEN THIS CALL
C . AND SUBSEQUENT CALLS TO F01BSF OR F04AXF.
C IKEEP(I,1),I=1,N HOLDS THE TOTAL LENGTH OF THE PART OF ROW I
C . IN THE DIAGONAL BLOCK.
C ROW IKEEP(I,2),I=1,N OF THE INPUT MATRIX IS THE ITH ROW IN
C . PIVOT ORDER.
C COLUMN IKEEP(I,3),I=1,N OF THE INPUT MATRIX IS THE ITH
C . COLUMN IN PIVOT ORDER.
C IKEEP(I,4),I=1,N HOLDS THE LENGTH OF THE PART OF ROW I IN
C . THE L PART OF THE L/U DECOMPOSITION.
C IKEEP(I,5),I=1,N HOLDS THE LENGTH OF THE PART OF ROW I IN
C . THE OFF-DIAGONAL BLOCKS. IF THERE IS ONLY ONE DIAGONAL
C . BLOCK, IKEEP(1,5) WILL BE SET TO -1.
C IW INTEGER ARRAY LENGTH 8*N.
C . USED AS WORKSPACE.
C W REAL ARRAY LENGTH N. USED BY F01BRQ BOTH AS WORKSPACE
C . AND TO RETURN GROWTH ESTIMATE IN W(1). THE USE OF THIS
C . ARRAY BY F01BRF IS THUS OPTIONAL DEPENDING ON THE VALUE
C . OF PARAMETER GROW.
C LBLOCK LOGICAL IF TRUE, F01BRY IS USED TO FIRST PERMUTE
C . THE MATRIX TO BLOCK LOWER TRIANGULAR FORM.
C GROW LOGICAL IF TRUE, THEN AN ESTIMATE OF THE INCREASE
C . IN SIZE OF MATRIX ELEMENTS DURING L/U DECOMPOSITION IS
C . GIVEN BY F01BRQ IN W(1).
C ABORT LOGICAL ARRAY LENGTH 4.
C IF ABORT(1)=TRUE, THE ROUTINE WILL EXIT IMMEDIATELY
C . ON DETECTING STRUCTURAL SINGULARITY.
C IF ABORT(2)=TRUE, THE ROUTINE WILL EXIT IMMEDIATELY
C . ON DETECTING NUMERICAL SINGULARITY.
C IF ABORT(3)=TRUE, THE ROUTINE WILL EXIT IMMEDIATELY WHEN THE
C . AVAILABLE SPACE IN A/ICN IS FILLED UP BY THE PREVIOUSLY
C . DECOMPOSED ACTIVE, AND UNDECOMPOSED PARTS OF THE MATRIX.
C IF ABORT(4)=TRUE, THE ROUTINE WILL EXIT IMMEDIATELY
C . ON DETECTING DUPLICATE ELEMENTS IN THE INPUT MATRIX.
C IDISP INTEGER ARRAY LENGTH 10. USED TO COMMUNICATE
C . INFORMATION ABOUT THE DECOMPOSITION TO THE USER AND
C . ALSO BETWEEN THIS CALL TO F01BRF AND SUBSEQUENT CALLS
C . TO F01BSF AND F04AXF.
C ON EXIT -
C IDISP(1) AND IDISP(2) INDICATE THE POSITION IN ARRAYS A AND
C . ICN OF THE FIRST AND LAST ELEMENTS IN THE L/U
C . DECOMPOSITION OF THE DIAGONAL BLOCKS RESPECTIVELY.
C IDISP(3)= IRNCP THE NUMBER OF COMPRESSES ON ARRAY IRN.
C IDISP(4)= ICNCP THE NUMBER OF COMPRESSES ON ARRAYS ICN/A.
C IDISP(5)= IRANK ESTIMATED RANK OF THE MATRIX.
C IDISP(6)= MINIRN MINIMUM LENGTH OF ARRAY IRN FOR SUCCESS ON
C . FUTURE RUNS.
C IDISP(7)= MINICN MINIMUM LENGTH OF ARRAYS ICN/A FOR SUCCESS
C . ON FUTURE RUNS.
C IDISP(8)= NUMNZ STRUCTURAL RANK OF MATRIX.
C IDISP(9)= NUM NUMBER OF DIAGONAL BLOCKS.
C IDISP(10)=LARGE SIZE OF LARGEST DIAGONAL BLOCK.
C IFAIL INTEGER VARIABLE USED AS ERROR FLAG BY ROUTINE.
C
C INTERNAL VARIABLES AND WORKSPACE USED IN F01BRF ARE DEFINED
C WITHIN THE SUBROUTINE IMMEDIATELY PRIOR TO THEIR FIRST USE.
SUBROUTINE F01BSF(N,NZ,A,LICN,IVECT,JVECT,ICN,IKEEP,IW,W,GROW,EPS,
* RMIN,ABORT,IDISP,IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C DERIVED FROM HARWELL LIBRARY ROUTINE MA28B
C
C DECOMPOSES A REAL SPARSE MATRIX USING THE PIVOTAL SEQUENCE
C PREVIOUSLY OBTAINED BY F01BRF WHEN A MATRIX OF THE SAME
C SPARSITY PATTERN WAS DECOMPOSED.
C
C THE PARAMETERS ARE AS FOLLOWS ...
C N INTEGER ORDER OF MATRIX NOT ALTERED BY SUBROUTINE.
C NZ INTEGER NUMBER OF NON-ZEROS IN INPUT MATRIX NOT
C . ALTERED BY SUBROUTINE.
C A REAL ARRAY LENGTH LICN. HOLDS NON-ZEROS OF MATRIX ON
C . ENTRY AND NON-ZEROS OF FACTORS ON EXIT. REORDERED
C . BY F01BSZ AND ALTERED BY SUBROUTINE F01BSY.
C LICN INTEGER LENGTH OF ARRAYS A AND ICN. NOT ALTERED BY
C . SUBROUTINE.
C IVECT,JVECT INTEGER ARRAYS LENGTH NZ. HOLD ROW AND COLUMN
C . INDICES OF NON-ZEROS RESPECTIVELY. NOT ALTERED BY
C . SUBROUTINE.
C ICN INTEGER ARRAY LENGTH LICN. SAME ARRAY AS OUTPUT FROM
C . F01BRF. UNCHANGED BY F01BSF.
C IKEEP INTEGER ARRAY LENGTH 5*N. SAME ARRAY AS OUTPUT FROM
C . F01BRF. UNCHANGED BY F01BSF.
C IW INTEGER ARRAY LENGTH 5*N.
C . USED AS WORKSPACE BY F01BSZ AND F01BSY.
C W REAL ARRAY LENGTH N. USED AS WORKSPACE
C . BY F01BSZ, F01BSY AND (OPTIONALLY) F01BRQ.
C GROW LOGICAL. IF TRUE, AN ESTIMATE OF THE INCREASE
C . IN SIZE OF ARRAY ELEMENTS DURING L/U DECOMPOSITION
C . IS GIVEN BY F01BRQ IN W(1).
C EPS REAL. USED TO TEST FOR SMALL PIVOTS. IF THE USER SETS
C . EPS.GT.1.0, NO CHECK IS MADE ON THE SIZE OF THE PIVOTS.
C RMIN REAL. GIVES THE USER SOME INFORMATION ABOUT THE
C . STABILITY OF THE DECOMPOSITION.
C ABORT LOGICAL. IF ABORT=TRUE, THE ROUTINE WILL EXIT
C . IMMEDIATELY ON DETECTING DUPLICATE ELEMENTS IN THE
C . INPUT MATRIX.
C IDISP INTEGER ARRAY LENGTH 2. IDISP(1) AND (2) MUST HAVE THE
C . SAME VALUES AS OUTPUT FROM F01BRF. UNCHANGED BY THE
C . ROUTINE.
C IFAIL INTEGER. USED AS ERROR FLAG BY THE ROUTINE.
C
SUBROUTINE F01BUF(N,M1,K,A,IA,W,IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C U*L*D*L**T*U**T DECOMPOSITION OF AN N*N POSITIVE DEFINITE
C MATRIX A OF BAND WIDTH 2*M+1 WITH M1=M+1. K DETERMINES
C THE SWITCHOVER POINT FROM U TO L AND MUST BE SUCH THAT
C K.LE.N, K.GE.M. IF K=N THEN AN L*D*L**T DECOMPOSITION IS
C OBTAINED. THE UPPER TRIANGULAR PART OF A IS
C PRESENTED AS AN M1*N ARRAY A WITH THE DIAGONAL
C ELEMENTS OF A IN THE M1-TH ROW. THE SPARE
C LOCATIONS IN THE TOP LEFT ARE NOT USED. U, L AND D
C ARE OVERWRITTEN ON A.
C D(I) OVERWRITES A(I,I), L(I,J) OVERWRITES A(J,I) AND U(I,J)
C OVERWRITES A(I,J), THE ELEMENT OF A(J,I) BEING STORED IN
C POSITION (J+M+1-I,I) OF ARRAY A.
C THE ALGORITHM WILL FAIL IF ANY COMPUTED ELEMENT OF D IS SUCH
C THAT D(I).LE.0.0.
C AN ADDITIONAL WORK SPACE OF M1 ELEMENTS IS USED.
C
C PDK, DNAC, NPL, TEDDINGTON. DEC 1977.
C NPL DNAC LIBRARY SUBROUTINE ULBAND.
C
SUBROUTINE F01BVF(N,MA1,MB1,M3,K,A,IA,B,IB,C,IC,W,IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C REDUCTION OF THE GENERAL SYMMETRIC EIGENPROBLEM
C A*X=LAMBDA*B*X WITH B POSITIVE DEFINITE, WHERE A AND B
C ARE BAND MATRICES OF ORDER N AND OF BANDWIDTHS 2*MA+1
C AND 2*MB+1 RESPECTIVELY WITH MA1=MA+1 AND
C MB1=MB+1, MA.GE.MB, TO THE EQUIVALENT
C STANDARD SYMMETRIC PROBLEM C*Z=LAMBDA*Z, WHERE C IS A
C BAND MATRIX OF WIDTH 2*MA+1. THE UPPER TRIANGLES OF
C A AND B ARE GIVEN IN THE ARRAYS A(MA1,N) AND
C B(MB1,N) WITH THEIR DIAGONAL ELEMENTS IN THE FINAL
C ROWS. THE MATRIX B MUST HAVE BEEN FIRSTLY
C DECOMPOSED USING THE SUBROUTINE F01BUF WITH K AS
C THE SWITCHOVER POINT FROM U TO L. THE UPPER TRIANGLE
C OF C OVERWRITES THE ELEMENTS OF ARRAY A, SO THE
C ORIGINAL A IS LOST. AN ADDITIONAL WORK SPACE OF
C (MA+MB+1)*(3*MA+MB) ELEMENTS IS USED.
C
C PDK, DNAC, NPL, TEDDINGTON. DEC 1977.
C NPL DNAC LIBRARY SUBROUTINE ULTRAN.
C
SUBROUTINE F01CKF(A,B,C,N,P,M,Z,IZ,OPT,IFAIL)
C MARK 14 RE-ISSUE. NAG COPYRIGHT 1989.
C
C RETURNS WITH THE RESULT OF THE MATRIX MULTIPLICATION OF B AND
C C
C IN THE MATRIX A, WITH THE OPTION TO OVERWRITE B OR C.
C **************************************************************
C ****
C MATRIX MULTIPLICATION
C A=B*C IF OPT= 1 A,B,C ASSUMED DISTINCT
C IF OPT=2 B IS OVERWRITTEN
C IF OPT=3 C IS OVERWRITTEN
C **************************************************************
C ****
SUBROUTINE F01CRF(A,M,N,MN,MOVE,IWRK,IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 14A REVISED. IER-684 (DEC 1989).
C *****
C ALGORITHM 380 - REVISED
C *****
C A IS A ONE-DIMENSIONAL ARRAY OF LENGTH MN = M*N, WHICH
C CONTAINS THE M X N MATRIX TO BE F01CRFPOSED
C (STORED COLUMNWISE).
C MOVE IS A ONE-DIMENSIONAL ARRAY OF LENGTH IWRK USED
C TO STORE INFORMATION TO SPEED UP THE PROCESS. THE
C VALUE IWRK = (M+N)/2 IS RECOMMENDED.
C IFAIL INDICATES THE SUCCESS OR FAILURE OF THE ROUTINE.
C NORMAL RETURN IFAIL = 0
C ERRORS IFAIL = 1, MN NOT EQUAL TO M*N
C IFAIL = 2, IWRK NEGATIVE OR ZERO
C IFAIL.LT.ZERO, (SHOULD NEVER OCCUR),
C IN THIS CASE WE SET IFAIL EQUAL TO THE FINAL VALUE OF I
C WHEN THE SEARCH IS COMPLETED BUT SOME LOOPS HAVE NOT BEEN
C MOVED NOTE * MOVE(I) WILL STAY ZERO FOR FIXED POINTS.
C CHECK ARGUMENTS AND INITIALIZE.
SUBROUTINE F01CTF(TRANSA,TRANSB,M,N,ALPHA,A,LDA,BETA,B,LDB,C,LDC,
* IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C MARK 14C REVISED. IER-883 (NOV 1990).
C
C Purpose
C =======
C
C F01CTF performs one of the matrix-matrix operations
C
C C := alpha*op( A ) + beta*op( B ),
C
C where op( X ) is one of
C
C op( X ) = X or op( X ) = X',
C
C alpha and beta are scalars, and A, B and C are matrices, with op( A )
C an m by n matrix, op( B ) an m by n matrix, and C an m by n matrix.
C
C Parameters
C ==========
C
C TRANSA - CHARACTER*1.
C On entry, TRANSA specifies the form of op( A ) to be used in
C the matrix addition as follows:
C
C TRANSA = 'N' or 'n', op( A ) = A.
C
C TRANSA = 'T' or 't', op( A ) = A'.
C
C TRANSA = 'C' or 'c', op( A ) = A'.
C
C Unchanged on exit.
C
C TRANSB - CHARACTER*1.
C On entry, TRANSB specifies the form of op( B ) to be used in
C the matrix addition as follows:
C
C TRANSB = 'N' or 'n', op( B ) = B.
C
C TRANSB = 'T' or 't', op( B ) = B'.
C
C TRANSB = 'C' or 'c', op( B ) = B'.
C
C Unchanged on exit.
C
C M - INTEGER.
C On entry, M specifies the number of rows of the matrices
C op( A ) and op( B ). M must be at least zero.
C Unchanged on exit.
C
C N - INTEGER.
C On entry, N specifies the number of columns of the matrices
C op( A ) and op( B ). N must be at least zero.
C Unchanged on exit.
C
C ALPHA - REAL .
C On entry, ALPHA specifies the scalar alpha. When ALPHA is
C supplied as zero then A need not be set on entry.
C Unchanged on exit.
C
C A - REAL array of DIMENSION ( LDA, ma ), where ma is
C m when TRANSA = 'N' or 'n', and is n otherwise.
C If alpha is zero, then A need not be set on entry.
C If alpha is not zero, then before entry with TRANSA = 'N' or
C 'n', the leading m by n part of the array A must contain the
C matrix A, otherwise the leading n by m part of the array A
C must contain the matrix A.
C Unchanged on exit.
C
C LDA - INTEGER.
C On entry, LDA specifies the first dimension of A as declared
C in the calling (sub) program. When TRANSA = 'N' or 'n' then
C LDA must be at least max( 1, m ), otherwise LDA must be at
C least max( 1, n ).
C Unchanged on exit.
C
C BETA - REAL .
C On entry, BETA specifies the scalar beta. When BETA is
C supplied as zero then B need not be set on entry.
C Unchanged on exit.
C
C B - REAL array of DIMENSION ( LDB, mb ), where mb is
C m when TRANSB = 'N' or 'n', and is n otherwise.
C If beta is zero, then B need not be set on entry.
C If beta is not zero, then before entry with TRANSB = 'N' or
C 'n', the leading m by n part of the array B must contain the
C matrix B, otherwise the leading n by m part of the array B
C must contain the matrix B.
C Unchanged on exit.
C
C LDB - INTEGER.
C On entry, LDB specifies the first dimension of B as declared
C in the calling (sub) program. When TRANSB = 'N' or 'n' then
C LDB must be at least max( 1, m ), otherwise LDB must be at
C least max( 1, n ).
C Unchanged on exit.
C
C C - REAL array of DIMENSION ( LDC, N ).
C On exit, the array C is overwritten by the m by n matrix
C alpha*op( A ) + beta*op( B ).
C
C LDC - INTEGER.
C On entry, LDC specifies the first dimension of C as declared
C in the calling (sub) program. LDC must be at least
C max( 1, m ).
C Unchanged on exit.
C
C
SUBROUTINE F01CWF(TRANSA,TRANSB,M,N,ALPHA,A,LDA,BETA,B,LDB,C,LDC,
* IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C MARK 15A REVISED. IER-909 (APR 1991).
C
C Purpose
C =======
C
C F01CWF performs one of the matrix-matrix operations
C
C C := alpha*op( A ) + beta*op( B ),
C
C where op( X ) is one of
C
C op( X ) = X or op( X ) = X',
C
C alpha and beta are scalars, and A, B and C are matrices, with op( A )
C an m by n matrix, op( B ) an m by n matrix, and C an m by n matrix.
C
C Parameters
C ==========
C
C TRANSA - CHARACTER*1.
C On entry, TRANSA specifies the form of op( A ) to be used in
C the matrix addition as follows:
C
C TRANSA = 'N' or 'n', op( A ) = A.
C
C TRANSA = 'T' or 't', op( A ) = A'.
C
C TRANSA = 'C' or 'c', op( A ) = conjg(A').
C
C Unchanged on exit.
C
C TRANSB - CHARACTER*1.
C On entry, TRANSB specifies the form of op( B ) to be used in
C the matrix addition as follows:
C
C TRANSB = 'N' or 'n', op( B ) = B.
C
C TRANSB = 'T' or 't', op( B ) = B'.
C
C TRANSB = 'C' or 'c', op( B ) = conjg(B').
C
C Unchanged on exit.
C
C M - INTEGER.
C On entry, M specifies the number of rows of the matrices
C op( A ) and op( B ). M must be at least zero.
C Unchanged on exit.
C
C N - INTEGER.
C On entry, N specifies the number of columns of the matrices
C op( A ) and op( B ). N must be at least zero.
C Unchanged on exit.
C
C ALPHA - COMPLEX .
C On entry, ALPHA specifies the scalar alpha. When ALPHA is
C supplied as zero then A need not be set on entry.
C Unchanged on exit.
C
C A - COMPLEX array of DIMENSION ( LDA, ma ), where ma is
C m when TRANSA = 'N' or 'n', and is n otherwise.
C If alpha is zero, then A need not be set on entry.
C If alpha is not zero, then before entry with TRANSA = 'N' or
C 'n', the leading m by n part of the array A must contain the
C matrix A, otherwise the leading n by m part of the array A
C must contain the matrix A.
C Unchanged on exit.
C
C LDA - INTEGER.
C On entry, LDA specifies the first dimension of A as declared
C in the calling (sub) program. When TRANSA = 'N' or 'n' then
C LDA must be at least max( 1, m ), otherwise LDA must be at
C least max( 1, n ).
C Unchanged on exit.
C
C BETA - COMPLEX .
C On entry, BETA specifies the scalar beta. When BETA is
C supplied as zero then B need not be set on entry.
C Unchanged on exit.
C
C B - COMPLEX array of DIMENSION ( LDB, mb ), where mb is
C m when TRANSB = 'N' or 'n', and is n otherwise.
C If beta is zero, then B need not be set on entry.
C If beta is not zero, then before entry with TRANSB = 'N' or
C 'n', the leading m by n part of the array B must contain the
C matrix B, otherwise the leading n by m part of the array B
C must contain the matrix B.
C Unchanged on exit.
C
C LDB - INTEGER.
C On entry, LDB specifies the first dimension of B as declared
C in the calling (sub) program. When TRANSB = 'N' or 'n' then
C LDB must be at least max( 1, m ), otherwise LDB must be at
C least max( 1, n ).
C Unchanged on exit.
C
C C - COMPLEX array of DIMENSION ( LDC, N ).
C On exit, the array C is overwritten by the m by n matrix
C alpha*op( A ) + beta*op( B ).
C
C LDC - INTEGER.
C On entry, LDC specifies the first dimension of C as declared
C in the calling (sub) program. LDC must be at least
C max( 1, m ).
C Unchanged on exit.
C
C
SUBROUTINE F01LEF(N,A,LAMBDA,B,C,TOL,D,IN,IFAIL)
C MARK 11 RELEASE. NAG COPYRIGHT 1983.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 14 REVISED. IER-728 (DEC 1989).
C
C F01LEF FACTORIZES THE MATRIX ( T - LAMBDA*I ), WHERE T IS AN
C N BY N TRIDIAGONAL MATRIX AND LAMBDA IS A SCALAR, AS
C
C T - LAMBDA*I = P*L*U ,
C
C WHERE P IS A PERMUTATION MATRIX, L IS A UNIT LOWER TRIANGULAR
C MATRIX WITH AT MOST ONE NON-ZERO SUB-DIAGONAL ELEMENT PER
C COLUMN AND U IS AN UPPER TRIANGULAR MATRIX WITH TWO NON-ZERO
C SUPER-DIAGONALS.
C
C FOR A DESCRIPTION OF THE PARAMETERS AND USE OF THIS ROUTINE SEE
C THE NAG LIBRARY MANUAL.
C
C -- WRITTEN ON 11-JANUARY-1983. S.J.HAMMARLING.
C
C NAG FORTRAN 66 GENERAL PURPOSE ROUTINE.
C
SUBROUTINE F01LHF(N,NBLOKS,BLKSTR,A,LENA,PIVOT,TOL,INDEX,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C MARK 13B REVISED. IER-661 (AUG 1988).
C MARK 14C REVISED. IER-884 (NOV 1990).
C
C Purpose:
C
C F01LHF decomposes a real almost block diagonal matrix.
C
C
C Specification:
C
C SUBROUTINE F01LHF(N,NBLOKS,BLKSTR,A,LENA,PIVOT,
C * TOL,INDEX,IFAIL)
C C INTEGER N,NBLOKS,BLKSTR(3,NBLOKS),LENA,PIVOT(N),
C C * INDEX,IFAIL
C C real A(LENA),TOL
C
C
C Description:
C
C The routine factorizes a real almost block diagonal matrix, A, by
C column elimination with alternate row and column pivoting such
C that no 'fill-in' is produced. The code, which is derived from
C ARCECO desribed in [1], uses Level 2 BLAS [2].
C No three successive diagonal blocks may have columns in
C common and therefore the almost block diagonal matrix must have
C the form shown in the following diagram.
C
C .------|
C | . |
C 1 |___.__|
C | .-----------|
C | . |
C | . |
C | . |
C 2 |___________.___|
C | .-------|
C | . |
C 3 |_________.___|
C .
C .
C .
C .
C .
C |-----.-----------|
C | . |
C | . |
C | . |
C NBLOKS-1 |_____________.___|
C | .---|
C | . |
C NBLOKS |_____________|
C
C
C References:
C
C [1] FORTRAN Packages for Solving Certain Almost Block Diagonal
C Linear Systems by Modified Alternate Row and Column Elimination,
C J.C.Diaz, G.Fairweather, P.Keast,
C ACM Transactions on Mathematical Software (1983) 9, 358-375
C
C [2] An Extended Set of Fortran Basic Linear Algebra Subprograms
C J.J.Dongarra, J.DuCroz, S.Hammarling, R.J.Hanson
C NAG Technical Report (1987) TR3/87
C
C
C Parameters:
C
C N - INTEGER
C On entry, N must specify the number of rows of the
C matrix A.
C N .gt. 0.
C Unchanged on exit.
C
C NBLOKS - INTEGER
C On entry, NBLOKS must specify the total number of blocks
C of the matrix A.
C 0 .lt. NBLOKS .le. N.
C Unchanged on exit.
C
C BLKSTR - INTEGER array of DIMENSION (3,NBLOKS)
C Before entry, BLKSTR must contain information which
C describes the block structure of A as follows:
C BLKSTR(1,k) must contain the number of rows in the kth
C block, k=1,2,..,NBLOKS;
C BLKSTR(2,k) must contain the number of columns in the kth
C block, k=1,2,..,NBLOKS;
C BLKSTR(3,k) must contain the number of columns of overlap
C between the kth and (k+1)th blocks,
C k=1,2,..,NBLOKS-1.
C BLKSTR(3,NBLOKS) need not be set.
C
C The following conditions delimit the structure of A.
C BLKSTR(1,k),BLKSTR(2,k) .gt. 0 k=1,2,..,NBLOKS
C BLKSTR(3,k) .ge. 0 k=1,2,..,NBLOKS
C (there must be at least one column and one row in each block
C and a non-negative number of columns of overlap)
C
C BLKSTR(3,k-1)+BLKSTR(3,k) .le.
C BLKSTR(2,k) k=2,3,..,NBLOKS-1
C (the total number of columns in overlaps in each block must
C not exceed the number of columns in that block)
C
C BLKSTR(2,1) .ge. BLKSTR(1,1)
C BLKSTR(2,1)+sum(BLKSTR(2,k)-BLKSTR(3,k-1);k=2,j) .ge.
C sum(BLKSTR(1,k);k=1,j) j=2,3,..,NBLOKS-1
C sum(BLKSTR(2,k)-BLKSTR(3,k);k=1,j) .le.
C sum(BLKSTR(1,k);k=1,j) j=1,2,..,NBLOKS-1
C (the index of the first column of the overlap between the
C jth and (j+1)th blocks must be .le. the index of the last
C row of the jth block, and the index of the last column of
C overlap must be .ge. the index of the last row of the jth
C block)
C
C sum(BLKSTR(1,k);k=1,NBLOKS) .eq. N
C BLKSTR(2,1)+sum(BLKSTR(2,k)-BLKSTR(3,k-1);k=2,NBLOKS) .eq. N
C (both the number of rows and of columns of A must equal N)
C
C Unchanged on exit.
C
C A - real array of DIMENSION (LENA)
C Before entry, A must contain the elements of the real almost
C block diagonal matrix stored block by block and each block
C stored column by column. The sizes of the blocks and the
C overlaps are defined by the parameter BLKSTR.
C If the first element of the kth block is in position (m,n)
C of the matrix A, then the (i,j)th element of the matrix A,
C if it is in the kth block, should be stored in location
C A(nbasek + (j+1-n)*nrowsk + (i+1-m))
C where
C nbasek = sum(BLKSTR(1,l)*BLKSTR(2,l);l=1,k-1)
C is the base address of the kth block, and
C nrowsk = BLKSTR(1,k)
C which is the number of rows of the kth block.
C On succesful exit, A contains the decomposed form of the
C matrix.
C
C LENA - INTEGER
C On entry, LENA must specify the dimension of array A as
C declared in the (sub)program from which F01LHF is called.
C LENA .ge. sum(BLKSTR(1,k)*BLKSTR(2,k);k=1,NBLOKS).
C Unchanged on exit.
C
C TOL - real
C Before entry, TOL must specify a relative tolerance to be
C used to indicate whether or not the matrix is singular. See
C 'Further Comments' on how TOL is used. If TOL is non-
C positive then TOL is reset to 10*eps, where eps is the
C relative machine precision (see NAG Library routine X02AJF).
C
C INDEX - INTEGER
C On succesful exit with IFAIL = 0, or on exit with
C IFAIL = -1, INDEX contains the value 0. On exit with
C IFAIL = 1 INDEX contains the value k, where k is the
C first position on the diagonal of the matrix A where too
C small a pivot or a zero pivot was detected.
C
C PIVOT - INTEGER array of DIMENSION (N)
C On exit, PIVOT details the interchanges.
C
C IFAIL - INTEGER
C
C Before entry, IFAIL must be set to 0, -1 or 1. For users
C not familiar with this parameter (described in chapter P01),
C the recommended value is 0.
C Unless the routine detects an error (see Section 6), IFAIL
C contains 0 on exit.
C
C
C Error Indicators:
C
C Errors detected by the routine:
C
C If on entry IFAIL = 0 or -1, explanatory error messages are
C output on the current error message unit (as defined by NAG
C Library routine X04AAF).
C
C IFAIL = -1 N .lt. 1 or
C NBLOKS .lt. 1 or
C N .lt. NBLOKS or
C LENA too small or
C illegal values detected in BLKSTR.
C
C IFAIL = 1 Decomposition complete but a small or zero pivot has
C been detected.
C
C
C Further Comments:
C
C Singularity or near singularity in A is determined by the
C parameter TOL. If the absolute value of any pivot is less than
C TOL * the maximum absolute element of A then A is said to be
C singular. The position on the diagonal of A of the first of any
C such pivots is indicated by the parameter INDEX.
C This routine may be followed by the routine F04LHF, which is
C designed to solve sets of linear equations AX = B. In general,
C more accurate solutions can be obtained by routine SUBSTB, but
C are more expensive in terms of arithmetic operations and require
C more storage.
C
C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C
C AUXILIARY ROUTINES
C ------------------
C F01LHY: TO PERFORM THE COLUMN ELIMINATIONS WITH ROW PIVOTING
C F01LHX: TO PERFORM THE COLUMN ELIMINATIONS WITH COLUMN PIVOTING
C F01LHZ: TO PERFORM CHECKS ON THE PARAMETER BLKSTR
C IDAMAX: TO FIND INDEX OF LARGEST ABSOLUTE ELEMEMT OF A VECTOR
C
C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C
SUBROUTINE F01MAF(N,NZ,A,LA,INI,LINI,INJ,DROPTL,DENSW,W,IK,IW,
* ABORT,INFORM,IFLAG)
C MARK 11 RELEASE. NAG COPYRIGHT 1983.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C F01MAF PERFORMS AN INCOMPLETE CHOLESKY FACTORIZATION OF
C A SPARSE SYMMETRIC POSITIVE DEFINITE MATRIX A.
C
C THE ROUTINE IS BASED UPON THE HARWELL ROUTINE MA31A BY
C N MUNKSGAARD.
C
C THE DEVELOPMENT OF MA31A WAS SUPPORTED BY THE DANISH
C NATURAL RESEARCH COUNCIL, GRANT NUMBER 511-10069
C
C FOR A DESCRIPTION OF THE PARAMETERS AND USE OF THIS ROUTINE
C SEE THE NAG LIBRARY MANUAL. PARAMETER ASSOCIATION IS AS BELOW.
C
C ROUTINE DOCUMENT
C N N
C NZ NZ
C A A
C LA LICN
C INI IRN
C LINI LIRN
C INJ ICN
C DROPTL DROPTL
C DENSW DENSW
C INFORM INFORM
C W WKEEP
C IK IKEEP
C IW IWORK
C ABORT ABORT
C IFLAG IFAIL
C
C ARRAYS MA31I, MA31J AND MA31K CORRESPOND TO COMMON BLOCKS
C OF THE SAME NAMES IN MA31A.
C
SUBROUTINE F01MCF(N,A,LAL,NROW,L,D,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 14 REVISED. IER-729 (DEC 1989).
C
C ******************************************************
C
C NPL ALGORITHMS LIBRARY ROUTINE VBCHFC
C
C CREATED 04 05 79. UPDATED 07 09 79. RELEASE 00/22
C
C AUTHOR ... MAURICE G. COX.
C NATIONAL PHYSICAL LABORATORY, TEDDINGTON,
C MIDDLESEX TW11 OLW, ENGLAND.
C
C ******************************************************
C
C F01MCF. AN ALGORITHM TO DETERMINE THE CHOLESKY
C FACTORIZATION A = LDU (WHERE L IS UNIT TRIANGULAR,
C D IS DIAGONAL AND U = L TRANSPOSED) OF A SYMMETRIC
C POSITIVE DEFINITE MATRIX A HAVING A VARIABLE BANDWIDTH.
C
C INPUT PARAMETERS
C N ORDER OF A
C A ELEMENTS WITHIN ENVELOPE OF A,
C IN ROW BY ROW ORDER
C LAL SMALLER OF THE DIMENSIONS OF A AND L. AT LEAST
C NROW(1) + NROW(2) + ... + NROW(N).
C NROW WIDTHS OF ROWS OF A
C
C OUTPUT (AND ASSOCIATED) PARAMETERS
C L ELEMENTS WITHIN ENVELOPE OF UNIT LOWER
C TRIANGULAR FACTOR L (INCLUDING UNIT
C DIAGONAL ELEMENTS), IN ROW BY ROW ORDER
C D DIAGONAL ELEMENTS OF D
C
C INPUT-OUTPUT PARAMETER
C IFAIL FAILURE INDICATOR
C
C ENTRY VALUES
C 0 - TERMINATE IMMEDIATELY IF MATRIX FOUND TO BE
C NON-DEFINITE
C NON-ZERO - CONTINUE UNLESS ZERO PIVOT FOUND
C
C EXIT VALUES
C 0 - SUCCESSFUL TERMINATION
C 1 - AT LEAST ONE OF THE FOLLOWING
C RESTRICTIONS RELATING TO N, LAL
C AND THE ARRAY NROW IS VIOLATED.
C N .GE. 1,
C NROW(1) .GE. 1 AND .LE. I,
C FOR I = 1, 2, ..., N,
C NROW(1) + NROW(2) + ...
C 2 - A, AS MODIFIED BY ROUNDING ERRORS,
C IS NOT POSITIVE DEFINITE. FACTORIZATION
C NOT COMPLETED.
C 3 - A, AS MODIFIED BY ROUNDING ERRORS,
C IS NOT POSITIVE DEFINITE. FACTORIZATION
C COMPLETED BUT MAY BE INACCURATE.
C
SUBROUTINE F01QGF(M,N,A,LDA,ZETA,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C 1. Purpose
C =======
C
C F01QGF reduces the m by n ( m.le.n ) upper trapezoidal matrix A
C to upper triangular form by means of orthogonal transformations.
C
C 2. Description
C ===========
C
C The m by n ( m .le. n ) upper trapezoidal matrix A given by
C
C A = ( U X ),
C
C where U is an m by m upper triangular matrix, is factorized as
C
C A = ( R 0 )*P',
C
C where P is an n by n orthogonal matrix and R is an m by m upper
C triangular matrix.
C
C The factorization is obtained by Householder's method. The kth
C transformation matrix, P( k ), which is used to introduce zeros into
C the ( m - k + 1 )th row of A is given in the form
C
C P( k ) = ( I 0 ),
C ( 0 T( k ) )
C
C where
C
C T( k ) = I - u( k )*u( k )', u( k ) = ( zeta ),
C ( 0 )
C ( z( k ) )
C
C zeta is a scalar and z( k ) is an ( n - m ) element vector.
C zeta and z( k ) are chosen to annhilate the elements of the kth row
C of X.
C
C The vector u( k ) is returned in the kth element of ZETA and in the
C kth row of A, such that zeta is in ZETA( k ) and the elements of
C z( k ) are in a( k, m + 1 ), ..., a( k, n ). The elements of R are
C returned in the upper triangular part of A.
C
C P is given by
C
C P = ( P( 1 )*P( 2 )*...*P( m ) )'.
C
C 3. Parameters
C ==========
C
C M - INTEGER.
C
C On entry, M specifies the number of rows of A. M must be at
C least 0. When M = 0 then an immediate return is effected.
C
C Unchanged on exit.
C
C N - INTEGER.
C
C On entry, N specifies the number of columns of A. N must be
C at least M.
C
C Unchanged on exit.
C
C A - REAL array of DIMENSION ( LDA, n ).
C
C Before entry, the leading M by N upper trapezoidal part of
C the array A must contain the matrix to be factorized.
C
C On exit, the M by M upper triangular part of A will contain
C the upper triangular matrix R and the remaining M by
C ( N - M ) upper trapezoidal part of A will contain details
C of the factorization as described above.
C
C LDA - INTEGER.
C
C On entry, LDA must specify the leading dimension of the
C array A as declared in the calling (sub) program. LDA must
C be at least M.
C
C Unchanged on exit.
C
C ZETA - REAL array of DIMENSION at least ( m ).
C
C On exit, ZETA( k ) contains the scalar zeta( k ) for the
C ( m - k + 1 )th transformation. If P( k ) = I then
C ZETA( k ) = 0.0, otherwise ZETA( k ) contains zeta( k ) as
C described above and zeta( k ) is always in the range
C ( 1.0, sqrt( 2.0 ) ).
C
C IFAIL - INTEGER.
C
C Before entry, IFAIL must contain one of the values -1 or 0
C or 1 to specify noisy soft failure or noisy hard failure or
C silent soft failure. ( See Chapter P01 for further details.)
C
C On successful exit IFAIL will be zero, otherwise IFAIL
C will be set to -1 indicating that an input parameter has
C been incorrectly set. See the next section for further
C details.
C
C 4. Diagnostic Information
C ======================
C
C IFAIL = -1
C
C One or more of the following conditions holds:
C
C M .lt. 0
C N .lt. M
C LDA .lt. M
C
C If on entry, IFAIL was either -1 or 0 then further diagnostic
C information will be output on the error message channel. ( See
C routine X04AAF. )
C
C
C Nag Fortran 77 Auxiliary linear algebra routine.
C
C -- Written on 19-November-1987.
C Sven Hammarling, Nag Central Office.
C
SUBROUTINE F01QJF(M,N,A,LDA,ZETA,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C 1. Purpose
C =======
C
C F01QJF finds the RQ factorization of the real m by n, m .le. n,
C matrix A, so that A is reduced to upper triangular form by means of
C orthogonal transformations from the right.
C
C 2. Description
C ===========
C
C The m by n matrix A is factorized as
C
C A = ( R 0 )*P' when m.lt.n,
C
C A = R*P' when m = n,
C
C where P is an n by n orthogonal matrix and R is an m by m upper
C triangular matrix.
C
C P is given as a sequence of Householder transformation matrices
C
C P = P( m )*...*P( 2 )*P( 1 ),
C
C the ( m - k + 1 )th transformation matrix, P( k ), being used to
C introduce zeros into the kth row of A. P( k ) has the form
C
C P( k ) = I - u( k )*u( k )',
C
C where
C
C u( k ) = ( w( k ) ),
C ( zeta( k ) )
C ( 0 )
C ( z( k ) )
C
C zeta( k ) is a scalar, w( k ) is an ( k - 1 ) element vector and
C z( k ) is an ( n - m ) element vector. u( k ) is chosen to annhilate
C the elements in the kth row of A.
C
C The vector u( k ) is returned in the kth element of ZETA and in the
C kth row of A, such that zeta( k ) is in ZETA( k ), the elements of
C w( k ) are in a( k, 1 ), ..., a( k, k - 1 ) and the elements of
C z( k ) are in a( k, m + 1 ), ..., a( k, n ). The elements of R are
C returned in the upper triangular part of A.
C
C 3. Parameters
C ==========
C
C M - INTEGER.
C
C On entry, M must specify the number of rows of A. M must be
C at least zero. When M = 0 then an immediate return is
C effected.
C
C Unchanged on exit.
C
C N - INTEGER.
C
C On entry, N must specify the number of columns of A. N must
C be at least m.
C
C Unchanged on exit.
C
C A - REAL array of DIMENSION ( LDA, n ).
C
C Before entry, the leading M by N part of the array A must
C contain the matrix to be factorized.
C
C On exit, the M by M upper triangular part of A will contain
C the upper triangular matrix R, and the M by M strictly
C lower triangular part of A and the M by ( N - M )
C rectangular part of A to the right of the upper triangular
C part will contain details of the factorization as described
C above.
C
C LDA - INTEGER.
C
C On entry, LDA must specify the leading dimension of the
C array A as declared in the calling (sub) program. LDA must
C be at least m.
C
C Unchanged on exit.
C
C ZETA - REAL array of DIMENSION at least ( m ).
C
C On exit, ZETA( k ) contains the scalar zeta( k ) for the
C ( m - k + 1 )th transformation. If P( k ) = I then
C ZETA( k ) = 0.0, otherwise ZETA( k ) contains zeta( k )
C as described above and zeta( k ) is always in the range
C ( 1.0, sqrt( 2.0 ) ).
C
C IFAIL - INTEGER.
C
C Before entry, IFAIL must contain one of the values -1 or 0
C or 1 to specify noisy soft failure or noisy hard failure or
C silent soft failure. ( See Chapter P01 for further details.)
C
C On successful exit IFAIL will be zero, otherwise IFAIL
C will be set to -1 indicating that an input parameter has
C been incorrectly set. See the next section for further
C details.
C
C 4. Diagnostic Information
C ======================
C
C IFAIL = -1
C
C One or more of the following conditions holds:
C
C M .lt. 0
C N .lt. M
C LDA .lt. M
C
C If on entry, IFAIL was either -1 or 0 then further diagnostic
C information will be output on the error message channel. ( See
C routine X04AAF. )
C
C 5. Further information
C ===================
C
C The first k rows of the orthogonal matrix P' can be obtained by
C by calling NAG Library routine F01QKF, which overwrites the k rows of
C P' on the first k rows of the array A. P' is obtained by the call:
C
C IFAIL = 0
C CALL F01QKF( 'Separate', M, N, K, A, LDA, ZETA, WORK, IFAIL )
C
C WORK must be a max( m - 1, k - m, 1 ) element array. If K is larger
C than M, then A must have been declared to have at least K rows.
C
C Operations involving the matrix R can readily be performed by the
C Level 2 BLAS routines DTRSV and DTRMV , (see Chapter F06), but note
C that no test for near singularity of R is incorporated in DTRSV .
C If R is singular, or nearly singular then the NAG Library routine
C F02WUF can be used to determine the singular value decomposition
C of R.
C
C
C Nag Fortran 77 Auxiliary linear algebra routine.
C
C -- Written on 17-November-1987.
C Sven Hammarling, Nag Central Office.
C
SUBROUTINE F01QKF(WHERET,M,N,NROWP,A,LDA,ZETA,WORK,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C 1. Purpose
C =======
C
C F01QKF returns the first nrowp rows of the n by n orthogonal matrix
C P', where P is given as the product of Householder transformation
C matrices.
C
C This routine is intended for use following NAG Fortran Library
C routine F01QJF.
C
C 2. Description
C ===========
C
C P is assumed to be given by
C
C P = P( m )*P( m - 1 )*...*P( 1 ),
C
C where
C
C P( k ) = I - u( k )*u( k )',
C
C u( k ) = ( w( k ) ),
C ( zeta( k ) )
C ( 0 )
C ( z( k ) )
C
C zeta( k ) is a scalar, w( k ) is a ( k - 1 ) element vector and
C z( k ) is an ( n - m ) element vector.
C
C w( k ) must be supplied in the kth row of A in elements a( k, 1 ),
C ..., a( k, k - 1 ). z( k ) must be supplied in the kth row of
C A in elements a( k, m + 1 ), ..., a( k, n ) and zeta( k ) must be
C supplied either in a( k, k ) or in ZETA( k ), depending upon the
C parameter WHERET.
C
C 3. Parameters
C ==========
C
C WHERET - CHARACTER*1.
C
C On entry, WHERET specifies where the elements of zeta are
C to be found as follows.
C
C WHERET = 'I' or 'i' ( In A )
C
C The elements of zeta are in A.
C
C WHERET = 'S' or 's' ( Separate )
C
C The elements of zeta are separate from A, in ZETA.
C
C Unchanged on exit.
C
C M - INTEGER.
C
C On entry, M must specify the number of rows of A. M must be
C at least zero.
C
C Unchanged on exit.
C
C N - INTEGER.
C
C On entry, N must specify the number of columns of A. N must
C be at least m.
C
C Unchanged on exit.
C
C NROWP - INTEGER.
C
C On entry, NROWP must specify the required number of rows
C of P. NROWP must be at least zero and not be larger than n.
C When NROWP = 0 then an immediate return is effected.
C
C Unchanged on exit.
C
C A - REAL array of DIMENSION ( LDA, n ).
C
C Before entry, the leading M by M strictly lower triangular
C part of the array A, and the M by ( N - M ) rectangular part
C of A with top left hand corner at element A( 1, M + 1 ) must
C contain details of the matrix P. In addition, when
C WHERET = 'I' or 'i' then the diagonal elements of A must
C contain the elements of zeta.
C
C On exit, the first NROWP rows of the array A are overwritten
C by the first NROWP rows of the n by n orthogonal matrix
C P'.
C
C Unchanged on exit.
C
C LDA - INTEGER.
C
C On entry, LDA must specify the leading dimension of the
C array A as declared in the calling (sub) program. LDA must
C be at least max(m,nrowp).
C
C Unchanged on exit.
C
C ZETA - REAL array of DIMENSION at least ( m ), when
C WHERET = 'S' or 's'.
C
C Before entry with WHERET = 'S' or 's', the array ZETA must
C contain the elements of zeta. If ZETA( k ) = 0.0 then
C P( k ) is assumed to be I, otherwise ZETA( k ) is assumed
C to contain zeta( k ).
C
C When WHERET = 'I' or 'i', the array ZETA is not referenced.
C
C Unchanged on exit.
C
C WORK - REAL array of DIMENSION at least ( lwork ),
C where lwork = max( m - 1, nrowp - m, 1 ).
C
C Used as internal workspace.
C
C IFAIL - INTEGER.
C
C Before entry, IFAIL must contain one of the values -1 or 0
C or 1 to specify noisy soft failure or noisy hard failure or
C silent soft failure. ( See Chapter P01 for further details.)
C
C On successful exit IFAIL will be zero, otherwise IFAIL
C will be set to -1 indicating that an input parameter has
C been incorrectly set. See the next section for further
C details.
C
C 4. Diagnostic Information
C ======================
C
C IFAIL = -1
C
C One or more of the following conditions holds:
C
C WHERET .ne. 'I' or 'i' or 'S' or 's'
C M .lt. 0
C N .lt. M
C NROWP .lt. 0 .or. NROWP .gt. N
C LDA .lt. M
C
C If on entry, IFAIL was either -1 or 0 then further diagnostic
C information will be output on the error message channel. ( See
C routine X04AAF. )
C
C
C Nag Fortran 77 Auxiliary linear algebra routine.
C
C -- Written on 3-December-1987.
C Sven Hammarling and Mick Pont, Nag Central Office.
C
SUBROUTINE F01RGF(M,N,A,LDA,THETA,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C 1. Purpose
C =======
C
C F01RGF reduces the m by n ( m.le.n ) upper trapezoidal matrix A
C to upper triangular form by means of unitary transformations.
C
C 2. Description
C ===========
C
C The m by n ( m .le. n ) upper trapezoidal matrix A given by
C
C A = ( U X ),
C
C where U is an m by m upper triangular matrix, is factorized as
C
C A = ( R 0 )*conjg( P' ),
C
C where P is an n by n unitary matrix and R is an m by m upper
C triangular matrix with real diagonal elements.
C
C P is given as a sequence of Householder transformation matrices
C
C P = P( m )*...*P( 2 )*P( 1 ),
C
C the ( m - k + 1 )th transformation matrix, P( k ), being used to
C introduce zeros into the kth row of A. P( k ) has the form
C
C P( k ) = ( I 0 ),
C ( 0 T( k ) )
C
C where
C
C T( k ) = I - gamma( k )*u( k )*conjg( u( k )' ),
C
C u( k ) = ( zeta( k ) ),
C ( 0 )
C ( z( k ) )
C
C gamma( k ) is a scalar for which real( gamma( k ) ) = 1.0, zeta( k )
C is a real scalar and z( k ) is an ( n - m ) element vector.
C gamma( k ), zeta( k ) and z( k ) are chosen to annhilate the elements
C of the kth row of X and to make the diagonal elements of R real.
C
C The scalar gamma( k ) and the vector u( k ) are returned in the
C kth element of THETA and in the kth row of A, such that theta( k ),
C given by
C
C theta( k ) = ( zeta( k ), aimag( gamma( k ) ) ),
C
C is in THETA( k ) and the elements of z( k ) are in a( k, m + 1 ),
C ..., a( k, n ). The elements of R are returned in the upper
C triangular part of A.
C
C 3. Parameters
C ==========
C
C M - INTEGER.
C
C On entry, M specifies the number of rows of A. M must be at
C least 0. When M = 0 then an immediate return is effected.
C
C Unchanged on exit.
C
C N - INTEGER.
C
C On entry, N specifies the number of columns of A. N must be
C at least M.
C
C Unchanged on exit.
C
C A - REAL array of DIMENSION ( LDA, n ).
C
C Before entry, the leading M by N upper trapezoidal part of
C the array A must contain the matrix to be factorized.
C
C On exit, the M by M upper triangular part of A will contain
C the upper triangular matrix R, with the imaginary parts of
C the diagonal elements set to zero, and the M by ( N - M )
C upper trapezoidal part of A will contain details of the
C factorization as described above.
C
C LDA - INTEGER.
C
C On entry, LDA must specify the leading dimension of the
C array A as declared in the calling (sub) program. LDA must
C be at least M.
C
C Unchanged on exit.
C
C THETA - COMPLEX array of DIMENSION at least ( m ).
C
C On exit, THETA( k ) contains the scalar theta( k ) for the
C ( m - k + 1 )th transformation. If T( k ) = I then
C THETA( k ) = 0.0, if
C
C T( k ) = ( alpha 0 ), real( alpha ) .lt. 0.0,
C ( 0 I )
C
C then THETA( k ) = alpha, otherwise THETA( k ) contains
C theta( k ) as described above and real( theta( k ) ) is
C always in the range ( 1.0, sqrt( 2.0 ) ).
C
C IFAIL - INTEGER.
C
C Before entry, IFAIL must contain one of the values -1 or 0
C or 1 to specify noisy soft failure or noisy hard failure or
C silent soft failure. ( See Chapter P01 for further details.)
C
C On successful exit IFAIL will be zero, otherwise IFAIL
C will be set to -1 indicating that an input parameter has
C been incorrectly set. See the next section for further
C details.
C
C 4. Diagnostic Information
C ======================
C
C IFAIL = -1
C
C One or more of the following conditions holds:
C
C M .lt. 0
C N .lt. M
C LDA .lt. M
C
C If on entry, IFAIL was either -1 or 0 then further diagnostic
C information will be output on the error message channel. ( See
C routine X04AAF. )
C
C
C Nag Fortran 77 Auxiliary linear algebra routine.
C
C -- Written on 19-November-1987.
C Sven Hammarling, Nag Central Office.
C
SUBROUTINE F01RJF(M,N,A,LDA,THETA,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C 1. Purpose
C =======
C
C F01RJF finds the RQ factorization of the complex m by n, m .le. n,
C matrix A, so that A is reduced to upper triangular form by means of
C unitary transformations from the right.
C
C 2. Description
C ===========
C
C The m by n matrix A is factorized as
C
C A = ( R 0 )*conjg( P' ) when m.lt.n,
C
C A = R*conjg( P' ) when m = n,
C
C where P is an n by n unitary matrix and R is an m by m upper
C triangular matrix with real diagonal elements.
C
C P is given as a sequence of Householder transformation matrices
C
C P = P( m )*...*P( 2 )*P( 1 ),
C
C the ( m - k + 1 )th transformation matrix, P( k ), being used to
C introduce zeros into the kth row of A. P( k ) has the form
C
C P( k ) = I - gamma( k )*u( k )*conjg( u( k )' ),
C
C where
C
C u( k ) = ( w( k ) ),
C ( zeta( k ) )
C ( 0 )
C ( z( k ) )
C
C gamma( k ) is a scalar for which real( gamma( k ) ) = 1.0, zeta( k )
C is a real scalar, w( k ) is an ( k - 1 ) element vector and z( k )
C is an ( n - m ) element vector. gamma( k ) and u( k ) are chosen to
C annhilate the elements in the kth row of A and to make the diagonal
C elements real.
C
C The scalar gamma( k ) and the vector u( k ) are returned in the kth
C element of THETA and in the kth row of A, such that theta( k ),
C given by
C
C theta( k ) = ( zeta( k ), aimag( gamma( k ) ) ),
C
C is in THETA( k ), the elements of w( k ) are in a( k, 1 ), ...,
C a( k, k - 1 ) and the elements of z( k ) are in a( k, m + 1 ), ...,
C a( k, n ). The elements of R are returned in the upper triangular
C part of A.
C
C 3. Parameters
C ==========
C
C M - INTEGER.
C
C On entry, M must specify the number of rows of A. M must be
C at least zero. When M = 0 then an immediate return is
C effected.
C
C Unchanged on exit.
C
C N - INTEGER.
C
C On entry, N must specify the number of columns of A. N must
C be at least m.
C
C Unchanged on exit.
C
C A - COMPLEX array of DIMENSION ( LDA, n ).
C
C Before entry, the leading M by N part of the array A must
C contain the matrix to be factorized.
C
C On exit, the M by M upper triangular part of A will contain
C the upper triangular matrix R, with the imaginary parts of
C the diagonal elements set to zero, and the M by M strictly
C lower triangular part of A and the M by ( N - M )
C rectangular part of A to the right of the upper triangular
C part will contain details of the factorization as described
C above.
C
C LDA - INTEGER.
C
C On entry, LDA must specify the leading dimension of the
C array A as declared in the calling (sub) program. LDA must
C be at least m.
C
C Unchanged on exit.
C
C THETA - COMPLEX array of DIMENSION at least ( m ).
C
C On exit, THETA( k ) contains the scalar theta( k ) for the
C ( m - k + 1 )th transformation. If P( k ) = I then
C THETA( k ) = 0.0, if
C
C P( k ) = ( I 0 0 ), real( alpha ) .lt. 0.0,
C ( 0 alpha 0 )
C ( 0 0 I )
C
C then THETA( k ) = alpha, otherwise THETA( k ) contains
C theta( k ) as described above and real( theta( k ) ) is
C always in the range ( 1.0, sqrt( 2.0 ) ).
C
C IFAIL - INTEGER.
C
C Before entry, IFAIL must contain one of the values -1 or 0
C or 1 to specify noisy soft failure or noisy hard failure or
C silent soft failure. ( See Chapter P01 for further details.)
C
C On successful exit IFAIL will be zero, otherwise IFAIL
C will be set to -1 indicating that an input parameter has
C been incorrectly set. See the next section for further
C details.
C
C 4. Diagnostic Information
C ======================
C
C IFAIL = -1
C
C One or more of the following conditions holds:
C
C M .lt. 0
C N .lt. M
C LDA .lt. M
C
C If on entry, IFAIL was either -1 or 0 then further diagnostic
C information will be output on the error message channel. ( See
C routine X04AAF. )
C
C 5. Further information
C ===================
C
C The first k rows of the unitary matrix conjg( P' ) can be obtained
C by calling NAG Library routine F01RKF, which overwrites the k rows
C of conjg( P' ) on the first k rows of the array A. conjg( P' ) is
C obtained by the call:
C
C IFAIL = 0
C CALL F01RKF( 'Separate', M, N, K, A, LDA, THETA, WORK, IFAIL )
C
C WORK must be a max( m - 1, k - m, 1 ) element array. If K is larger
C than M, then A must have been declared to have at least K rows.
C
C Operations involving the matrix R can readily be performed by the
C Level 2 BLAS routines ZTRSV and ZTRMV , (see Chapter F06), but note
C that no test for near singularity of R is incorporated in ZTRSV .
C If R is singular, or nearly singular then the NAG Library routine
C F02XUF can be used to determine the singular value decomposition
C of R.
C
C
C Nag Fortran 77 Auxiliary linear algebra routine.
C
C -- Written on 17-November-1987.
C Sven Hammarling, Nag Central Office.
C
SUBROUTINE F01RKF(WHERET,M,N,NROWP,A,LDA,THETA,WORK,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C 1. Purpose
C =======
C
C F01RKF returns the first nrowp rows of the n by n unitary matrix
C conjg( P' ), where P is given as the product of Householder
C transformation matrices.
C
C This routine is intended for use following NAG Fortran Library
C routine F01RJF.
C
C 2. Description
C ===========
C
C P is assumed to be given by
C
C P = P( m )*P( m - 1 )*...*P( 1 ),
C
C where
C
C P( k ) = I - gamma( k )*u( k )*conjg( u( k )' ),
C
C u( k ) = ( w( k ) ),
C ( zeta( k ) )
C ( 0 )
C ( z( k ) )
C
C gamma( k ) is a scalar for which real( gamma( k ) ) = 1.0, zeta( k )
C is a real scalar, w( k ) is a ( k - 1 ) element vector and z( k )
C is an ( n - m ) element vector.
C
C w( k ) must be supplied in the kth row of A in elements a( k, 1 ),
C ..., a( k, k - 1 ). z( k ) must be supplied in the kth row of
C A in elements a( k, m + 1 ), ..., a( k, n ) and theta( k ), given by
C
C theta( k ) = ( zeta( k ), aimag( gamma( k ) ) ),
C
C must be supplied either in a( k, k ) or in THETA( k ), depending upon
C the parameter WHERET.
C
C 3. Parameters
C ==========
C
C WHERET - CHARACTER*1.
C
C On entry, WHERET specifies where the elements of theta are
C to be found as follows.
C
C WHERET = 'I' or 'i' ( In A )
C
C The elements of theta are in A.
C
C WHERET = 'S' or 's' ( Separate )
C
C The elements of theta are separate from A, in THETA.
C
C Unchanged on exit.
C
C M - INTEGER.
C
C On entry, M must specify the number of rows of A. M must be
C at least zero.
C
C Unchanged on exit.
C
C N - INTEGER.
C
C On entry, N must specify the number of columns of A. N must
C be at least m.
C
C Unchanged on exit.
C
C NROWP - INTEGER.
C
C On entry, NROWP must specify the required number of rows
C of P. NROWP must be at least zero and not be larger than n.
C When NROWP = 0 then an immediate return is effected.
C
C Unchanged on exit.
C
C A - COMPLEX array of DIMENSION ( LDA, n ).
C
C Before entry, the leading M by M strictly lower triangular
C part of the array A, and the M by ( N - M ) rectangular part
C of A with top left hand corner at element A( 1, M + 1 ) must
C contain details of the matrix P. In addition, when
C WHERET = 'I' or 'i' then the diagonal elements of A must
C contain the elements of theta.
C
C On exit, the first NROWP rows of the array A are overwritten
C by the first NROWP rows of the n by n unitary matrix
C conjg( P' ).
C
C Unchanged on exit.
C
C LDA - INTEGER.
C
C On entry, LDA must specify the leading dimension of the
C array A as declared in the calling (sub) program. LDA must
C be at least max(m,nrowp).
C
C Unchanged on exit.
C
C THETA - COMPLEX array of DIMENSION at least ( m ), when WHERET = 'S'
C or 's'.
C
C Before entry with WHERET = 'S' or 's', the array THETA must
C contain the elements of theta. If THETA( k ) = 0.0 then
C P( k ) is assumed to be I, if THETA( k ) = alpha, with
C real( alpha ) .lt. 0.0 then P( k ) is assumed to be of
C the form
C
C P( k ) = ( I 0 0 ),
C ( 0 alpha 0 )
C ( 0 0 I )
C
C otherwise THETA( k ) is assumed to contain theta( k ) given
C by theta( k ) = ( zeta( k ), aimag( gamma( k ) ) ).
C
C When WHERET = 'I' or 'i', the array THETA is not referenced.
C
C Unchanged on exit.
C
C WORK - COMPLEX array of DIMENSION at least ( lwork ), where
C lwork = max( m - 1, nrowp - m, 1 ).
C
C Used as internal workspace.
C
C IFAIL - INTEGER.
C
C Before entry, IFAIL must contain one of the values -1 or 0
C or 1 to specify noisy soft failure or noisy hard failure or
C silent soft failure. ( See Chapter P01 for further details.)
C
C On successful exit IFAIL will be zero, otherwise IFAIL
C will be set to -1 indicating that an input parameter has
C been incorrectly set. See the next section for further
C details.
C
C 4. Diagnostic Information
C ======================
C
C IFAIL = -1
C
C One or more of the following conditions holds:
C
C WHERET .ne. 'I' or 'i' or 'S' or 's'
C M .lt. 0
C N .lt. M
C NROWP .lt. 0 .or. NROWP .gt. N
C LDA .lt. M
C
C If on entry, IFAIL was either -1 or 0 then further diagnostic
C information will be output on the error message channel. ( See
C routine X04AAF. )
C
C
C Nag Fortran 77 Auxiliary linear algebra routine.
C
C -- Written on 3-December-1987.
C Sven Hammarling and Mick Pont, Nag Central Office.
C
SUBROUTINE F01ZAF(JOB,UPLO,DIAG,N,A,LDA,B,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C Copies a real triangular matrix from packed vector storage to
C a two-dimensional array, or vice-versa.
C
C Parameters:
C
C JOB - CHARACTER*1.
C JOB = 'P' or 'p' : 'Pack' - the contents of matrix A are to be
C packed into vector B.
C JOB = 'U' or 'u' : 'Unpack' - the contents of vector B are to
C be unpacked into matrix A.
C
C UPLO - CHARACTER*1.
C UPLO = 'U' or 'u' : 'Upper' - the matrix is upper triangular,
C and stored by column in the packed vector.
C UPLO = 'L' or 'l' : 'Upper' - the matrix is lower triangular,
C and stored by column in the packed vector.
C
C DIAG - CHARACTER*1.
C DIAG = 'N' or 'n' : 'Non-unit' - the triangular matrix has
C non-unit diagonal elements, which are packed or unpacked with
C the rest of the matrix.
C DIAG = 'U' or 'u' : 'Unit' - the triangular matrix is assumed
C to have unit diagonal elements which are not stored, in the
C format being copied from, but are inserted into the format
C being copied to.
C DIAG = ' ' : 'Blank' - the matrix is assumed to be strictly
C upper or strictly lower triangular. No reference is made
C either to the diagonal elements of matrix A or to the
C corresponding elements of vector B, although space must be
C allocated for them in the packed vector.
C
C N - INTEGER.
C The order of the matrix. N > 0.
C
C A(LDA, N) - real array.
C If JOB = 'P' or 'p', then on entry A must contain the matrix
C stored in the upper triangle if UPLO = 'U' or 'u', and stored
C in the lower triangle if UPLO = 'L' or 'l'. The opposite
C triangle of A may be undefined.
C If JOB = 'U' or 'u', then on exit A will contain the matrix
C stored in the upper triangle if UPLO = 'U' or 'u', and stored
C in the lower triangle if UPLO = 'L' or 'l'. The opposite
C triangle is left untouched.
C
C LDA - INTEGER.
C The leading dimension of array A as declared in the calling
C (sub)program.
C
C B((N*(N+1))/2) - real array.
C If JOB = 'U' or 'u', then on entry B must contain the triangular
C matrix packed by column.
C If JOB = 'P' or 'p', then on exit B will contain the triangular
C matrix packed by column.
C
C IFAIL - INTEGER.
C Error flag. On exit,
C IFAIL = 1 if JOB <> 'P', 'p', 'U' or 'u'.
C IFAIL = 2 if UPLO <> 'U', 'u', 'L' or 'l'.
C IFAIL = 3 if DIAG <> 'N', 'n', 'U', 'u', 'B' or 'b'.
C IFAIL = 4 if N < 1.
C IFAIL = 5 if LDA < N.
C
SUBROUTINE F01ZBF(JOB,UPLO,DIAG,N,A,LDA,B,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C Copies a complex triangular matrix from packed vector storage to
C a two-dimensional array, or vice-versa.
C
C Parameters:
C
C JOB - CHARACTER*1.
C JOB = 'P' or 'p' : 'Pack' - the contents of matrix A are to be
C packed into vector B.
C JOB = 'U' or 'u' : 'Unpack' - the contents of vector B are to
C be unpacked into matrix A.
C
C UPLO - CHARACTER*1.
C UPLO = 'U' or 'u' : 'Upper' - the matrix is upper triangular,
C and stored by column in the packed vector.
C UPLO = 'L' or 'l' : 'Upper' - the matrix is lower triangular,
C and stored by column in the packed vector.
C
C DIAG - CHARACTER*1.
C DIAG = 'N' or 'n' : 'Non-unit' - the triangular matrix has
C non-unit diagonal elements, which are packed or unpacked with
C the rest of the matrix.
C DIAG = 'U' or 'u' : 'Unit' - the triangular matrix is assumed
C to have unit diagonal elements which are not stored, in the
C format being copied from, but are inserted into the format
C being copied to.
C DIAG = ' ' : 'Blank' - the matrix is assumed to be strictly
C upper or strictly lower triangular. No reference is made
C either to the diagonal elements of matrix A or to the
C corresponding elements of vector B, although space must be
C allocated for them in the packed vector.
C
C N - INTEGER.
C The order of the matrix. N > 0.
C
C A(LDA, N) - complex array.
C If JOB = 'P' or 'p', then on entry A must contain the matrix
C stored in the upper triangle if UPLO = 'U' or 'u', and stored
C in the lower triangle if UPLO = 'L' or 'l'. The opposite
C triangle of A may be undefined.
C If JOB = 'U' or 'u', then on exit A will contain the matrix
C stored in the upper triangle if UPLO = 'U' or 'u', and stored
C in the lower triangle if UPLO = 'L' or 'l'. The opposite
C triangle is left untouched.
C
C LDA - INTEGER.
C The leading dimension of array A as declared in the calling
C (sub)program.
C
C B((N*(N+1))/2) - complex array.
C If JOB = 'U' or 'u', then on entry B must contain the triangular
C matrix packed by column.
C If JOB = 'P' or 'p', then on exit B will contain the triangular
C matrix packed by column.
C
C IFAIL - INTEGER.
C Error flag. On exit,
C IFAIL = 1 if JOB <> 'P', 'p', 'U' or 'u'.
C IFAIL = 2 if UPLO <> 'U', 'u', 'L' or 'l'.
C IFAIL = 3 if DIAG <> 'N', 'n', 'U', 'u', 'B' or 'b'.
C IFAIL = 4 if N < 1.
C IFAIL = 5 if LDA < N.
C
SUBROUTINE F01ZCF(JOB,M,N,KL,KU,A,LDA,B,LDB,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C Converts a real banded matrix from packed storage to unpacked,
C or vice-versa.
C
C Parameters:
C
C JOB - CHARACTER*1.
C JOB = 'P' or 'p' : 'Pack' - the contents of matrix A are to be
C packed into matrix B.
C JOB = 'U' or 'u' : 'Unpack' - the contents of matrix B are to
C be unpacked into matrix A.
C
C M - INTEGER.
C On entry, M specifies the number of rows of the matrix A.
C
C N - INTEGER.
C On entry, N specifies the number of columns of the matrix A.
C
C KL - INTEGER.
C On entry, KL specifies the number of sub-diagonals of the
C matrix A. KL >= 0.
C
C KU - INTEGER.
C On entry, KU specifies the number of super-diagonals of the
C matrix A. KU >= 0.
C
C A(LDA, N) - real array.
C If JOB = 'P' or 'p', then on entry the leading M by N part
C of array A must contain the band matrix stored in unpacked
C form. Elements of A that lie above the KUth super-diagonal
C and below the KLth sub-diagonal are not referenced and need
C not be defined.
C If JOB = 'U' or 'u', then on entry the array A may be
C undefined. On exit, A contains the band matrix stored in
C unpacked form. Elements of the leading M by N part of A that
C lie above the KUth super-diagonal or below the KLth
C sub-diagonal are assigned the value zero.
C
C LDA - INTEGER.
C On entry, LDA specifies the first dimension of A as declared
C in the calling (sub)program. LDA >= M.
C
C B(LDA, MIN(M+KU,N)) - real array.
C If JOB = 'U' or 'u', then on entry the leading (KL + KU + 1)
C by min(M+KU,N) part of array B must contain the band matrix,
C packed column by column, with the leading diagonal of the
C matrix in row (KU + 1) of the array, the first super-diagonal
C starting at position 2 in row KU, the first sub-diagonal
C starting at position 1 in row (KU + 2), and so on.
C Elements in the array B that do not correspond to elements
C in the band matrix (such as the top left KU by KU triangle)
C are not referenced.
C If JOB = 'P', then on entry B may be undefined, and on exit
C B contains the band matrix, packed as described above.
C Elements in the array B that do not correspond to elements
C in the band matrix (such as the top left KU by KU triangle)
C are not referenced.
C
C LDB - INTEGER.
C On entry, LDB specifies the first dimension of B as declared
C in the calling (sub)program. LDB >= (KL + KU + 1).
C
C IFAIL - INTEGER.
C Error flag. On exit,
C IFAIL = 1 if JOB <> 'P', 'p', 'U' or 'u'.
C IFAIL = 2 if KL < 0.
C IFAIL = 3 if KU < 0.
C IFAIL = 4 if LDA < M.
C IFAIL = 5 if LDB < KL + KU + 1.
C
SUBROUTINE F01ZDF(JOB,M,N,KL,KU,A,LDA,B,LDB,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C Converts a complex banded matrix from packed storage to unpacked,
C or vice-versa.
C
C Parameters:
C
C JOB - CHARACTER*1.
C JOB = 'P' or 'p' : 'Pack' - the contents of matrix A are to be
C packed into matrix B.
C JOB = 'U' or 'u' : 'Unpack' - the contents of matrix B are to
C be unpacked into matrix A.
C
C M - INTEGER.
C On entry, M specifies the number of rows of the matrix A.
C
C N - INTEGER.
C On entry, N specifies the number of columns of the matrix A.
C
C KL - INTEGER.
C On entry, KL specifies the number of sub-diagonals of the
C matrix A. KL >= 0.
C
C KU - INTEGER.
C On entry, KU specifies the number of super-diagonals of the
C matrix A. KU >= 0.
C
C A(LDA, N) - complex array.
C If JOB = 'P' or 'p', then on entry the leading M by N part
C of array A must contain the band matrix stored in unpacked
C form. Elements of A that lie above the KUth super-diagonal
C and below the KLth sub-diagonal are not referenced and need
C not be defined.
C If JOB = 'U' or 'u', then on entry the array A may be
C undefined. On exit, A contains the band matrix stored in
C unpacked form. Elements of the leading M by N part of A that
C lie above the KUth super-diagonal or below the KLth
C sub-diagonal are assigned the value zero.
C
C LDA - INTEGER.
C On entry, LDA specifies the first dimension of A as declared
C in the calling (sub)program. LDA >= M.
C
C B(LDA, MIN(M+KU,N)) - complex array.
C If JOB = 'U' or 'u', then on entry the leading (KL + KU + 1)
C by min(M+KU,N) part of array B must contain the band matrix,
C packed column by column, with the leading diagonal of the
C matrix in row (KU + 1) of the array, the first super-diagonal
C starting at position 2 in row KU, the first sub-diagonal
C starting at position 1 in row (KU + 2), and so on.
C Elements in the array B that do not correspond to elements
C in the band matrix (such as the top left KU by KU triangle)
C are not referenced.
C If JOB = 'P', then on entry B may be undefined, and on exit
C B contains the band matrix, packed as described above.
C Elements in the array B that do not correspond to elements
C in the band matrix (such as the top left KU by KU triangle)
C are unreferenced.
C
C LDB - INTEGER.
C On entry, LDB specifies the first dimension of B as declared
C in the calling (sub)program. LDB >= (KL + KU + 1).
C
C IFAIL - INTEGER.
C Error flag. On exit,
C IFAIL = 1 if JOB <> 'P', 'p', 'U' or 'u'.
C IFAIL = 2 if KL < 0.
C IFAIL = 3 if KU < 0.
C IFAIL = 4 if LDA < M.
C IFAIL = 5 if LDB < KL + KU + 1.
C
SUBROUTINE F02BBF(A,IA,N,ALB,UB,M,MM,R,V,IV,D,E,E2,X,G,C,ICOUNT,
* IFAIL)
C NAG COPYRIGHT 1976. MARK 5 RELEASE.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C MARK 14 REVISED. IER-740 (DEC 1989).
C WRITTEN BY W.PHILLIPS 1ST OCTOBER 1975
C OXFORD UNIVERSITY COMPUTING LABORATORY.
C THIS ROUTINE REPLACES F02ACF.
C
C SELECTED EIGENVALUES AND EIGENVECTORS OF A REAL SYMMETRIC
C MATRIX
C
SUBROUTINE F02BCF(A,IA,N,RLB,RUB,M,MM,RR,RI,VR,IVR,VI,IVI,INTGER,
* ICNT,C,B,IB,U,V,LFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C W.PHILLIPS. OXFORD UNIVERSITY COMPUTING SERVICE. 1 JUN 1977
C
C SELECTED EIGENVALUES AND EIGENVECTORS OF A REAL UNSYMMETRIC
C MATRIX
C
C THIS ROUTINE REPLACES F02AHF.
C
SUBROUTINE F02BDF(AR,IAR,AI,IAI,N,RLB,RUB,M,MM,RR,RI,VR,IVR,VI,
* IVI,INTGER,C,BR,IBR,BI,IBI,U,V,LFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C W.PHILLIPS. OXFORD UNIVERSITY COMPUTING SERVICE. 1 JUN 1977
C
C SELECTED EIGENVALUES AND EIGENVECTORS OF A COMPLEX
C UNSYMMETRIC MATRIX
C
C THIS ROUTINE REPLACES F02ALF.
C
SUBROUTINE F02BJF(N,A,IA,B,IB,EPS1,ALFR,ALFI,BETA,MATZ,Z,IZ,ITER,
* IFAIL)
C MARK 6 RELEASE. NAG COPYRIGHT 1977
C MARK 6A REVISED IER-99
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C THIS SUBROUTINE PERFORMS THE QZ ALGORITHM FOR SOLVING THE
C GENERALIZED MATRIX EIGENVALUE PROBLEM, A*Z = LAMBDA*B*Z
C WHERE A AND B ARE REAL SQUARE MATRICES.
C REFERENCE
C SIAM J.NUMER.ANAL., 10, PP241-256, 1973.
C C. B. MOLER AND G. W. STEWART.
C
C THIS SUBROUTINE WAS WRITTEN BY
C W. PHILLIPS OXFORD UNIVERSITY COMPUTING SERVICE.
C IT CALLS F02BJW, F02BJX, F02BJY AND F02BJZ WHICH
C WERE OBTAINED FROM EISPACK.
C
SUBROUTINE F02EAF(JOB,N,A,LDA,WR,WI,Z,LDZ,WORK,LWORK,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
C
C F02EAF computes all the eigenvalues, the Schur form, or the
C complete Schur factorization, of a real general matrix A.
C
C F02EAF is a driver routine which calls computational routines
C from LAPACK in Chapter F08.
C
SUBROUTINE F02EBF(JOB,N,A,LDA,WR,WI,VR,LDVR,VI,LDVI,WORK,LWORK,
* IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
C
C F02EBF computes all the eigenvalues, and optionally all the
C eigenvectors, of a real general matrix A.
C
C F02EBF is a driver routine which calls computational routines
C from LAPACK in Chapter F08.
C
SUBROUTINE F02FAF(JOB,UPLO,N,A,LDA,W,WORK,LWORK,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
C
C F02FAF computes all the eigenvalues, and optionally all the
C eigenvectors, of a real symmetric matrix A.
C
C F02FAF is a driver routine which calls computational routines from
C LAPACK in Chapter F08.
C
SUBROUTINE F02FDF(ITYPE,JOB,UPLO,N,A,LDA,B,LDB,W,WORK,LWORK,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
C
C F02FDF computes all the eigenvalues, and optionally all the
C eigenvectors, of a real symmetric-definite generalized
C eigenproblem, of one of the types:
C
C 1. A*z = lambda*B*z
C 2. A*B*z = lambda*z
C 3. B*A*z = lambda*z.
C
C Here A and B are symmetric and B is also positive-definite.
C
C F02FDF is a driver routine which calls computational routines from
C LAPACK in Chapter F08.
C
SUBROUTINE F02FHF(N,MA,A,NRA,MB,B,NRB,D,WORK,LWORK,IFAIL)
C MARK 11 RELEASE. NAG COPYRIGHT 1983.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C
C F02FHF FINDS THE EIGENVALUES FOR THE GENERALIZED BAND SYMMETRIC
C EIGENVALUE PROBLEM
C
C A*X = LAMBDA*B*X ,
C
C WHERE A AND B ARE N BY N BAND SYMMETRIC MATRICES OF HALF-BAND
C WIDTHS MA AND MB RESPECTIVELY AND B IS POSITIVE DEFINITE, BY
C A VARIANT OF THE METHOD OF CRAWFORD.
C
C FOR A DESCRIPTION OF THE PARAMETERS AND USE OF THIS ROUTINE SEE
C THE NAG LIBRARY MANUAL.
C
C -- WRITTEN ON 20-DECEMBER-1982. S.J.HAMMARLING.
C
C NAG FORTRAN 66 BLACK BOX ROUTINE.
C
C
SUBROUTINE F02FJF(N,EM,P,KM,EPS,IP,OP,INF,NOVECS,X,MN,D,WORK,
* LWORK,RWORK,LRWORK,IWORK,LIWORK,IFAIL)
C MARK 11 RELEASE. NAG COPYRIGHT 1983.
C MARK 11B REVISED. IER-458 (SEP 1984).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 12A REVISED. IER-509 (AUG 1986).
C
C **************************************************************
C *********
C
C F02FJF - ITERATIVE COMPUTATION OF EIGENVALUES LARGEST IN
C MAGNI-
C TUDE AND CORRESPONDING EIGENVECTORS OF A REAL GENERAL-
C IZED SYMMETRIC MATRIX
C
C PURPOSE
C
C A REAL N-SQUARE MATRIX C IS B-SYMMETRIC RELATIVE TO AN
C N-SQUARE
C SYMMETRIC POSITIVE DEFINITE MATRIX B WHEN
C BC = ( C - TRANSPOSED )B.
C GIVEN AS OPTIONAL INPUT A SET OF P INITIAL APPROXIMATE
C EIGENVECTORS OF A REAL N-SQUARE B-SYMMETRIC MATRIX C CORRES-
C PONDING TO P EIGENVALUES OF C LARGEST IN MAGNITUDE, F02FJF
C COM-
C PUTES EM EIGENVALUES AND EM CORRESPONDING EIGENVECTORS TO A
C PRECISION DEPENDENT ON THE STRUCTURE OF B AND C AND ON A GIVEN
C TOLERANCE EPS. THE MATRIX B IS PRESENTED TO F02FJF AS AN
C ALGO-
C RITHM FOR CALCULATING THE STANDARD INNER PRODUCT ( W, BZ ) =
C ( W - TRANSPOSED )BZ GIVEN COLUMN N-VECTORS W AND Z
C IMPLEMENTED
C AS A FORTRAN COMPATIBLE REAL FUNCTION SUBPROGRAM. THE MATRIX C
C IS PRESENTED AS A SUBROUTINE SUBPROGRAM WHICH GIVEN A COLUMN
C N-VECTOR Z CALCULATES ITS IMAGE W = CZ UNDER THE MATRIX C.
C DEPENDING ON THE CHOICE OF B AND C, F02FJF APPLIES TO A WIDE
C VARIETY OF SYMMETRIC EIGENPROBLEMS.
C
C METHOD
C
C F02FJF REPRESENTS RESULTS OF EXTENSIVE MODIFICATIONS AND TESTS
C OF SUBROUTINE RITZIT, AN ANSI FORTRAN TRANSLATION OF THE
C ALGOL 60 PROCEDURE OF THE SAME NAME. THE BASIC RUTISHAUSER
C -REINSCH ALGORITHM IS PRESERVED.
C
C PAUL J. NIKOLAI
C US AIR FORCE FLIGHT DYNAMICS LABORATORY
C AFFDL/FBR
C WRIGHT - PATTERSON AFB, OHIO 45433
C (513) - 255 - 5350
C
C FURTHER MODIFIED AT NAG CENTRAL OFFICE BY SVEN.
C CALLS TO SVD ROUTINES REPLACE CALLS AT EACH ITERATION TO
C TRED2 AND IMTQL2. THIS AVOIDS FORMING R( R**T ).
C WORKSPACE ASSIGNED TO A VECTOR AND REDUCED SLIGHTLY.
C SOME MINOR MODIFICATIONS TO IMPROVE PERFORMANCE ON PAGED
C MACHINES
C
C FOR A DESCRIPTION OF THE PARAMETERS AND USE OF THIS ROUTINE
C SEE THE
C NAG LIBRARY MANUAL. PARAMETER ASSOCIATION IS AS BELOW.
C
C ROUTINE DOCUMENT
C N N
C EM M
C P K
C KM NOITS
C EPS TOL
C IP DOT
C OP IMAGE
C INF MONIT
C NOVECS NOVECS
C X X
C MN NRX
C D D
C WORK WORK
C LWORK LWORK
C RWORK RWORK
C LRWORK LRWORK
C IWORK IWORK
C LIWORK LIWORK
C IFAIL IFAIL
C
C **************************************************************
C *********
C
C INF, OP
C
C THE LOCAL VARIABLE ARRAYS FROM RITZIT ARE ASSIGNED TO THE
C VARIABLE ARRAY WORK AS FOLLOWS
C
C B - WORK( I ), I = 1, ..., P**2.
C U - WORK( I ), I = 1, ..., N.
C W - WORK( I ), I = N + 1, ..., 2*N.
C CX - WORK( I ), I = CX1, ..., CX1 + P - 1
C WHERE CX1 = MAX( P**2, 2*N ) + 1
C F - WORK( I ), I = CX1 + P, ..., CX1 + 2*P - 1
C RQ - WORK( I ), I = CX1 + 2*P, ..., CX1 + 3*P - 1
C
C V, R AND Q ARE NOT NEEDED.
C
C TEST THE INPUT PARAMETERS.
C
SUBROUTINE F02GAF(JOB,N,A,LDA,W,Z,LDZ,RWORK,WORK,LWORK,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
C
C F02GAF computes all the eigenvalues, the Schur form, or the
C complete Schur factorization, of a complex general matrix A.
C
C F02GAF is a driver routine which calls computational routines
C from LAPACK in Chapter F08.
C
SUBROUTINE F02GBF(JOB,N,A,LDA,W,V,LDV,RWORK,WORK,LWORK,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
C
C F02GBF computes all the eigenvalues, and optionally all the
C eigenvectors, of a complex general matrix A.
C
C F02GBF is a driver routine which calls computational routines
C from LAPACK in Chapter F08.
C
SUBROUTINE F02GJF(N,AR,IAR,AI,IAI,BR,IBR,BI,IBI,EPS1,ALFR,ALFI,
* BETA,MATZ,ZR,IZR,ZI,IZI,ITER,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C THIS ROUTINE IS THE COMPLEX ANALOGUE OF THE QZ ALGORITHM FOR
C SOLVING THE GENERALIZED MATRIX EIGENVALUE PROBLEM,
C A*Z = LAMBDA*B*Z
C WHERE A AND B ARE REAL SQUARE MATRICES.
C REFERENCE
C SIAM J. NUMER. ANAL., 10, PP241-256, 1973.
C C.B. MOLER AND G.W. STEWART.
C
C THIS BLACK BOX ROUTINE WAS WRITTEN BY
C W. PHILLIPS OXFORD UNIVERSITY COMPUTING SERVICE
C
C IT CALLS F02GJX, F02GJY AND F02GJZ WHICH
C WERE OBTAINED FROM EISPACK.
C
SUBROUTINE F02HAF(JOB,UPLO,N,A,LDA,W,RWORK,WORK,LWORK,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
C
C F02HAF computes all the eigenvalues, and optionally all the
C eigenvectors, of a complex Hermitian matrix A.
C
C F02HAF is a driver routine which calls computational routines from
C LAPACK in Chapter F08.
C
SUBROUTINE F02HDF(ITYPE,JOB,UPLO,N,A,LDA,B,LDB,W,RWORK,WORK,LWORK,
* IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
C
C F02HDF computes all the eigenvalues, and optionally all the
C eigenvectors, of a complex Hermitian-definite generalized
C eigenproblem, of one of the types:
C
C 1. A*z = lambda*B*z
C 2. A*B*z = lambda*z
C 3. B*A*z = lambda*z.
C
C Here A and B are Hermitian and B is also positive-definite.
C
C F02HDF is a driver routine which calls computational routines from
C LAPACK in Chapter F08.
C
SUBROUTINE F02SDF(N,MA1,MB1,A,IA,B,IB,SYM,RELEP,RMU,VEC,D,INT,
* WORK,LWORK,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C
C PDK, DNACS, NPL, TEDDINGTON, JAN 1979
C NPL DNACS LIBRARY SUBROUTINE BABIT
C
C FINDS AN EIGENVECTOR CORRESPONDING TO A GIVEN EIGENVALUE
C BY INVERSE ITERATION FOR THE GENERAL EIGENPROBLEM
C A * X = RMU * B * X
C WHERE BOTH A AND B ARE BAND MATRICES OF ORDER N
C WITH BANDWIDTHS 2*MA1-1 AND 2*MB1-1 RESPECTIVELY.
C
C THE ROUTINE PROVIDES SPECIAL OPTIONS FOR THE CASES WHERE
C - BOTH A AND B ARE SYMMETRIC (SPECIFIED BY SYM = .TRUE.)
C - B IS A UNIT MATRIX (SPECIFIED BY MB1.LE.0).
C IN THE LATTER CASE THE ARRAY B IS NOT REFERENCED.
C
C FOR AN UNSYMMETRIC PROBLEM (SYM = .FALSE.) THE DIAGONAL
C LINES OF THE MATRICES A AND B MUST BE STORED IN THE ROWS
C OF THE ARRAYS A AND B, WITH THE LOWEST SUBDIAGONAL IN
C ROW 1 AND THE MAIN DIAGONAL IN ROW MA1 OR MB1. EACH ROW
C OF THE MATRICES IS STORED IN THE CORRESPONDING COLUMN
C OF THE ARRAYS. THE ELEMENTS IN THE UPPER LEFT AND LOWER
C RIGHT CORNERS NEED NOT BE SET, BUT THEY ARE SET TO ZERO
C AT THE START OF THE ROUTINE.
C
C FOR A SYMMETRIC PROBLEM (SYM = .TRUE.) ONLY THE LOWER
C TRIANGLES OF THE MATRICES NEED BE STORED, IN THE FIRST
C MA1 OR MB1 ROWS OF THE ARRAYS. NOTE THAT THE ARRAY A
C MUST ALWAYS HAVE AT LEAST 2*MA1-1 ROWS, BUT THE ARRAY B
C NEED ONLY HAVE MB1 ROWS IF SYM IS .TRUE..
C
C THE COMPUTED EIGENVECTOR IS STORED IN THE ARRAY VEC(N),
C NORMALISED SO THAT THE LARGEST ELEMENT IS 1.0.
C RELEP IS THE RELATIVE ERROR OF THE GIVEN DATA. IF IT IS LESS
C THAN THE VALUE RETURNED BY X02AJF, THEN THE LATTER VALUE
C IS USED INSTEAD.
C
C ON ENTRY D(1) PROVIDES INFORMATION ON THE NATURE OF THE
C PROBLEM WHICH IS USED TO DETERMINE THE COURSE OF THE
C ALGORITHM. ON EXIT THE ARRAY D CONTAINS THE SUCCESSIVE
C CORRECTIONS TO RMU IF D(1).NE.0.0 ON ENTRY AND
C THE FINAL CORRECTION TO RMU IS ALSO ALWAYS STORED IN
C D(30).
C
SUBROUTINE F02WDF(M,N,A,NRA,WANTB,B,TOL,SVD,IRANK,Z,SV,WANTR,R,
* NRR,WANTPT,PT,NRPT,WORK,LWORK,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C WRITTEN BY S. HAMMARLING, MIDDLESEX POLYTECHNIC (SVDGN4)
C
C F02WDF RETURNS THE HOUSEHOLDER QU FACTORIZATION OF A
C AND, IF EITHER A IS NOT OF FULL RANK OR IF SVD IS TRUE,
C PART OR ALL OF THE SINGULAR VALUE DECOMPOSITION (SVD) OF
C A, WHERE A IS AN M*N (M.GE.N) MATRIX.
C
C A IS FIRST FACTORIZED AS
C
C A = Q*(U) ,
C (0)
C
C WHERE Q IS ORTHOGONAL AND U IS UPPER TRIANGULAR.
C
C IF EITHER U IS SINGULAR OR SVD IS TRUE THEN PART OR ALL
C OF THE SVD OF U GIVEN BY
C
C U = R*D*(P**T) ,
C
C WHERE R AND P ARE ORTHOGONAL AND D IS DIAGONAL WITH
C NON-NEGATIVE DIAGONAL ELEMENTS, IS OBTAINED.
C
C THE SVD OF A IS THEN GIVEN BY
C
C A = Q*(R 0)*(D)*(P**T) ,
C (0 I) (0)
C
C THE DIAGONAL ELEMENTS OF D BEING THE SINGULAR VALUES OF A.
C
C IF WANTB IS TRUE THEN (Q**T)*B IS FORMED FOLLOWED, IF
C APPROPRIATE, BY (R**T 0)*(Q**T)*B.
C ( 0 I)
C
C INPUT PARAMETERS.
C
C M - NUMBER OF ROWS OF A. M MUST BE AT LEAST N.
C
C N - NUMBER OF COLUMNS OF A. N MUST BE AT LEAST UNITY.
C
C A - M*N MATRIX TO BE FACTORIZED.
C
C NRA - ROW DIMENSION OF A AS DECLARED IN THE
C CALLING PROGRAM. NRA MUST BE AT LEAST M.
C
C WANTB - MUST BE .TRUE. IF (Q**T)*B OR, IF APPROPRIATE,
C (R**T 0)*(Q**T)*B IS REQUIRED.
C ( 0 I)
C IF WANTB IS .FALSE. THEN B IS NOT REFERENCED.
C
C B - AN M ELEMENT VECTOR.
C
C TOL - A RELATIVE TOLERANCE USED TO DETERMINE THE RANK OF A.
C TOL SHOULD BE CHOSEN AS APPROXIMATELY THE LARGEST
C RELATIVE ERROR IN THE ELEMENTS OF A.
C LETTING EPS DENOTE THE SMALLEST REAL FOR WHICH
C 1.0+EPS.GT.1.0 ON THE MACHINE, IF TOL IS OUTSIDE THE
C RANGE (EPS,1) THEN THE VALUE EPS IS USED IN
C PLACE OF TOL. U IS REGARDED AS SINGULAR IF
C L(U)*L(U**(-1))*TOL.GT.1, WHERE L(U) DENOTES
C THE EUCLIDEAN LENGTH OF U.
C IF THE SINGULAR VALUES OF A ARE COMPUTED
C THEN THE RANK OF A IS DETERMINED BY
C NEGLECTING SINGULAR VALUES FOR WHICH
C SV(1)*TOL.GE.SV(I), WHERE SV(1) IS THE
C LARGEST SINGULAR VALUE.
C
C SVD - MUST BE .TRUE. IF PART OR ALL OF THE SVD IS
C REQUIRED EVEN IF A IS OF FULL RANK.
C
C WANTR - MUST BE .TRUE. IF THE ORTHOGONAL MATRIX R IS REQUIRED
C IF WANTR IS .FALSE. THEN R IS NOT REFERENCED.
C
C NRR - ROW DIMENSION OF R AS DECLARED IN THE CALLING PROGRAM
C NRR MUST BE AT LEAST N.
C
C WANTPT - MUST BE .TRUE. IF THE ORTHOGONAL MATRIX P**T
C IS REQUIRED. NOTE THAT PT IS REFERENCED EVEN
C WHEN WANTPT IS .FALSE., BUT SEE OUTPUT
C PARAMETER PT BELOW.
C
C NRPT - ROW DIMENSION OF PT AS DECLARED IN THE
C CALLING PROGRAM. NRPT MUST BE AT LEAST N.
C
C IFAIL - THE USUAL FAILURE PARAMETER. IF IN DOUBT SET IFAIL TO
C ZERO BEFORE CALLING F02WDF.
C
C OUTPUT PARAMETERS.
C
C A - A, TOGETHER WITH THE VECTOR Z, WILL CONTAIN
C DETAILS OF THE QU FACTORIZATION AS RETURNED
C FROM ROUTINE F01QAF, UNLESS F02WDF IS CALLED
C WITH PT=A AND SVD IS RETURNED AS .TRUE..
C
C B - IF WANTB IS .TRUE. THEN IF A IS OF FULL RANK AND SVD
C IS RETURNED AS .FALSE. B IS OVERWRITTEN BY (Q**T)*B,
C OTHERWISE B IS OVERWRITTEN BY (R**T 0)*(Q**T)*B.
C ( 0 I)
C IF WANTB IS .FALSE. THEN B IS NOT REFERENCED.
C
C SVD - SVD IS RETURNED AS .TRUE. IF THE SINGULAR VALUES OF A
C HAVE BEEN OBTAINED AND IS RETURNED AS
C .FALSE. IF ONLY THE QU FACTORIZATION OF A
C HAS BEEN OBTAINED.
C
C IRANK - THE RANK OF THE MATRIX A.
C
C Z - N ELEMENT VECTOR. SEE OUTPUT PARAMETER A ABOVE.
C IF Z IS NOT REQUIRED THEN THE ROUTINE MAY BE
C CALLED WITH Z=SV.
C
C SV - N ELEMENT VECTOR. IF A IS NOT OF FULL RANK OR IF SVD
C IS RETURNED AS .TRUE. THEN SV WILL CONTAIN
C THE SINGULAR VALUES OF A ARRANGED IN
C DESCENDING ORDER.
C
C R - N*N MATRIX. IF WANTR IS .TRUE. AND SVD IS RETURNED AS
C .TRUE. THEN R RETURNS THE LEFT HAND
C ORTHOGONAL MATRIX OF THE SVD OF U.
C IF WANTR IS .FALSE. THEN R IS NOT REFERENCED.
C
C PT - N*N MATRIX. IF WANTPT IS .TRUE. AND SVD IS
C RETURNED AS .TRUE. THEN PT RETURNS THE
C MATRIX P**T.
C THE ROUTINE MAY BE CALLED WITH PT=A.
C IF WANTPT IS .FALSE. BUT WANTR IS .TRUE.
C THEN THE ROUTINE MAY BE CALLED WITH PT=R.
C
C IFAIL - ON NORMAL RETURN IFAIL WILL BE ZERO.
C IN THE UNLIKELY EVENT THAT THE QR-ALGORITHM
C FAILS TO FIND THE SINGULAR VALUES IN 50*N
C ITERATIONS THEN IFAIL WILL BE 2 OR MORE AND
C SUCH THAT SV(1),SV(2),..., SV(IFAIL-1) MAY
C NOT HAVE BEEN FOUND.
C IF AN INPUT PARAMETER IS INCORRECTLY
C SUPPLIED THEN IFAIL IS SET TO UNITY.
C
C WORKSPACE PARAMETERS.
C
C WORK - A 3*N ELEMENT VECTOR.
C IF SVD IS RETURNED AS .FALSE. THEN WORK(1) WILL
C CONTAIN THE CONDITION NUMBER, L(U)*L(U**(-1)), OF THE
C UPPER TRIANGULAR MATRIX U.
C IF SVD IS RETURNED AS .TRUE. THEN WORK(1)
C WILL CONTAIN THE TOTAL NUMBER OF ITERATIONS
C TAKEN BY THE QR-ALGORITHM.
C
C LWORK - LENGTH OF THE VECTOR WORK. LWORK MUST BE AT
C LEAST 3*N.
C
SUBROUTINE F02WEF(M,N,A,LDA,NCOLB,B,LDB,WANTQ,Q,LDQ,SV,WANTP,PT,
* LDPT,WORK,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C MARK 14 REVISED. IER-789 (DEC 1989).
C MARK 14B REVISED. IER-834 (MAR 1990).
C
C 1. Purpose
C =======
C
C F02WEF returns all, or part, of the singular value decomposition of
C a general real matrix.
C
C 2. Description
C ===========
C
C The m by n matrix A is factorized as
C
C A = Q*D*P',
C
C where
C
C D = ( S ), m .gt. n,
C ( 0 )
C
C D = S, m .eq. n,
C
C D = ( S 0 ), m .lt. n,
C
C Q is an m by m orthogonal matrix, P is an n by n orthogonal matrix
C and S is a min( m, n ) by min( m, n ) diagonal matrix with non-
C negative diagonal elements, sv( 1 ), sv( 2 ), ..., sv( min( m, n ) ),
C ordered such that
C
C sv( 1 ) .ge. sv( 2 ) .ge. ... .ge. sv( min( m, n ) ) .ge. 0.
C
C The first min( m, n ) columns of Q are the left-hand singular vectors
C of A, the diagonal elements of S are the singular values of A and the
C first min( m, n ) columns of P are the right-hand singular vectors of
C A.
C
C Either or both of the left-hand and right-hand singular vectors of A
C may be requested and the matrix
C C given by
C
C C = Q'*B,
C
C where B is an m by ncolb given matrix, may also be requested.
C
C The routine obtains the singular value decomposition by first
C reducing A to upper triangular form by means of Householder
C transformations, from the left when m .ge. n and from the right when
C m .lt. n. The upper triangular form is then reduced to bidiagonal
C form by Givens plane rotations and finally the QR algorithm is used
C to obtain the singular value decomposition of the bidiagonal form.
C
C 3. Parameters
C ==========
C
C M - INTEGER.
C
C On entry, M must specify the number of rows of the matrix A.
C M must be at least zero. When M = 0 then an immediate
C return is effected.
C
C Unchanged on exit.
C
C N - INTEGER.
C
C On entry, N must specify the number of columns of the matrix
C A. N must be at least zero. When N = 0 then an immediate
C return is effected.
C
C Unchanged on exit.
C
C A - REAL array of DIMENSION ( LDA, n ).
C
C Before entry, the leading M by N part of the array A must
C contain the matrix A whose singular value decomposition is
C required.
C
C If m .ge. n and WANTQ is .TRUE. then on exit, the M by N
C part of A will contain the first n columns of the orthogonal
C matrix Q.
C
C If m .lt. n and WANTP is .TRUE. then on exit, the M by N
C part of A will contain the first m rows of the orthogonal
C matrix P'.
C
C If m .ge. n and WANTQ is .FALSE. and WANTP is .TRUE.
C then on exit, the min( M, N ) by N part of A will contain
C the first min( m, n ) rows of the orthogonal matrix P'.
C
C Otherwise the M by N part of A is used as internal
C workspace.
C
C LDA - INTEGER.
C
C On entry, LDA must specify the leading dimension of the
C array A as declared in the calling (sub) program. LDA must
C be at least m.
C
C Unchanged on exit.
C
C NCOLB - On entry, NCOLB must specify the number of columns of the
C matrix B and must be at least zero. When NCOLB = 0 the
C array B is not referenced.
C
C B - REAL array of DIMENSION ( LDB, ncolb ).
C
C Before entry with NCOLB .gt. 0, the leading M by NCOLB part
C of the array B must contain the matrix to be transformed
C and on exit, B is overwritten by the m by ncolb matrix
C Q'*B.
C
C LDB - INTEGER.
C
C On entry, LDB must specify the leading dimension of the
C array B as declared in the calling (sub) program. When
C NCOLB .gt. 0 then LDB must be at least m.
C
C Unchanged on exit.
C
C WANTQ - LOGICAL.
C
C On entry, WANTQ must be .TRUE. if the left-hand singular
C vectors are required. If WANTQ is .FALSE. then the array
C Q is not referenced.
C
C Unchanged on exit.
C
C Q - REAL array of DIMENSION ( LDQ, m ).
C
C On exit with M .lt. N and WANTQ as .TRUE., the leading
C M by M part of the array Q will contain the orthogonal
C matrix Q. Otherwise the array Q is not referenced.
C
C LDQ - INTEGER.
C
C On entry, LDQ must specify the leading dimension of the
C array Q as declared in the calling (sub) program. When
C M .lt. N and WANTQ is .TRUE., LDQ must be at least m.
C
C Unchanged on exit.
C
C SV - REAL array of DIMENSION at least ( min( m, n ) ).
C
C On exit, the array SV will contain the min( m, n ) diagonal
C elements of the matrix S.
C
C WANTP - LOGICAL.
C
C On entry, WANTP must be .TRUE. if the right-hand singular
C vectors are required. If WANTP is .FALSE. then the array
C PT is not referenced.
C
C Unchanged on exit.
C
C PT - REAL array of DIMENSION ( LDPT, n ).
C
C On exit with M .ge. N and WANTQ and WANTP as .TRUE., the
C leading N by N part of the array PT will contain the
C orthogonal matrix P'. Otherwise the array PT is not
C referenced.
C
C LDPT - INTEGER.
C
C On entry, LDPT must specify the leading dimension of the
C array PT as declared in the calling (sub) program. When
C M .ge. N and WANTQ and WANTP are .TRUE., LDPT must be at
C least n.
C
C Unchanged on exit.
C
C WORK - REAL array of DIMENSION at least ( lwork ), where lwork must
C be as given in the following table:
C
C M .ge. N
C
C WANTQ is .TRUE. and WANTP is .TRUE.
C
C lwork = max(n**2 + 5*( n - 1 ), n + ncolb, 4)
C
C WANTQ is .TRUE. and WANTP is .FALSE.
C
C lwork = max(n**2 + 4*( n - 1 ) + 1, n + ncolb, 4)
C
C WANTQ is .FALSE. and WANTP is .TRUE.
C
C lwork = max( 3*( n - 1 ), 2 ) when NCOLB = 0
C lwork = max( 5*( n - 1 ), 2 ) when NCOLB .gt. 0
C
C WANTQ is .FALSE. and WANTP is .FALSE.
C
C lwork = max( 2*( n - 1 ), 2 ) when NCOLB = 0
C lwork = max( 3*( n - 1 ), 2 ) when NCOLB .gt. 0
C
C M .lt. N
C
C WANTQ is .TRUE. and WANTP is .TRUE.
C
C lwork = max( m**2 + 5*( m - 1 ), 2 )
C
C WANTQ is .TRUE. and WANTP is .FALSE.
C
C lwork = max( 3*( m - 1 ), 1 )
C
C WANTQ is .FALSE. and WANTP is .TRUE.
C
C lwork = max( m**2 + 3*( m - 1 ), 2 )
C when NCOLB = 0
C lwork = max( m**2 + 5*( m - 1 ), 2 )
C when NCOLB .gt. 0
C
C WANTQ is .FALSE. and WANTP is .FALSE.
C
C lwork = max( 2*( m - 1 ), 1 ) when NCOLB = 0
C lwork = max( 3*( m - 1 ), 1 ) when NCOLB .gt. 0
C
C The array WORK is used as internal workspace by F02WEF.
C On exit, WORK( min( m, n ) ) contains the total number of
C iterations taken by the QR algorithm.
C
C IFAIL - INTEGER.
C
C Before entry, IFAIL must contain one of the values -1 or 0
C or 1 to specify noisy soft failure or noisy hard failure or
C silent soft failure. ( See Chapter P01 for further details.)
C
C On successful exit IFAIL will be zero, otherwise IFAIL
C will be set to a non-zero value indicating either that an
C input parameter has been incorrectly set, or that the QR
C algorithm is not converging. See the next section for
C further details.
C
C 4. Diagnostic Information
C ======================
C
C IFAIL = -1
C
C One or more of the following conditions holds:
C
C M .lt. 0
C N .lt. 0
C LDA .lt. M
C NCOLB .lt. 0
C LDB .lt. M and NCOLB .gt. 0
C LDQ .lt. M and M .lt. N and WANTQ is true
C LDPT .lt. N and M .ge. N and WANTQ is true
C and WANTP is true
C
C IFAIL .gt. 0
C
C The QR algorithm has failed to converge in 50*min( M, N )
C iterations. In this case sv( 1 ), sv( 2 ), ..., sv( IFAIL ) may
C not have been found correctly and the remaining singular values
C may not be the smallest. The matrix A will nevertheless have been
C factorized as A = Q*E*P', where the leading min( m, n ) by
C min( m, n ) part of E is a bidiagonal matrix with sv( 1 ),
C sv( 2 ), ..., sv( min( m, n ) ) as the diagonal elements and
C work( 1 ), work( 2 ), ..., work( min( m, n ) - 1 ) as the super-
C diagonal elements.
C
C This failure is not likely to occur.
C
C If on entry, IFAIL was either -1 or 0 then further diagnostic
C information will be output on the error message channel. ( See
C routine X04AAF. )
C
C 5. Further information
C ===================
C
C Following the use of this routine the rank of A may be estimated by
C a call to the INTEGER function F06KLF. The statement:
C
C IRANK = F06KLF( MIN( M, N ), SV, 1, TOL )
C
C returns the value ( k - 1 ), in IRANK, where k is the smallest
C integer for which sv( k ) .lt. tol*sv( 1 ), where tol is the
C tolerance supplied in TOL, so that IRANK is an estimate of the rank
C of S and thus also of A. If TOL is supplied as negative then the
C relative machine precision ( see routine X02AJF ) is used in place of
C TOL.
C
C
C Nag Fortran 77 Auxiliary linear algebra routine.
C
C -- Written on 12-January-1988.
C Sven Hammarling, Nag Central Office.
C
SUBROUTINE F02WUF(N,A,LDA,NCOLB,B,LDB,WANTQ,Q,LDQ,SV,WANTP,WORK,
* IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C 1. Purpose
C =======
C
C F02WUF returns all, or part, of the singular value decomposition of
C a real upper triangular matrix.
C
C 2. Description
C ===========
C
C The n by n upper triangular matrix R is factorized as
C
C R = Q*S*P',
C
C where Q and P are n by n orthogonal matrices and S is an n by n
C diagonal matrix with non-negative diagonal elements, sv( 1 ),
C sv( 2 ), ..., sv( n ), ordered such that
C
C sv( 1 ) .ge. sv( 2 ) .ge. ... .ge. sv( n ) .ge. 0.
C
C The columns of Q are the left-hand singular vectors of R, the
C diagonal elements of S are the singular values of R and the columns
C of P are the right-hand singular vectors of R.
C
C Either or both of Q and P' may be requested and the matrix C given by
C
C C = Q'*B,
C
C where B is an n by ncolb given matrix, may also be requested.
C
C The routine obtains the singular value decomposition by first
C reducing R to bidiagonal form by means of Givens plane rotations
C and then using the QR algorithm to obtain the singular value
C decomposition of the bidiagonal form.
C
C 3. Parameters
C ==========
C
C N - INTEGER.
C
C On entry, N must specify the order of the matrix R. N must
C be at least zero. When N = 0 then an immediate return is
C effected.
C
C Unchanged on exit.
C
C A - REAL array of DIMENSION ( LDA, n ).
C
C Before entry, the leading N by N upper triangular part of
C the array A must contain the upper triangular matrix R.
C
C If WANTP is .TRUE. then on exit, the N by N part of A will
C contain the n by n orthogonal matrix P', otherwise the
C N by N upper triangular part of A is used as internal
C workspace, but the strictly lower triangular part of A
C is not referenced.
C
C LDA - INTEGER.
C
C On entry, LDA must specify the leading dimension of the
C array A as declared in the calling (sub) program. LDA must
C be at least n.
C
C Unchanged on exit.
C
C NCOLB - On entry, NCOLB must specify the number of columns of the
C matrix B and must be at least zero. When NCOLB = 0 the
C array B is not referenced.
C
C B - REAL array of DIMENSION ( LDB, ncolb ).
C
C Before entry with NCOLB .gt. 0, the leading N by NCOLB part
C of the array B must contain the matrix to be transformed
C and on exit, B is overwritten by the n by ncolb matrix
C Q'*B.
C
C LDB - INTEGER.
C
C On entry, LDB must specify the leading dimension of the
C array B as declared in the calling (sub) program. When
C NCOLB .gt. 0 then LDB must be at least n.
C
C Unchanged on exit.
C
C WANTQ - LOGICAL.
C
C On entry, WANTQ must be .TRUE. if the matrix Q is required.
C If WANTQ is .FALSE. then the array Q is not referenced.
C
C Unchanged on exit.
C
C Q - REAL array of DIMENSION ( LDQ, n ).
C
C On exit with WANTQ as .TRUE., the leading N by N part of
C the array Q will contain the orthogonal matrix Q. Otherwise
C the array Q is not referenced.
C
C LDQ - INTEGER.
C
C On entry, LDQ must specify the leading dimension of the
C array Q as declared in the calling (sub) program. When
C WANTQ is .TRUE., LDQ must be at least n.
C
C Unchanged on exit.
C
C SV - REAL array of DIMENSION at least ( n ).
C
C On exit, the array SV will contain the n diagonal elements
C of the matrix S.
C
C WANTP - LOGICAL.
C
C On entry, WANTP must be .TRUE. if the matrix P' is required,
C in which case P' is overwritten on the array A, otherwise
C WANTP must be .FALSE..
C
C Unchanged on exit.
C
C WORK - REAL array of DIMENSION at least ( max( 1, lwork ) ), where
C lwork must satisfy:
C
C lwork = 2*( n - 1 ) when
C ncolb = 0 and WANTQ and WANTP are .FALSE.,
C
C lwork = 3*( n - 1 ) when
C either ncolb = 0 and WANTQ is .FALSE. and
C WANTP is .TRUE., or WANTP is .FALSE. and one or
C both of ncolb .gt. 0 and WANTQ is .TRUE.
C
C lwork = 5*( n - 1 ) otherwise.
C
C The array WORK is used as internal workspace by F06XUF.
C On exit, WORK( n ) contains the total number of iterations
C taken by the QR algorithm.
C
C IFAIL - INTEGER.
C
C Before entry, IFAIL must contain one of the values -1 or 0
C or 1 to specify noisy soft failure or noisy hard failure or
C silent soft failure. ( See Chapter P01 for further details.)
C
C On successful exit IFAIL will be zero, otherwise IFAIL
C will be set to a non-zero value indicating either that an
C input parameter has been incorrectly set, or that the QR
C algorithm is not converging. See the next section for
C further details.
C
C 4. Diagnostic Information
C ======================
C
C IFAIL = -1
C
C One or more of the following conditions holds:
C
C N .lt. 0
C LDA .lt. N
C NCOLB .lt. 0
C LDB .lt. N and NCOLB .gt. 0
C LDQ .lt. N and WANTQ is true
C
C IFAIL .gt. 0
C
C The QR algorithm has failed to converge in 50*N iterations. In
C this case sv( 1 ), sv( 2 ), ..., sv( IFAIL ) may not have been
C found correctly and the remaining singular values may not be the
C smallest. The matrix R will nevertheless have been factorized as
C R = Q*E*P', where E is a bidiagonal matrix with sv( 1 ), sv( 2 ),
C ..., sv( n ) as the diagonal elements and work( 1 ), work( 2 ),
C ..., work( n - 1 ) as the super-diagonal elements.
C
C This failure is not likely to occur.
C
C If on entry, IFAIL was either -1 or 0 then further diagnostic
C information will be output on the error message channel. ( See
C routine X04AAF. )
C
C 5. Further information
C ===================
C
C Following the use of this routine the rank of R may be estimated by
C a call to the INTEGER function F06KLF. The statement:
C
C IRANK = F06KLF( N, SV, 1, TOL )
C
C returns the value ( k - 1 ), in IRANK, where k is the smallest
C integer for which sv( k ) .lt. tol*sv( 1 ), where tol is the
C tolerance supplied in TOL, so that IRANK is an estimate of the rank
C of S and thus also of R. If TOL is supplied as negative then the
C relative machine precision ( see routine X02AJF ) is used in place of
C TOL.
C
C
C Nag Fortran 77 Auxiliary linear algebra routine.
C
C -- Written on 10-January-1988.
C Sven Hammarling, Nag Central Office.
C
SUBROUTINE F02XEF(M,N,A,LDA,NCOLB,B,LDB,WANTQ,Q,LDQ,SV,WANTP,PH,
* LDPH,RWORK,CWORK,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C MARK 14 REVISED. IER-791 (DEC 1989).
C MARK 14B REVISED. IER-835 (MAR 1990).
C
C 1. Purpose
C =======
C
C F02XEF returns all, or part, of the singular value decomposition of
C a general complex matrix.
C
C 2. Description
C ===========
C
C The m by n matrix A is factorized as
C
C A = Q*D*conjg( P' ),
C
C where
C
C D = ( S ), m .gt. n,
C ( 0 )
C
C D = S, m .eq. n,
C
C D = ( S 0 ), m .lt. n,
C
C Q is an m by m unitary matrix, P is an n by n unitary matrix and S is
C a min( m, n ) by min( m, n ) diagonal matrix with real non-negative
C diagonal elements, sv( 1 ), sv( 2 ), ..., sv( min( m, n ) ), ordered
C such that
C
C sv( 1 ) .ge. sv( 2 ) .ge. ... .ge. sv( min( m, n ) ) .ge. 0.
C
C The first min( m, n ) columns of Q are the left-hand singular vectors
C of A, the diagonal elements of S are the singular values of A and the
C first min( m, n ) columns of P are the right-hand singular vectors of
C A.
C
C Either or both of the left-hand and right-hand singular vectors of A
C may be requested and the matrix
C C given by
C
C C = conjg( Q' )*B,
C
C where B is an m by ncolb given matrix, may also be requested.
C
C The routine obtains the singular value decomposition by first
C reducing A to upper triangular form by means of Householder
C transformations, from the left when m .ge. n and from the right when
C m .lt. n. The upper triangular form is then reduced to bidiagonal
C form by Givens plane rotations and finally the QR algorithm is used
C to obtain the singular value decomposition of the bidiagonal form.
C
C 3. Parameters
C ==========
C
C M - INTEGER.
C
C On entry, M must specify the number of rows of the matrix A.
C M must be at least zero. When M = 0 then an immediate
C return is effected.
C
C Unchanged on exit.
C
C N - INTEGER.
C
C On entry, N must specify the number of columns of the matrix
C A. N must be at least zero. When N = 0 then an immediate
C return is effected.
C
C Unchanged on exit.
C
C A - COMPLEX array of DIMENSION ( LDA, n ).
C
C Before entry, the leading M by N part of the array A must
C contain the matrix A whose singular value decomposition is
C required.
C
C If m .ge. n and WANTQ is .TRUE. then on exit, the M by N
C part of A will contain the first n columns of the unitary
C matrix Q.
C
C If m .lt. n and WANTP is .TRUE. then on exit, the M by N
C part of A will contain the first m rows of the unitary
C matrix conjg( P' ).
C
C If m .ge. n and WANTQ is .FALSE. and WANTP is .TRUE.
C then on exit, the min( M, N ) by N part of A will contain
C the first min( m, n ) rows of the unitary matrix
C conjg( P' ).
C
C Otherwise the M by N part of A is used as internal
C workspace.
C
C LDA - INTEGER.
C
C On entry, LDA must specify the leading dimension of the
C array A as declared in the calling (sub) program. LDA must
C be at least m.
C
C Unchanged on exit.
C
C NCOLB - On entry, NCOLB must specify the number of columns of the
C matrix B and must be at least zero. When NCOLB = 0 the
C array B is not referenced.
C
C B - COMPLEX array of DIMENSION ( LDB, ncolb ).
C
C Before entry with NCOLB .gt. 0, the leading M by NCOLB part
C of the array B must contain the matrix to be transformed
C and on exit, B is overwritten by the m by ncolb matrix
C conjg( Q' )*B.
C
C LDB - INTEGER.
C
C On entry, LDB must specify the leading dimension of the
C array B as declared in the calling (sub) program. When
C NCOLB .gt. 0 then LDB must be at least m.
C
C Unchanged on exit.
C
C WANTQ - LOGICAL.
C
C On entry, WANTQ must be .TRUE. if the left-hand singular
C vectors are required. If WANTQ is .FALSE. then the array
C Q is not referenced.
C
C Unchanged on exit.
C
C Q - COMPLEX array of DIMENSION ( LDQ, m ).
C
C On exit with M .lt. N and WANTQ as .TRUE., the leading
C M by M part of the array Q will contain the unitary matrix
C Q. Otherwise the array Q is not referenced.
C
C LDQ - INTEGER.
C
C On entry, LDQ must specify the leading dimension of the
C array Q as declared in the calling (sub) program. When
C M .lt. N and WANTQ is .TRUE., LDQ must be at least m.
C
C Unchanged on exit.
C
C SV - REAL array of DIMENSION at least ( min( m, n ) ).
C
C On exit, the array SV will contain the min( m, n ) diagonal
C elements of the matrix S.
C
C WANTP - LOGICAL.
C
C On entry, WANTP must be .TRUE. if the right-hand singular
C vectors are required. If WANTP is .FALSE. then the array
C PH is not referenced.
C
C Unchanged on exit.
C
C PH - COMPLEX array of DIMENSION ( LDPH, n ).
C
C On exit with M .ge. N and WANTQ and WANTP as .TRUE., the
C leading N by N part of the array PH will contain the unitary
C matrix conjg( P' ). Otherwise the array PH is not
C referenced.
C
C LDPH - INTEGER.
C
C On entry, LDPH must specify the leading dimension of the
C array PH as declared in the calling (sub) program. When
C M .ge. N and WANTQ and WANTP are .TRUE., LDPH must be at
C least n.
C
C Unchanged on exit.
C
C RWORK - REAL array of DIMENSION at least ( max( 1, lrwork ) ), where
C lrwork must satisfy:
C
C lrwork = 2*( min( m, n ) - 1 ) when
C ncolb = 0 and WANTQ and WANTP are .FALSE.,
C
C lrwork = 3*( min( m, n ) - 1 ) when
C either ncolb = 0 and WANTQ is .FALSE. and
C WANTP is .TRUE., or WANTP is .FALSE. and one or
C both of ncolb .gt. 0 and WANTQ is .TRUE.
C
C lrwork = 5*( min( m, n ) - 1 ) otherwise.
C
C The array RWORK is used as internal workspace by F06XUF.
C On exit, RWORK( min( m, n ) ) contains the total number of
C iterations taken by the QR algorithm.
C
C CWORK - COMPLEX array of DIMENSION at least ( lcwork ), where lcwork
C must satisfy:
C
C lcwork = n + max( n**2, ncolb ) when
C m .ge. n and WANTQ and WANTP are both .TRUE.
C
C lcwork = n + max( n**2 + n, ncolb ) when
C m .ge. n and WANTQ is .TRUE., but WANTP is .FALSE.
C
C lcwork = n + max( n, ncolb ) when
C m .ge. n and WANTQ is .FALSE.
C
C lcwork = m**2 + m when
C m .lt. n and WANTP is .TRUE.
C
C lcwork = m when
C m .lt. n and WANTP is .FALSE.
C
C The array CWORK is used as internal workspace by F06XEF.
C
C IFAIL - INTEGER.
C
C Before entry, IFAIL must contain one of the values -1 or 0
C or 1 to specify noisy soft failure or noisy hard failure or
C silent soft failure. ( See Chapter P01 for further details.)
C
C On successful exit IFAIL will be zero, otherwise IFAIL
C will be set to a non-zero value indicating either that an
C input parameter has been incorrectly set, or that the QR
C algorithm is not converging. See the next section for
C further details.
C
C 4. Diagnostic Information
C ======================
C
C IFAIL = -1
C
C One or more of the following conditions holds:
C
C M .lt. 0
C N .lt. 0
C LDA .lt. M
C NCOLB .lt. 0
C LDB .lt. M and NCOLB .gt. 0
C LDQ .lt. M and M .lt. N and WANTQ is true
C LDPH .lt. N and M .ge. N and WANTQ is true
C and WANTP is true
C
C IFAIL .gt. 0
C
C The QR algorithm has failed to converge in 50*min( M, N )
C iterations. In this case sv( 1 ), sv( 2 ), ..., sv( IFAIL ) may
C not have been found correctly and the remaining singular values
C may not be the smallest. The matrix A will nevertheless have been
C factorized as A = Q*E*conjg( P' ), where the leading min( m, n )
C by min( m, n ) part of E is a bidiagonal matrix with sv( 1 ),
C sv( 2 ), ..., sv( min( m, n ) ) as the diagonal elements and
C rwork( 1 ), rwork( 2 ), ..., rwork( min( m, n ) - 1 ) as the
C super-diagonal elements.
C
C This failure is not likely to occur.
C
C If on entry, IFAIL was either -1 or 0 then further diagnostic
C information will be output on the error message channel. ( See
C routine X04AAF. )
C
C 5. Further information
C ===================
C
C Following the use of this routine the rank of A may be estimated by
C a call to the INTEGER function F06KLF. The statement:
C
C IRANK = F06KLF( MIN( M, N ), SV, 1, TOL )
C
C returns the value ( k - 1 ), in IRANK, where k is the smallest
C integer for which sv( k ) .lt. tol*sv( 1 ), where tol is the
C tolerance supplied in TOL, so that IRANK is an estimate of the rank
C of S and thus also of A. If TOL is supplied as negative then the
C relative machine precision ( see routine X02AJF ) is used in place of
C TOL.
C
C
C Nag Fortran 77 Auxiliary linear algebra routine.
C
C -- Written on 26-November-1987.
C Sven Hammarling, Nag Central Office.
C
SUBROUTINE F02XUF(N,A,LDA,NCOLB,B,LDB,WANTQ,Q,LDQ,SV,WANTP,RWORK,
* CWORK,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C MARK 14 REVISED. IER-792 (DEC 1989).
C
C 1. Purpose
C =======
C
C F02XUF returns all, or part, of the singular value decomposition of
C a complex upper triangular matrix.
C
C 2. Description
C ===========
C
C The n by n upper triangular matrix R is factorized as
C
C R = Q*S*conjg( P' ),
C
C where Q and P are n by n unitary matrices and S is an n by n diagonal
C matrix with real non-negative diagonal elements, sv( 1 ), sv( 2 ),
C ..., sv( n ), ordered such that
C
C sv( 1 ) .ge. sv( 2 ) .ge. ... .ge. sv( n ) .ge. 0.
C
C The columns of Q are the left-hand singular vectors of R, the
C diagonal elements of S are the singular values of R and the columns
C of P are the right-hand singular vectors of R.
C
C Either or both of Q and conjg( P' ) may be requested and the matrix
C C given by
C
C C = conjg( Q' )*B,
C
C where B is an n by ncolb given matrix, may also be requested.
C
C The routine obtains the singular value decomposition by first
C reducing R to bidiagonal form by means of Givens plane rotations
C and then using the QR algorithm to obtain the singular value
C decomposition of the bidiagonal form.
C
C 3. Parameters
C ==========
C
C N - INTEGER.
C
C On entry, N must specify the order of the matrix R. N must
C be at least zero. When N = 0 then an immediate return is
C effected.
C
C Unchanged on exit.
C
C A - COMPLEX array of DIMENSION ( LDA, n ).
C
C Before entry, the leading N by N upper triangular part of
C the array A must contain the upper triangular matrix R.
C
C If WANTP is .TRUE. then on exit, the N by N part of A will
C contain the n by n unitary matrix conjg( P' ), otherwise
C the N by N upper triangular part of A is used as internal
C workspace, but the strictly lower triangular part of A
C is not referenced.
C
C LDA - INTEGER.
C
C On entry, LDA must specify the leading dimension of the
C array A as declared in the calling (sub) program. LDA must
C be at least n.
C
C Unchanged on exit.
C
C NCOLB - On entry, NCOLB must specify the number of columns of the
C matrix B and must be at least zero. When NCOLB = 0 the
C array B is not referenced.
C
C B - COMPLEX array of DIMENSION ( LDB, ncolb ).
C
C Before entry with NCOLB .gt. 0, the leading N by NCOLB part
C of the array B must contain the matrix to be transformed
C and on exit, B is overwritten by the n by ncolb matrix
C conjg( Q' )*B.
C
C LDB - INTEGER.
C
C On entry, LDB must specify the leading dimension of the
C array B as declared in the calling (sub) program. When
C NCOLB .gt. 0 then LDB must be at least n.
C
C Unchanged on exit.
C
C WANTQ - LOGICAL.
C
C On entry, WANTQ must be .TRUE. if the matrix Q is required.
C If WANTQ is .FALSE. then the array Q is not referenced.
C
C Unchanged on exit.
C
C Q - COMPLEX array of DIMENSION ( LDQ, n ).
C
C On exit with WANTQ as .TRUE., the leading N by N part of
C the array Q will contain the unitary matrix Q. Otherwise
C the array Q is not referenced.
C
C LDQ - INTEGER.
C
C On entry, LDQ must specify the leading dimension of the
C array Q as declared in the calling (sub) program. When
C WANTQ is .TRUE., LDQ must be at least n.
C
C Unchanged on exit.
C
C SV - REAL array of DIMENSION at least ( n ).
C
C On exit, the array SV will contain the n diagonal elements
C of the matrix S.
C
C WANTP - LOGICAL.
C
C On entry, WANTP must be .TRUE. if the matrix conjg( P' ) is
C required, in which case conjg( P' ) is overwritten on the
C array A, otherwise WANTP must be .FALSE..
C
C Unchanged on exit.
C
C RWORK - REAL array of DIMENSION at least ( max( 1, lrwork ) ), where
C lrwork must satisfy:
C
C lrwork = 2*( n - 1 ) when
C ncolb = 0 and WANTQ and WANTP are .FALSE.,
C
C lrwork = 3*( n - 1 ) when
C either ncolb = 0 and WANTQ is .FALSE. and
C WANTP is .TRUE., or WANTP is .FALSE. and one or
C both of ncolb .gt. 0 and WANTQ is .TRUE.
C
C lrwork = 5*( n - 1 ) otherwise.
C
C The array RWORK is used as internal workspace by F06XUF.
C On exit, RWORK( n ) contains the total number of iterations
C taken by the QR algorithm.
C
C CWORK - COMPLEX array of DIMENSION at least ( max( 1, n - 1 ) ).
C
C The array CWORK is used as internal workspace by F06XUF.
C
C IFAIL - INTEGER.
C
C Before entry, IFAIL must contain one of the values -1 or 0
C or 1 to specify noisy soft failure or noisy hard failure or
C silent soft failure. ( See Chapter P01 for further details.)
C
C On successful exit IFAIL will be zero, otherwise IFAIL
C will be set to a non-zero value indicating either that an
C input parameter has been incorrectly set, or that the QR
C algorithm is not converging. See the next section for
C further details.
C
C 4. Diagnostic Information
C ======================
C
C IFAIL = -1
C
C One or more of the following conditions holds:
C
C N .lt. 0
C LDA .lt. N
C NCOLB .lt. 0
C LDB .lt. N and NCOLB .gt. 0
C LDQ .lt. N and WANTQ is true
C
C IFAIL .gt. 0
C
C The QR algorithm has failed to converge in 50*N iterations. In
C this case sv( 1 ), sv( 2 ), ..., sv( IFAIL ) may not have been
C found correctly and the remaining singular values may not be the
C smallest. The matrix R will nevertheless have been factorized as
C R = Q*E*conjg( P' ), where E is a bidiagonal matrix with
C sv( 1 ), sv( 2 ), ..., sv( n ) as the diagonal elements and
C rwork( 1 ), rwork( 2 ), ..., rwork( n - 1 ) as the super-diagonal
C elements.
C
C This failure is not likely to occur.
C
C If on entry, IFAIL was either -1 or 0 then further diagnostic
C information will be output on the error message channel. ( See
C routine X04AAF. )
C
C 5. Further information
C ===================
C
C Following the use of this routine the rank of R may be estimated by
C a call to the INTEGER function F06KLF. The statement:
C
C IRANK = F06KLF( N, SV, 1, TOL )
C
C returns the value ( k - 1 ), in IRANK, where k is the smallest
C integer for which sv( k ) .lt. tol*sv( 1 ), where tol is the
C tolerance supplied in TOL, so that IRANK is an estimate of the rank
C of S and thus also of R. If TOL is supplied as negative then the
C relative machine precision ( see routine X02AJF ) is used in place of
C TOL.
C
C
C Nag Fortran 77 Auxiliary linear algebra routine.
C
C -- Written on 25-November-1987.
C Sven Hammarling, Nag Central Office.
C
SUBROUTINE F03AAF(A,IA,N,DET,WKSPCE,IFAIL)
C MARK 15 RE-ISSUE. NAG COPYRIGHT 1991.
C
C Determinant of real matrix.
C 1st August 1971
C
C Rewritten to call F07ADG, a modified version of LAPACK routine
C SGETRF/F07ADF; new IFAIL exit inserted for illegal input
C parameters; error messages inserted. February 1991.
C
SUBROUTINE F03ABF(A,IA,N,DET,WKSPCE,IFAIL)
C MARK 15 RE-ISSUE. NAG COPYRIGHT 1991.
C
C Determinant of real symmetric positive definite matrix
C 1st August 1971
C
C Rewritten to call LAPACK routine SPOTRF/F07FDF;
C new IFAIL exit inserted for illegal input parameters;
C error messages inserted. February 1991.
C
SUBROUTINE F03ACF(A,IA,N,M,DET,RL,IL,M1,LFAIL)
C MARK 2 RELEASE. NAG COPYRIGHT 1972
C MARK 3 REVISED.
C MARK 4.5 REVISED
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C
C DETERMINANT OF REAL POSITIVE DEFINITE SYMMETRIC BAND MATRIX
C 1ST AUGUST 1971
C
SUBROUTINE F03ADF(A,IA,N,DETR,DETI,WKSPCE,IFAIL)
C MARK 15 RE-ISSUE. NAG COPYRIGHT 1991.
C
C Determinant of complex matrix
C 1st August 1971
C
C Rewritten to call F07ARG, a modified version of LAPACK routine
C CGETRF/F07ARF; new IFAIL exit inserted for illegal input
C parameters; error messages inserted. February 1991.
C
SUBROUTINE F03AEF(N,A,IA,P,D1,ID,IFAIL)
C MARK 15 RE-ISSUE. NAG COPYRIGHT 1991.
C
C CHOLDET1
C The upper triangle of a positive definite symmetric matrix,
C A, is stored in the upper triangle of an N*N array A(I,J),
C I=1,N, J=1,N. The Cholesky decomposition A = LU, where
C U is the transpose of L, is performed and stored in the
C remainder of the array A except for the reciprocals of the
C diagonal elements which are stored in P(I), I = 1,N,
C instead of the elements themselves. A is retained so that
C the solution obtained can be subsequently improved. The
C determinant, D1 * 2.0**ID, of A is also computed. The
C subroutine will fail if A, modified by the rounding errors,
C is not positive definite.
C 1st December 1971
C
C Rewritten to call LAPACK routine SPOTRF/F07ADF; new IFAIL
C exit inserted for illegal input parameters; error messages
C inserted. February 1991.
C
SUBROUTINE F03AFF(N,EPS,A,IA,D1,ID,P,IFAIL)
C MARK 15 RE-ISSUE. NAG COPYRIGHT 1991.
C
C UNSYMDET
C The unsymmetric matrix, A, is stored in the N*N array A(I,J),
C I=1,N, J=1,N. The decomposition A=LU, where L is a
C lower triangular matrix and U is a unit upper triangular
C matrix, is performed and overwritten on A, omitting the unit
C diagonal of U. A record of any interchanges made to the rows
C of A is kept in P(I), I=1,N, such that the I-th row and
C the P(I)-th row were interchanged at the I-th step. The
C determinant, D1 * 2.0**ID, of A is also computed. The
C subroutine will fail if A, modified by the rounding errors, is
C singular or almost singular.
C 1st December 1971
C
C Rewritten to call F07ADG, a modified version of LAPACK routine
C SGETRF/F07ADF; new IFAIL exit inserted for illegal input
C parameters; error messages inserted. February 1991.
C
SUBROUTINE F04AAF(A,IA,B,IB,N,M,C,IC,WKSPCE,IFAIL)
C MARK 15 RE-ISSUE. NAG COPYRIGHT 1991.
C
C Approximate solution of a set of real linear equations
C with multiple right hand sides.
C 1st August 1971
C
C Rewritten to call F07ADG and F07AEG, modified versions of LAPACK
C routines SGETRF/F07ADF and SGETRS/F07AEF; new IFAIL exit inserted
C for illegal input parameters; error messages inserted.
C February 1991.
C
SUBROUTINE F04ABF(A,IA,B,IB,N,M,C,IC,WKSPCE,BB,IBB,IFAIL)
C MARK 15 RE-ISSUE. NAG COPYRIGHT 1991.
C
C Accurate solution of a set of real symmetric positive
C definite linear equations with multiple right hand sides.
C 1st August 1971
C
C Rewritten to call LAPACK routine SPOTRF/F07FDF;
C new IFAIL exit inserted for illegal input parameters;
C error messages inserted. February 1991.
C
SUBROUTINE F04ACF(A,IA,B,IB,N,M,LR,C,IC,RL,IL,M1,LFAIL)
C MARK 2 RELEASE. NAG COPYRIGHT 1972
C MARK 3 REVISED.
C MARK 4.5 REVISED
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C APPROXIMATE SOLUTION OF A SET OF REAL SYMMETRIC POSITIVE
C DEFINITE BAND LINEAR EQUATIONS WITH MULTIPLE RIGHT
C HAND SIDES.
C 1ST AUGUST 1971
C
SUBROUTINE F04ADF(A,IA,B,IB,N,M,C,IC,WKSPCE,IFAIL)
C MARK 15 RE-ISSUE. NAG COPYRIGHT 1991.
C
C Approximate solution of a set of complex linear
C equations with multiple right hand sides.
C 1st August 1971
C
C Rewritten to call F07ARG and F07ASG, modified versions of LAPACK
C routines CGETRF/F07ARF and CGETRS/F07ASF; new IFAIL exit inserted
C for illegal input parameters; error messages inserted.
C February 1991.
C
SUBROUTINE F04AEF(A,IA,B,IB,N,M,C,IC,WKSPCE,AA,IAA,BB,IBB,IFAIL)
C MARK 15 RE-ISSUE. NAG COPYRIGHT 1991.
C
C Accurate solution of a set of real linear equations
C with multiple right hand sides.
C 1st August 1971
C
C Rewritten to call F07ADG, a modified version of LAPACK routine
C SGETRF/F07ADF; new IFAIL exit inserted for illegal input
C parameters; error messages inserted. February 1991.
C
SUBROUTINE F04AFF(N,IR,A,IA,P,B,IB,EPS,X,IX,BB,IBB,L,IFAIL)
C MARK 2 RELEASE. NAG COPYRIGHT 1972
C MARK 3 REVISED.
C MARK 4.5 REVISED
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. IER-637 (APR 1988).
C ACCSOLVE
C SOLVES AX=B WHERE A IS AN N*N POSITIVE DEFINITE SYMMETRIC
C MATRIX AND B IS AN N*IR MATRIX OF RIGHT HAND SIDES, USING THE
C SUBROUTINE F04AGF . THE SUBROUTINE MUST BY PRECEDED BY
C F03AEF IN WHICH L IS PRODUCED IN A(I,J) AND P(I). THE
C RESIDUALS BB=B-AX ARE CALCULATED AND AD=BB IS SOLVED,
C OVERWRITING D ON BB. THE REFINEMENT IS REPEATED, AS LONG AS
C THE MAXIMUM CORRECTION AT ANY STAGE IS LESS THAN HALF THAT AT
C THE PREVIOUS STAGE, UNTIL THE MAXIMUM CORRECTION IS LESS
C THAN 2 EPS TIMES THE MAXIMUM X. EXITS WITH IFAIL = 1 IF
C THE SOLUTION FAILS TO IMPROVE. L IS THE NUMBER OF
C ITERATIONS.
C ADDITIONAL PRECISION INNERPRODUCTS ARE ABSOLUTELY NECESSARY.
C 1ST DECEMBER 1971
C
SUBROUTINE F04AGF(N,IR,A,IA,P,B,IB,X,IX)
C MARK 2 RELEASE. NAG COPYRIGHT 1972
C MARK 4.5 REVISED
C MARK 11 REVISED. VECTORISATION (JAN 1984).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 12 REVISED. EXTENDED BLAS (JUNE 1986)
C
C CHOLSOL1
C SOLVES AX=B, WHERE A IS A POSITIVE DEFINITE SYMMETRIC MATRIX
C AND B IS AN N*IR MATRIX OF IR RIGHT-HAND SIDES. THE
C SUBROUTINE
C F04AGF MUST BY PRECEDED BY F03AEF IN WHICH L IS
C PRODUCED IN A(I,J) AND P(I), FROM A. AX=B IS SOLVED IN TWO
C STEPS, LY=B AND UX=Y, AND X IS OVERWRITTEN ON Y.
C 1ST AUGUST 1971
C
SUBROUTINE F04AHF(N,IR,A,IA,AA,IAA,P,B,IB,EPS,X,IX,BB,IBB,L,IFAIL)
C MARK 2 RELEASE. NAG COPYRIGHT 1972
C MARK 3 REVISED.
C MARK 4 REVISED.
C MARK 4.5 REVISED
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C UNSYMACCSOLVE
C SOLVES AX=B WHERE A IS AN N*N UNSYMMETRIC MATRIX AND B IS AN
C N*IR MATRIX OF RIGHT HAND SIDES, USING THE SUBROUTINE F04AJF.
C THE SUBROUTINE MUST BY PRECEDED BY F03AFF IN WHICH L AND U
C ARE PRODUCED IN AA(I,J) AND THE INTERCHANGES IN P(I). THE
C RESIDUALS BB=B-AX ARE CALCULATED AND AD=BB IS SOLVED, OVER-
C WRITING D ON BB. THE REFINEMENT IS REPEATED, AS LONG AS THE
C MAXIMUM CORRECTION AT ANY STAGE IS LESS THAN HALF THAT AT THE
C PREVIOUS STAGE, UNTIL THE MAXIMUM CORRECTION IS LESS THAN 2
C EPS TIMES THE MAXIMUM X. SETS IFAIL = 1 IF THE SOLUTION FAILS
C TO IMPROVE, ELSE IFAIL = 0. L IS THE NUMBER OF ITERATIONS.
C ADDITIONAL PRECISION INNERPRODUCTS ARE ABSOLUTELY NECESSARY.
C 1ST DECEMBER 1971
C
SUBROUTINE F04AJF(N,IR,A,IA,P,B,IB)
C MARK 2 RELEASE. NAG COPYRIGHT 1972
C MARK 4 REVISED.
C MARK 4.5 REVISED
C MARK 11 REVISED. VECTORISATION (JAN 1984).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 12 REVISED. EXTENDED BLAS (JUNE 1986)
C
C UNSYMSOL
C SOLVES AX=B, WHERE A IS AN UNSYMMETRIC MATRIX AND B IS AN
C N*IR
C MATRIX OF IR RIGHT-HAND SIDES. THE SUBROUTINE F04AJF MUST BY
C PRECEDED BY F03AFF IN WHICH L AND U ARE PRODUCED IN A(I,J),
C FROM A, AND THE RECORD OF THE INTERCHANGES IS PRODUCED IN
C P(I). AX=B IS SOLVED IN THREE STEPS, INTERCHANGE THE
C ELEMENTS OF B, LY=B AND UX=Y. THE MATRICES Y AND THEN X ARE
C OVERWRITTEN ON B.
C 1ST AUGUST 1971
C
SUBROUTINE F04AMF(A,IA,X,IX,B,IB,M,N,IP,ETA,QR,IQR,ALPHA,E,Y,Z,R,
* IPIVOT,IFAIL)
C MARK 2 RELEASE. NAG COPYRIGHT 1972
C MARK 3 REVISED.
C MARK 4.5 REVISED
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 16A REVISED. IER-1003 (JUN 1993).
C LEAST SQUARES SOLUTION
C THE ARRAY A(M,N) CONTAINS THE GIVEN MATRIX OF AN
C OVERDETERMINED SYSTEM OF M LINEAR EQUATIONS IN N UNKNOWNS
C (M.GE.N). FOR THE IP RIGHT HAND SIDES GIVEN AS THE COLUMNS
C OF THE ARRAY B(M,IP), THE LEAST SQUARES SOLUTIONS ARE
C COMPUTED AND STORED AS THE COLUMNS OF THE ARRAY X(N,IP).
C IF RANK(A).LT.N THEN THE PROBLEM IS LEFT UNSOLVED AND IFAIL
C IS SET EQUAL TO 1. IN EITHER CASE A AND B ARE
C LEFT INTACT. ETA IS THE RELATIVE MACHINE PRECISION.
C ADDITIONAL PRECISION INNERPRODUCTS ARE ABSOLUTELY NECESSARY.
C 1ST. MARCH 1972
C
SUBROUTINE F04ARF(A,IA,B,N,C,WKS,IFAIL)
C MARK 15 RE-ISSUE. NAG COPYRIGHT 1991.
C
C Approximate solution of a set of real linear
C equations with one right hand side.
C 1st. April 1973
C
C Rewritten to call F07ADG and F07AEG, modified versions of LAPACK
C routines SGETRF/F07ADF and SGETRS/F07AEF; new IFAIL exit inserted
C for illegal input parameters; error messages inserted.
C February 1991.
C
SUBROUTINE F04ASF(A,IA,B,N,C,WK1,WK2,IFAIL)
C MARK 15 RE-ISSUE. NAG COPYRIGHT 1991.
C MARK 16 REVISED. IER-1004 (JUN 1993).
C
C Accurate solution of a set of real symmetric positive
C definite linear equations with one right hand side.
C 1st April 1973
C
C Rewritten to call LAPACK routine SPOTRF/F07FDF;
C new IFAIL exit inserted for illegal input parameters;
C error messages inserted. February 1991.
C
SUBROUTINE F04ATF(A,IA,B,N,C,AA,IAA,WKS1,WKS2,IFAIL)
C MARK 15 RE-ISSUE. NAG COPYRIGHT 1991.
C
C Accurate solution of a set of real linear equations
C with one right side.
C 1st. April 1973
C
C Rewritten to call F07ADG, a modified version of LAPACK routine
C SGETRF/F07ADF; new IFAIL exit inserted for illegal input
C parameters; error messages inserted. February 1991.
C
SUBROUTINE F04AXF(N,A,LICN,ICN,IKEEP,RHS,W,MTYPE,IDISP,RESID)
C MARK 7 RELEASE. NAG COPYRIGHT 1978
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C DERIVED FROM HARWELL LIBRARY ROUTINE MA28C
C
C THE PARAMETERS ARE AS FOLLOWS ....
C N INTEGER ORDER OF MATRIX NOT ALTERED BY SUBROUTINE.
C A REAL ARRAY LENGTH LICN. THE SAME ARRAY AS WAS USED
C . IN THE MOST RECENT CALL TO F01BRF OR F01BSF.
C LICN INTEGER LENGTH OF ARRAYS A AND ICN. NOT ALTERED BY
C . SUBROUTINE.
C ICN INTEGER ARRAY LENGTH LICN. SAME ARRAY AS OUTPUT FROM
C . F01BRF. UNCHANGED BY F04AXF.
C IKEEP INTEGER ARRAY LENGTH 5*N. SAME ARRAY AS OUTPUT FROM
C . F01BRF. UNCHANGED BY F04AXF.
C RHS REAL ARRAY LENGTH N. ON ENTRY, IT HOLDS THE
C . RIGHT HAND SIDE. ON EXIT, THE SOLUTION VECTOR.
C W REAL ARRAY LENGTH N. USED AS WORKSPACE BY
C . F04AXZ.
C MTYPE INTEGER USED TO TELL F04AXZ TO SOLVE THE DIRECT
C . EQUATION (MTYPE=1) OR ITS TRANSPOSE (MTYPE.NE.1).
C IDISP INTEGER ARRAY LENGTH 2. SAME ARRAY AS OUTPUT BY
C . F01BRF. IT IS UNCHANGED BY F04AXF.
C RESID REAL VARIBLE. RETURNS MAXIMUM RESIDUAL OF EQUATIONS
C . WHERE PIVOT WAS ZERO.
C
C
C F04AXZ PREFORMS THE SOLUTION OF THE SET OF EQUATIONS
SUBROUTINE F04EAF(N,D,DU,DL,B,IFAIL)
C MARK 11 RELEASE. NAG COPYRIGHT 1983.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 14A REVISED. IER-688 (DEC 1989).
C
C F04EAF SOLVES THE EQUATIONS
C
C T*X = B ,
C
C WHERE T IS AN N BY N TRIDIAGONAL MATRIX, BY GAUSSIAN ELIMINATION
C WITH PARTIAL PIVOTING.
C
C FOR A DESCRIPTION OF THE PARAMETERS AND USE OF THIS ROUTINE SEE
C THE NAG LIBRARY MANUAL.
C
C -- WRITTEN ON 14-JANUARY-1983. S.J.HAMMARLING.
C
C NAG FORTRAN 66 BLACK BOX ROUTINE.
C
SUBROUTINE F04FAF(JOB,N,D,E,B,IFAIL)
C MARK 11 RELEASE. NAG COPYRIGHT 1983.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 14 REVISED. IER-741 (DEC 1989).
C
C F04FAF SOLVES THE EQUATIONS
C
C T*X = B ,
C
C WHERE T IS AN N BY N SYMMETRIC POSITIVE DEFINITE TRIDIAGONAL
C MATRIX, BY A MODIFIED CHOLESKY ALGORITHM.
C
C FOR A DESCRIPTION OF THE PARAMETERS AND USE OF THIS ROUTINE SEE
C THE NAG LIBRARY MANUAL.
C
C -- WRITTEN ON 17-JANUARY-1983. S.J.HAMMARLING.
C
SUBROUTINE F04FEF(N,T,X,WANTP,P,WANTV,V,VLAST,WORK,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C -- Written on 9-February-1990.
C This version dated 29-December-1990.
C Sven Hammarling, Nag Ltd.
C
SUBROUTINE F04FFF(N,T,B,X,WANTP,P,WORK,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C -- Written on 1-December-1990.
C This version dated 29-December-1990.
C Sven Hammarling, Nag Ltd.
C
SUBROUTINE F04JAF(M,N,A,NRA,B,TOL,SIGMA,IRANK,WORK,LWORK,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 16 REVISED. IER-1005 (JUN 1993).
C WRITTEN BY S. HAMMARLING, MIDDLESEX POLYTECHNIC (SVDLS1)
C
C F04JAF RETURNS THE N ELEMENT VECTOR X, OF MINIMAL
C LENGTH, THAT MINIMIZES THE EUCLIDEAN LENGTH OF THE M
C ELEMENT VECTOR R GIVEN BY
C
C R = B-A*X ,
C
C WHERE A IS AN M*N (M.GE.N) MATRIX AND B IS AN M ELEMENT
C VECTOR. X IS OVERWRITTEN ON B.
C
C THE SOLUTION IS OBTAINED VIA A SINGULAR VALUE
C DECOMPOSITION (SVD) OF THE MATRIX A GIVEN BY
C
C A = Q*(D)*(P**T) ,
C (0)
C
C WHERE Q AND P ARE ORTHOGONAL AND D IS A DIAGONAL MATRIX WITH
C NON-NEGATIVE DIAGONAL ELEMENTS, THESE BEING THE SINGULAR
C VALUES OF A.
C
C INPUT PARAMETERS.
C
C M - NUMBER OF ROWS OF A. M MUST BE AT LEAST N.
C
C N - NUMBER OF COLUMNS OF A. N MUST BE AT LEAST UNITY.
C
C A - AN M*N REAL MATRIX.
C
C NRA - ROW DIMENSION OF A AS DECLARED IN THE CALLING PROGRAM.
C NRA MUST BE AT LEAST M.
C
C B - AN M ELEMENT REAL VECTOR.
C
C TOL - A RELATIVE TOLERANCE USED TO DETERMINE THE RANK OF A.
C TOL SHOULD BE CHOSEN AS APPROXIMATELY THE
C LARGEST RELATIVE ERROR IN THE ELEMENTS OF A.
C FOR EXAMPLE IF THE ELEMENTS OF A ARE CORRECT
C TO ABOUT 4 SIGNIFICANT FIGURES THEN TOL
C SHOULD BE CHOSEN AS ABOUT 5.0*10.0**(-4).
C
C IFAIL - THE USUAL FAILURE PARAMETER. IF IN DOUBT SET
C IFAIL TO ZERO BEFORE CALLING THIS ROUTINE.
C
C OUTPUT PARAMETERS.
C
C A - THE TOP N*N PART OF A WILL CONTAIN THE
C ORTHOGONAL MATRIX P**T OF THE SVD.
C THE REMAINDER OF A IS USED FOR INTERNAL WORKSPACE.
C
C B - THE FIRST N ELEMENTS OF B WILL CONTAIN THE
C MINIMAL LEAST SQUARES SOLUTION VECTOR X.
C
C SIGMA - IF M IS GREATER THAN IRANK THEN SIGMA WILL CONTAIN THE
C STANDARD ERROR GIVEN BY
C SIGMA=L(R)/SQRT(M-IRANK), WHERE L(R) DENOTES
C THE EUCLIDEAN LENGTH OF THE RESIDUAL VECTOR
C R. IF M=IRANK THEN SIGMA IS RETURNED AS ZERO.
C
C IRANK - THE RANK OF THE MATRIX A.
C
C IFAIL - ON NORMAL RETURN IFAIL WILL BE ZERO.
C IN THE UNLIKELY EVENT THAT THE QR-ALGORITHM
C FAILS TO FIND THE SINGULAR VALUES IN 50*N
C ITERATIONS THEN IFAIL IS SET TO 2.
C IF AN INPUT PARAMETER IS INCORRECTLY SUPPLIED
C THEN IFAIL IS SET TO UNITY.
C
C WORKSPACE PARAMETERS.
C
C WORK - A 4*N ELEMENT VECTOR.
C ON RETURN THE FIRST N ELEMENTS OF WORK WILL
C CONTAIN THE SINGULAR VALUES OF A ARRANGED IN
C DESCENDING ORDER.
C WORK(N+1) WILL CONTAIN THE TOTAL NUMBER OF ITERATIONS
C TAKEN BY THE QR-ALGORITHM.
C
C LWORK - THE LENGTH OF THE VECTOR WORK. LWORK MUST BE
C AT LEAST 4*N.
C
SUBROUTINE F04JDF(M,N,A,NRA,B,TOL,SIGMA,IRANK,WORK,LWORK,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C WRITTEN BY S. HAMMARLING, MIDDLESEX POLYTECHNIC (SVDLS2)
C
C F04JDF RETURNS THE N ELEMENT VECTOR X, OF MINIMAL
C LENGTH, THAT MINIMIZES THE EUCLIDEAN LENGTH OF THE M
C ELEMENT VECTOR R GIVEN BY
C
C R = B-A*X ,
C
C WHERE A IS AN M*N (M.LE.N) MATRIX AND B IS AN M ELEMENT
C VECTOR. X IS OVERWRITTEN ON B.
C
C THE SOLUTION IS OBTAINED VIA A SINGULAR VALUE
C DECOMPOSITION (SVD) OF THE MATRIX A GIVEN BY
C
C A = Q*(D 0)*(P**T) ,
C
C WHERE Q AND P ARE ORTHOGONAL AND D IS A DIAGONAL MATRIX WITH
C NON-NEGATIVE DIAGONAL ELEMENTS, THESE BEING THE SINGULAR
C VALUES OF A.
C
C INPUT PARAMETERS.
C
C M - NUMBER OF ROWS OF A. M MUST BE AT LEAST UNITY.
C
C N - NUMBER OF COLUMNS OF A. N MUST BE AT LEAST M.
C
C A - AN M*N REAL MATRIX.
C
C NRA - ROW DIMENSION OF A AS DECLARED IN THE CALLING PROGRAM.
C NRA MUST BE AT LEAST M.
C
C B - AN N ELEMENT REAL VECTOR.
C THE FIRST M ELEMENTS OF B MUST CONTAIN THE
C VECTOR B OF THE LEAST SQUARES PROBLEM.
C
C
C TOL - A RELATIVE TOLERANCE USED TO DETERMINE THE RANK OF A.
C TOL SHOULD BE CHOSEN AS APPROXIMATELY THE
C LARGEST RELATIVE ERROR IN THE ELEMENTS OF A.
C FOR EXAMPLE IF THE ELEMENTS OF A ARE CORRECT
C TO ABOUT 4 SIGNIFICANT FIGURES THEN TOL
C SHOULD BE CHOSEN AS ABOUT 5.0*10.0**(-4).
C
C IFAIL - THE USUAL FAILURE PARAMETER. IF IN DOUBT SET
C IFAIL TO ZERO BEFORE CALLING THIS ROUTINE.
C
C OUTPUT PARAMETERS.
C
C A - A WILL CONTAIN THE FIRST M ROWS OF THE
C ORTHOGONAL MATRIX P**T OF THE SVD.
C
C B - B WILL CONTAIN THE MINIMAL LEAST SQUARES
C SOLUTION VECTOR X.
C
C SIGMA - IF M IS GREATER THAN IRANK THEN SIGMA WILL CONTAIN THE
C STANDARD ERROR GIVEN BY
C SIGMA=L(R)/SQRT(M-IRANK), WHERE L(R) DENOTES
C THE EUCLIDEAN LENGTH OF THE RESIDUAL VECTOR
C R. IF M=IRANK THEN SIGMA IS RETURNED AS ZERO.
C
C IRANK - THE RANK OF THE MATRIX A.
C
C IFAIL - ON NORMAL RETURN IFAIL WILL BE ZERO.
C IN THE UNLIKELY EVENT THAT THE QR-ALGORITHM
C FAILS TO FIND THE SINGULAR VALUES IN 50*M
C ITERATIONS THEN IFAIL IS SET TO 2.
C IF AN INPUT PARAMETER IS INCORRECTLY SUPPLIED
C THEN IFAIL IS SET TO UNITY.
C
C WORKSPACE PARAMETERS.
C
C WORK - AN (M*M+4*M) ELEMENT VECTOR.
C ON RETURN THE FIRST M ELEMENTS OF WORK WILL
C CONTAIN THE SINGULAR VALUES OF A ARRANGED IN
C DESCENDING ORDER. WORK(M+1) WILL CONTAIN THE
C TOTAL NUMBER OF ITERATIONS TAKEN BY THE
C QR-ALGORITHM.
C
C LWORK - THE LENGTH OF THE VECTOR WORK.
C LWORK MUST BE AT LEAST M*M+4*M.
C
SUBROUTINE F04JGF(M,N,A,NRA,B,TOL,SVD,SIGMA,IRANK,WORK,LWORK,
* IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C WRITTEN BY S. HAMMARLING, MIDDLESEX POLYTECHNIC (SVDLS4)
C
C F04JGF RETURNS THE N ELEMENT VECTOR X, OF MINIMAL
C LENGTH, THAT MINIMIZES THE EUCLIDEAN LENGTH OF THE M
C ELEMENT VECTOR R GIVEN BY
C
C R = B-A*X ,
C
C WHERE A IS AN M*N (M.GE.N) MATRIX AND B IS AN M ELEMENT
C VECTOR. X IS OVERWRITTEN ON B.
C
C THE HOUSEHOLDER QU FACTORIZATION OF A GIVEN BY
C
C A = Q*(U) ,
C (0)
C
C WHERE Q IS ORTHOGONAL AND U IS UPPER TRIANGULAR, IS
C COMPUTED FIRST.
C
C IF U IS NON-SINGULAR THEN THE LEAST SQUARES SOLUTION, X,
C IS OBTAINED VIA THE QU FACTORIZATION.
C
C ON THE OTHER HAND IF U IS SINGULAR THEN THE SINGULAR VALUE
C DECOMPOSITION (SVD) OF U IS COMPUTED. THIS, TOGETHER
C WITH THE QU FACTORIZATION, GIVES THE SVD OF A GIVEN BY
C
C A = Q1*(D)*(P**T) ,
C (0)
C
C WHERE Q1 AND P ARE ORTHOGONAL AND D IS A DIAGONAL MATRIX WITH
C NON-NEGATIVE DIAGONAL ELEMENTS, THESE BEING THE SINGULAR
C VALUES OF A.
C
C IN THIS CASE THE MINIMAL LEAST SQUARES SOLUTION, X, IS
C OBTAINED VIA THE SVD OF A.
C
C DECISIONS ON SINGULARITY AND RANK ARE BASED UPON THE
C USER SUPPLIED PARAMETER TOL.
C
C INPUT PARAMETERS.
C
C M - NUMBER OF ROWS OF A. M MUST BE AT LEAST N.
C
C N - NUMBER OF COLUMNS OF A. N MUST BE AT LEAST UNITY.
C
C A - AN M*N REAL MATRIX.
C
C NRA - ROW DIMENSION OF A AS DECLARED IN THE CALLING PROGRAM.
C NRA MUST BE AT LEAST M.
C
C B - AN M ELEMENT REAL VECTOR.
C
C TOL - A RELATIVE TOLERANCE USED TO DETERMINE THE RANK OF A.
C TOL SHOULD BE CHOSEN AS APPROXIMATELY THE
C LARGEST RELATIVE ERROR IN THE ELEMENTS OF A.
C FOR EXAMPLE IF THE ELEMENTS OF A ARE CORRECT
C TO ABOUT 4 SIGNIFICANT FIGURES THEN TOL
C SHOULD BE CHOSEN AS ABOUT 5.0*10.0**(-4).
C
C IFAIL - THE USUAL FAILURE PARAMETER. IF IN DOUBT SET
C IFAIL TO ZERO BEFORE CALLING THIS ROUTINE.
C
C
C OUTPUT PARAMETERS.
C
C A - IF SVD IS RETURNED AS .TRUE. THEN THE TOP N*N
C PART OF A CONTAINS THE ORTHOGONAL MATRIX P**T
C OF THE SVD OF A. IN THIS CASE THE REMAINDER
C OF A IS USED FOR INTERNAL WORKSPACE.
C IF SVD IS RETURNED AS .FALSE. THEN A,
C TOGETHER WITH THE FIRST N ELEMENTS OF THE
C VECTOR WORK, CONTAINS DETAILS OF THE
C HOUSEHOLDER QU FACTORIZATION OF A AS RETURNED
C FROM ROUTINE F01QAF.
C
C B - THE FIRST N ELEMENTS OF B WILL CONTAIN THE
C SOLUTION VECTOR X.
C
C SVD - WILL BE .TRUE. IF THE SVD OF A HAS BEEN
C COMPUTED AND WILL BE .FALSE. IF ONLY THE QU
C FACTORIZATION OF A HAS BEEN COMPUTED.
C IT SHOULD BE NOTED THAT IT IS POSSIBLE FOR SVD TO BE
C .TRUE. AND FOR IRANK TO BE N. SUCH AN
C OCCURANCE MEANS THAT THE MATRIX U ONLY JUST
C FAILED THE TEST FOR NON-SINGULARITY.
C
C SIGMA - IF M IS GREATER THAN IRANK THEN SIGMA WILL CONTAIN THE
C STANDARD ERROR GIVEN BY
C SIGMA=L(R)/SQRT(M-IRANK), WHERE L(R) DENOTES
C THE EUCLIDEAN LENGTH OF THE RESIDUAL VECTOR
C R. IF M=IRANK THEN SIGMA IS RETURNED AS ZERO.
C
C IRANK - THE RANK OF THE MATRIX A.
C
C IFAIL - ON NORMAL RETURN IFAIL WILL BE ZERO.
C IN THE UNLIKELY EVENT THAT THE QR ALGORITHM
C FAILS TO FIND THE SINGULAR VALUES IN 50*N
C ITERATIONS THEN IFAIL IS SET TO 2. THIS
C FAILURE CANNOT OCCUR IF THE SVD OF A IS NOT
C COMPUTED.
C IF AN INPUT PARAMETER IS INCORRECTLY SUPPLIED
C THEN IFAIL IS SET TO UNITY.
C
C WORKSPACE PARAMETERS.
C
C WORK - A 4*N ELEMENT VECTOR.
C IF SVD IS RETURNED AS .TRUE. THEN THE FIRST N
C ELEMENTS OF WORK WILL CONTAIN THE SINGULAR
C VALUES OF A ARRANGED IN DESCENDING ORDER.
C IF SVD IS RETURNED AS .FALSE. THEN THE FIRST
C N ELEMENTS OF WORK WILL CONTAIN INFORMATION
C ON THE QU FACTORIZATION OF A. SEE OUTPUT
C PARAMETER A ABOVE.
C IF SVD IS RETURNED AS .FALSE. THEN WORK(N+1)
C WILL CONTAIN THE CONDITION NUMBER
C L(U)*L(U**(-1)) OF THE UPPER TRIANGULAR
C MATRIX U.
C IF SVD IS RETURNED AS .TRUE. THEN WORK(N+1) WILL
C CONTAIN THE TOTAL NUMBER OF ITERATIONS TAKEN BY THE
C QR-ALGORITHM.
C
C LWORK - THE LENGTH OF THE VECTOR WORK. LWORK MUST BE
C AT LEAST 4*N.
C
SUBROUTINE F04LEF(JOB,N,A,B,C,D,IN,Y,TOL,IFAIL)
C MARK 11 RELEASE. NAG COPYRIGHT 1983.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 14 REVISED. IER-743 (DEC 1989).
C MARK 16A REVISED. IER-1006 (JUN 1993).
C
C F04LEF SOLVES ONE OF THE SYSTEMS OF EQUATIONS
C
C T*X = Y , ( T**T )*X = Y , U*X = Y ,
C
C WHERE T IS AN N BY N TRIDIAGONAL MATRIX THAT HAS BEEN
C FACTORIZED AS
C
C T = P*L*U ,
C
C BY ROUTINE F01LEF.
C
C FOR A DESCRIPTION OF THE PARAMETERS AND USE OF THIS ROUTINE SEE
C THE NAG LIBRARY MANUAL.
C
C -- WRITTEN ON 19-JANUARY-1983. S.J.HAMMARLING.
C
C NAG FORTRAN 66 GENERAL PURPOSE ROUTINE.
C
SUBROUTINE F04LHF(TRANS,N,NBLOKS,BLKSTR,A,LENA,PIVOT,B,LDB,IR,
* IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C MARK 13B REVISED. IER-665 (AUG 1988).
C MARK 5 REVISED. IER-930 (APR 1991).
C
C Purpose:
C
C F04LHF calculates the approximate solution of a set of real linear
C equations with multiple right hand sides, AX = B or A(trans)X = B,
C where A is an almost block diagonal matrix which has been
C decomposed using F01LHF.
C
C
C Specification:
C
C SUBROUTINE F04LHF(TRANS,N,NBLOKS,BLKSTR,A,LENA,PIVOT,
C * B,LDB,IR,IFAIL)
C C INTEGER N,NBLOKS,BLKSTR(3,NBLOKS),LENA,PIVOT(N),
C C * LDB,IR,IFAIL
C C real A(LENA),B(LDB,IR)
C C CHARACTER*1 TRANS
C
C
C Description:
C
C The routine solves a set of real linear equations AX = B or
C A(trans)X = B, where
C A is almost block diagonal. A must first be decomposed by
C F01LHF. F04LHFthen computes X by forward and backward
C substitution over the blocks. The code utilizes Level 3 BLAS [2].
C
C
C References:
C
C [1] FORTRAN Packages for Solving Certain Almost Block Diagonal
C Linear Systems by Modified Alternate Row and Column Elimination,
C J.C.Diaz, G.Fairweather, P.Keast,
C ACM Transactions on Mathematical Software (1983) 9, 358-375
C
C [2] A Proposal for a Set of Level 3 Basic Linear Algebra
C Subprograms
C J.J.Dongarra, J.DuCroz, I.Duff, S.Hammarling
C NAG Technical Report (1987) In preparation
C
C
C Parameters:
C
C TRANS - CHARACTER*1.
C On entry, TRANS must specify what solutions are required as
C follows:
C TRANS = 'N' or 'n' AX = B
C TRANS = 'T' or 't' A(trans)X=B
C Unchanged on exit.
C
C N - INTEGER
C On entry, N must specify the number of linear equations
C defined by the matrix A. This must be the same parameter
C N supplied to F01LHF when decomposing A.
C N .gt. 0.
C Unchanged on exit.
C
C NBLOKS - INTEGER
C On entry, NBLOKS must specify the total number of blocks
C of the matrix A. This must be the same parameter NBLOKS
C supplied to F01LHF.
C 0 .lt. NBLOKS .le. N.
C Unchanged on exit.
C
C BLKSTR - INTEGER array of DIMENSION (3,NBLOKS)
C Before entry, BLKSTR must contain information which
C describes the block structure of A. It must be unchanged
C since the last call to F01LHF.
C Unchanged on exit.
C
C A - real array of DIMENSION (LENA)
C Before entry, A must contain the elements in the
C decomposition of A as output by F01LHF. It must be
C unchanged since the last call to F01LHF.
C Unchanged on exit.
C
C LENA - INTEGER
C On entry, LENA specifies the dimension of array A as
C declared in the (sub)program from which F04LHF is called.
C This must be the same parameter LENA supplied to F01LHF.
C LENA .ge. sum(BLKSTR(1,k)*BLKSTR(2,k);k=1,NBLOKS)
C Unchanged on exit.
C
C PIVOT - INTEGER array of DIMENSION (N)
C Before entry, PIVOT must contain the pivot information
C about the decomposition as output from F01LHF. It must be
C unchanged since the last call to F01LHF.
C Unchanged on exit.
C
C B - real array of DIMENSION (LDB,s) where s.ge.IR .
C Before entry, B must contain the elements of the IR right
C hand sides stored as columns.
C On succesful exit, B will contain the IR solution vectors.
C
C LDB - INTEGER
C On entry, LDB must specify the first dimension of array B
C as declared in the (sub)program from which F04LHF is called.
C LDB .ge. N.
C Unchanged on exit.
C
C IR - INTEGER
C On entry, IR must specify the number of right hand sides.
C IR .gt. 0.
C Unchanged on exit.
C
C IFAIL - INTEGER
C Before entry, IFAIL must be set to 0, -1 or 1. For users
C not familiar with this parameter (described in chapter P01),
C the recommended value is 0.
C Unless the routine detects an error (see section 6), IFAIL
C contains 0 on exit.
C
C
C Error Indicators:
C
C If on entry IFAIL = 0 or -1, explanatory error messages are output
C on the current error message unit (as defined by the NAG Library
C routine X04AAF).
C
C IFAIL = 1 N .lt. 1 or
C NBLOKS .lt. 1 or
C IR .lt. 1 or
C LDB .lt. N or
C N .lt. NBLOKS or
C LENA too small or
C illegal values detected in BLKSTR or
C illegal value for TRANS.
C
C
C Further Comments:
C
C None.
C
C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C
C AUXILIARY ROUTINES
C ------------------
C SSWAP : TO INTERCHANGE TWO VECTORS
C SGEMM : TO PERFORM MATRIX X MATRIX TYPE OPERATIONS
C STRSM : TO PERFORM TRIANGULAR SOLVES
C F04LHX: TO PERFORM TRAPEZOIDAL SOLVES
C F01LHZ: TO PERFORM CHECKS ON THE PARAMETER BLKSTR
C
C SGEMM AND STRSM ARE EXISTING LEVEL 3 BLAS
C (THEY CALL SGEMV AND STRSV RESPECTIVELY)
C F04LHX/STXSM IS A PROPOSED LEVEL 3 BLAS
C (IT CALLS F04LHW/STXSV, A PROPOSED LEVEL 2 BLAS)
C
C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C
C NOTE THAT THERE IS SPECIAL CODE TO HANDLE THE CASE WHEN IR = 1 .
C
C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C
SUBROUTINE F04MAF(N,NZ,A,LA,INI,LINI,INJ,B,EPS,KMAX,W,W1,IK,
* INFORM,IFLAG)
C MARK 11 RELEASE. NAG COPYRIGHT 1983.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 16A REVISED. IER-1007 (JUN 1993).
C
C F04MAF SOLVES THE EQUATIONS A*X = B, WHERE A IS A SPARSE
C SYMMETRIC POSITIVE DEFINITE MATRIX, USING A PRECONDITIONED
C CONJUGATE GRADIENT METHOD. A MUST HAVE BEEN FACTORIZED BY
C THE NAG LIBRARY ROUTINE F01MAF.
C
C THE ROUTINE IS BASED UPON THE HARWELL ROUTINE MA31B BY
C N MUNKSGAARD.
C
C THE DEVELOPMENT OF MA31B WAS SUPPORTED BY THE DANISH
C NATURAL RESEARCH COUNCIL, GRANT NUMBER 511-10069
C
C FOR A DESCRIPTION OF THE PARAMETERS AND USE OF THIS ROUTINE
C SEE THE NAG LIBRARY MANUAL. PARAMETER ASSOCIATION IS AS
C BELOW.
C
C ROUTINE DOCUMENT
C N N
C NZ NZ
C A A
C LA LA
C INI INDI
C LINI LINDI
C INJ INDJ
C B B
C EPS ACC
C KMAX NOITS
C W WORK
C W1 WORK1
C IK IWORK
C INFORM INFORM
C IFLAG IFAIL
C
C **************************************************
C
C
SUBROUTINE F04MBF(N,B,X,APROD,MSOLVE,PRECON,SHIFT,RTOL,ITNLIM,
* MSGLVL,ITN,ANORM,ACOND,RNORM,XNORM,WORK,RWORK,
* LRWORK,IWORK,LIWORK,INFORM,IFAIL)
C MARK 11 RELEASE. NAG COPYRIGHT 1983.
C MARK 11B REVISED. IER-459 (SEP 1984).
C MARK 11D REVISED. IER-472 (NOV 1985).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13A REVISED. IER-622 (APR 1988).
C
C DESCRIPTION
C -----------
C
C F04MBF IS DESIGNED TO SOLVE THE SYSTEM OF LINEAR EQUATIONS
C
C A*X = B
C
C WHERE A IS AN N*N SYMMETRIC MATRIX AND B IS A GIVEN VECTOR.
C THE MATRIX A IS NOT REQUIRED TO BE POSITIVE DEFINITE.
C ( IF A IS KNOWN TO BE DEFINITE, THE METHOD OF CONJUGATE
C GRADIENTS WILL USUALLY BE SOMEWHAT MORE EFFICIENT. )
C
C
C THE MATRIX A IS INTENDED TO BE LARGE AND SPARSE. IT IS ACCESSED
C BY MEANS OF A SUBROUTINE CALL OF THE FORM
C
C CALL APROD ( IFLAG, N, X, Y,
C RWORK, LRWORK, IWORK, LIWORK )
C
C WHICH MUST RETURN THE PRODUCT Y = A*X FOR ANY GIVEN VECTOR X.
C
C
C MORE GENERALLY, F04MBF IS DESIGNED TO SOLVE THE SYSTEM
C
C ( A - SHIFT*I )*X = B
C
C WHERE SHIFT IS A SPECIFIED SCALAR VALUE. IF SHIFT AND B
C ARE SUITABLY CHOSEN, THE COMPUTED VECTOR X MAY APPROXIMATE AN
C EIGENVECTOR OF A, AS IN THE METHODS OF INVERSE ITERATION
C AND/OR RAYLEIGH-QUOTIENT ITERATION.
C
C
C AGAIN, THE MATRIX ( A - SHIFT*I ) NEED NOT BE POSITIVE DEFINITE.
C THE WORK PER ITERATION IS VERY SLIGHTLY LESS IF SHIFT = 0.
C
C A FURTHER OPTION IS THAT OF PRECONDITIONING , WHICH MAY REDUCE
C THE NUMBER OF ITERATIONS REQUIRED. IF M IS A POSITIVE
C DEFINITE MATRIX WHICH IS KNOWN TO APPROXIMATE ( A - SHIFT*I )
C IN SOME SENSE, AND IF SYSTEMS OF THE FORM M*Y = X CAN BE
C SOLVED EFFICIENTLY, THE PARAMETERS PRECON AND MSOLVE MAY BE
C USEFUL ( SEE BELOW ). WHEN PRECON = .TRUE. , F04MBF WILL
C IMPLICITLY SOLVE THE SYSTEM OF EQUATIONS
C
C M**( -1/2 )*( A - SHIFT*I )*M**( -1/2 )*Y = M**( -1/2 )*B,
C
C AND RETURN THE SOLUTION X = M**( -1/2 )*Y.
C
C PARAMETERS
C ----------
C
C N INPUT THE DIMENSION OF THE MATRIX A.
C
C B( N ) INPUT THE RHS VECTOR B.
C
C X( N ) OUTPUT RETURNS THE COMPUTED SOLUTION X.
C
C APROD EXTERNAL A SUBROUTINE DEFINING THE MATRIX A.
C FOR A GIVEN VECTOR X, THE STATEMENT
C
C CALL APROD ( IFLAG, N, X, Y,
C RWORK, LRWORK, IWORK, LIWORK )
C
C SHOULD RETURN THE PRODUCT Y = A*X.
C APROD MUST NOT ALTER THE VECTOR X.
C
C MSOLVE EXTERNAL AN OPTIONAL SUBROUTINE DEFINING A
C PRECONDITIONING MATRIX M, WHICH WILL USUALLY
C APPROXIMATE ( A - SHIFT*I ) IN SOME SENSE.
C M MUST BE POSITIVE DEFINITE.
C FOR A GIVEN VECTOR X, THE STATEMENT
C
C CALL MSOLVE( IFLAG, N, X, Y,
C RWORK, LRWORK, IWORK, LIWORK )
C
C SHOULD SOLVE THE LINEAR SYSTEM M*Y = X.
C MSOLVE MUST NOT ALTER THE VECTOR X.
C
C NOTE. THE PROGRAM CALLING F04MBF MUST
C DECLARE APROD AND MSOLVE TO BE EXTERNAL.
C
C PRECON INPUT IF PRECON = .TRUE. , PRECONDITIONING WILL
C BE INVOKED. OTHERWISE, SUBROUTINE MSOLVE
C WILL NOT BE REFERENCED. IN THIS CASE THE
C ACTUAL PARAMETER CORRESPONDING TO MSOLVE MAY
C BE THE SAME AS THAT CORRESPONDING TO APROD .
C
C SHIFT INPUT SHOULD BE ZERO IF THE SYSTEM A*X = B IS TO
C BE SOLVED. OTHERWISE SHIFT COULD BE AN
C APPROXIMATION TO AN EIGENVALUE OF A,
C E.G. THE RAYLEIGH QUOTIENT
C B( T )*A*B/( B( T )*B )
C CORRESPONDING TO THE VECTOR B.
C
C RTOL INPUT A USER-SPECIFIED TOLERANCE. F04MBF TERMINATES
C IF IT APPEARS THAT NORM( RBAR ) IS SMALLER
C THAN RTOL*NORM( ABAR )*NORM( Y ), WHERE
C ABAR IS THE TRANSFORMED MATRIX OPERATOR
C
C ABAR = M**( -1/2 )*( A - SHIFT*I )*M**( -1/2 )
C
C AND RBAR IS THE TRANSFORMED RESIDUAL VECTOR
C
C RBAR = M**( -1/2 )*( B - ( A - SHIFT*I )*X )
C
C ( AND M = I IF PRECON = .FALSE. ).
C
C ITNLIM INPUT AN UPPER LIMIT ON THE NUMBER OF ITERATIONS.
C
C MSGLVL INPUT DETERMINES THE LEVEL OF PRINTING.
C IF MSGLVL .GT. 1 THEN PROGRESS IS MONITERED
C BY PRINTING OUT A SUMMARY LINE PERIODICALLY.
C IF MSGLVL .GT. 0 THEN A SUMMARY IS PRINTED
C AT THE END OF THE ROUTINE.
C OUTPUT IS ON THE UNIT GIVEN BY X04ABF.
C
C INFORM OUTPUT AN INTEGER GIVING THE REASON FOR TERMINATION...
C
C 0 B = 0, SO THE EXACT SOLUTION IS X = 0.
C NO ITERATIONS WERE PERFORMED.
C
C 1 NORM( RBAR ) APPEARS TO BE LESS THAN
C THE VALUE RTOL*NORM( ABAR )*NORM( Y ).
C
C 2 NORM( RBAR ) APPEARS TO BE LESS THAN
C THE VALUE EPS*NORM( ABAR )*NORM( Y ).
C THIS MEANS THAT THE RESIDUAL IS AS SMALL AS
C SEEMS REASONABLE ON THIS MACHINE.
C
C 3 NORM( ABAR )*NORM( Y ) EXCEEDS NORM( B )/EPS,
C WHICH SHOULD INDICATE THAT X HAS ESSENTIALLY
C CONVERGED TO AN EIGENVECTOR OF A
C CORRESPONDING TO THE EIGENVALUE SHIFT.
C
C 4 THE ITERATION LIMIT WAS REACHED BEFORE ANY OF
C THE PREVIOUS CRITERIA WERE SATISFIED.
C
C 5 THE PRECONDITIONING MATRIX M DOES NOT APPEAR
C TO BE POSITIVE DEFINITE. X WILL NOT BE AN
C ACCEPTABLE SOLUTION.
C
C ITN OUTPUT THE NUMBER OF ITERATIONS PERFORMED.
C
C ANORM OUTPUT AN ESTIMATE OF THE NORM OF THE MATRIX OPERATOR
C ABAR = M**( -1/2 )*( A - SHIFT*I )*M**( -1/2 ).
C
C ACOND OUTPUT AN ESTIMATE OF THE CONDITION OF ABAR ABOVE.
C THIS WILL USUALLY BE A SUBSTANTIAL
C UNDER-ESTIMATE OF THE TRUE CONDITION.
C
C RNORM OUTPUT THE NORM OF THE FINAL RESIDUAL VECTOR,
C R = B - ( A - SHIFT*I )*X.
C
C XNORM OUTPUT THE NORM OF THE FINAL SOLUTION VECTOR X.
C
C WORK WORKSPACE OF LENGTH AL LEAST 5*N.
C
C REFERENCES
C ----------
C
C THIS ROUTINE IS DERIVED FROM THE SUBROUTINE SYMMLQ DESCRIBED IN
C THE FOLLOWING REFERENCES...
C
C C.C. PAIGE AND M.A. SAUNDERS, SOLUTION OF SPARSE INDEFINITE
C SYSTEMS OF EQUATIONS AND LEAST SQUARES PROBLEMS,
C TECHNICAL REPORT STAN-CS-73-399, COMPUTER SCIENCE DEPARTMENT,
C STANFORD UNIVERSITY, STANFORD, CALIFORNIA, NOVEMBER 1973.
C
C C.C. PAIGE AND M.A. SAUNDERS, SOLUTION OF SPARSE INDEFINITE
C SYSTEMS OF LINEAR EQUATIONS,
C SIAM J. NUMER. ANAL. 12, 4, SEPTEMBER 1975, PP. 617-629.
C
C J.G. LEWIS, ALGORITHMS FOR SPARSE MATRIX EIGENVALUE PROBLEMS,
C TECHNICAL REPORT STAN-CS-77-595, COMPUTER SCIENCE DEPARTMENT,
C STANFORD UNIVERSITY, STANFORD, CALIFORNIA, MARCH 1977.
C
C FURTHER COMMENTS
C ----------------
C
C SYMMLQ. VERSION DATED 13 OCT 1981. MAS.
C F04MBF. THIS VERSION DATED 11 OCT 1983. S.J.HAMMARLING
C F04MBFT. THIS VERSION DATED 3 JAN 1984. S.J.HAMMARLING
C SETS W(1)BAR = V(1).
C
C THE VECTORS Y, V, R1, R2 AND W OF SYMMLQ HAVE BEEN COMBINED INTO
C THE ARRAY WORK IN F04MBF AS FOLLOWS.
C
C Y( I ) = WORK( I, 1 ), I = 1, 2, ..., N
C V( I ) = WORK( I, 2 ), I = 1, 2, ..., N
C R1( I ) = WORK( I, 3 ), I = 1, 2, ..., N
C R2( I ) = WORK( I, 4 ), I = 1, 2, ..., N
C W( I ) = WORK( I, 5 ), I = 1, 2, ..., N.
C
C IN ALL THE EMBEDDED COMMENTS IN THE CODE TAKE M = I IF
C PRECON = .FALSE..
C
C SUBROUTINES AND FUNCTIONS
C
C USER APROD , MSOLVE
C BLAS AXPY , COPY , DOT , NRM2
C NAG BLAS CNSV , NROT , NROTG , SCMV
C NAG X02ZAZ
C FORTRAN ABS , AMAX1 , AMIN1 , MOD , SQRT
C
C APROD, MSOLVE
SUBROUTINE F04MCF(N,L,LL,D,NROW,P,B,NRB,ISELCT,X,NRX,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 14 REVISED. IER-746 (DEC 1989).
C
C ******************************************************
C
C NPL ALGORITHMS LIBRARY ROUTINE VBSOLC
C
C CREATED 16 05 79. UPDATED 07 09 79. RELEASE 00/12
C
C AUTHOR ... MAURICE G. COX.
C NATIONAL PHYSICAL LABORATORY, TEDDINGTON,
C MIDDLESEX TW11 OLW, ENGLAND.
C
C ******************************************************
C
C F04MCF. AN ALGORITHM TO DETERMINE THE SOLUTION OF THE
C SYSTEM AX = B (AND RELATED SYSTEMS), WHERE A IS A
C SYMMETRIC POSITIVE DEFINITE VARIABLE BANDWIDTH MATRIX,
C FOLLOWING AN LDU FACTORIZATION OF A (WHERE
C U = L TRANSPOSED).
C
C INPUT PARAMETERS
C N ORDER OF L
C L ELEMENTS WITHIN ENVELOPE OF UNIT LOWER
C TRIANGULAR MATRIX L (INCLUDING UNIT
C DIAGONAL ELEMENTS), IN ROW BY ROW ORDER
C LL DIMENSION OF L. AT LEAST
C NROW(1) + NROW(2) + ... + NROW(N).
C D DIAGONAL ELEMENTS OF DIAGONAL MATRIX D
C NROW WIDTHS OF ROWS OF L
C P NUMBER OF COLUMNS OF B.
C GREATER THAN OR EQUAL TO 1.
C B RIGHT HAND SIDE VECTORS, STORED BY COLUMNS
C NRB ROW DIMENSION OF B, AS DECLARED IN CALLING
C PROGRAM UNIT
C ISELCT SELECTION INDICATOR
C 1 - SOLVE LDUX = B (NORMAL APPLICATION)
C 2 - SOLVE LDX = B (LOWER TRIANGULAR
C SOLVER)
C 3 - SOLVE DUX = B (UPPER TRIANGULAR
C SOLVER)
C 4 - SOLVE LUX = B
C 5 - SOLVE LX = B (UNIT LOWER
C TRIANGULAR SOLVER)
C 6 - SOLVE UX = B (UNIT UPPER
C TRIANGULAR SOLVER),
C WHERE U = L TRANSPOSED.
C
C OUTPUT (AND ASSOCIATED) PARAMETERS
C X SOLUTION VECTORS, STORED BY COLUMNS
C NRX ROW DIMENSION OF X, AS DECLARED IN CALLING
C PROGRAM UNIT
C
C FAILURE INDICATORS
C IFAIL FAILURE INDICATOR
C 0 - SUCCESSFUL TERMINATION
C 1 - AT LEAST ONE OF THE FOLLOWING
C RESTRICTIONS RELATING TO N, LL
C AND THE ARRAY NROW IS VIOLATED.
C N .GE. 1,
C NROW(1) .GE. 1 AND .LE. I,
C FOR I = 1, 2, ..., N,
C NROW(1) + NROW(2) + ...
C ... + NROW(N) .LE. LL.
C 2 - AT LEAST ONE OF THE FOLLOWING
C RESTRICTIONS RELATING TO P,
C NRB AND NRX IS VIOLATED.
C P .GE. 1,
C NRB .GE. N,
C NRX .GE. N
C 3 - ISELCT IS LESS THAN 1 OR GREATER THAN 6
C 4 - D SINGULAR
C (APPLIES ONLY IF ISELCT .LE. 3)
C 5 - L NOT U N I T TRIANGULAR, AS EXPECTED
C
SUBROUTINE F04MEF(N,T,X,V,WORK,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C -- Written on 9-February-1990.
C This version dated 8-January-1991.
C Sven Hammarling, Nag Ltd.
C
SUBROUTINE F04MFF(N,T,B,X,P,WORK,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C -- Written on 9-February-1990.
C This version dated 8-January-1991.
C Sven Hammarling, Nag Ltd.
C
SUBROUTINE F04QAF(M,N,B,X,SE,APROD,DAMP,ATOL,BTOL,CONLIM,ITNLIM,
* MSGLVL,ITN,ANORM,ACOND,RNORM,ARNORM,XNORM,WORK,
* RWORK,LRWORK,IWORK,LIWORK,INFORM,IFAIL)
C MARK 11 RELEASE. NAG COPYRIGHT 1983.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C DESCRIPTION
C -----------
C
C F04QAF FINDS A SOLUTION X TO THE FOLLOWING PROBLEMS...
C
C 1. UNSYMMETRIC EQUATIONS -- SOLVE A*X = B
C
C 2. LINEAR LEAST SQUARES -- SOLVE A*X = B
C IN THE LEAST-SQUARES SENSE
C
C 3. DAMPED LEAST SQUARES -- SOLVE ( A )*X = ( B )
C ( DAMP*I ) ( 0 )
C IN THE LEAST-SQUARES SENSE
C
C WHERE A IS A MATRIX WITH M ROWS AND N COLUMNS, B IS AN
C M-VECTOR, AND DAMP IS A SCALAR ( ALL QUANTITIES REAL ).
C THE MATRIX A IS INTENDED TO BE LARGE AND SPARSE. IT IS ACCESSED
C BY MEANS OF SUBROUTINE CALLS OF THE FORM
C
C CALL APROD ( MODE, M, N, X, Y, RWORK, LRWORK, IWORK, LIWORK )
C
C WHICH MUST PERFORM THE FOLLOWING FUNCTIONS...
C
C IF MODE = 1, COMPUTE Y = Y + A*X.
C IF MODE = 2, COMPUTE X = X + A( TRANSPOSE )*Y.
C
C THE VECTORS X AND Y ARE INPUT PARAMETERS IN BOTH CASES.
C IF MODE = 1, Y SHOULD BE ALTERED WITHOUT CHANGING X.
C IF MODE = 2, X SHOULD BE ALTERED WITHOUT CHANGING Y.
C THE PARAMETERS RWORK, LRWORK, IWORK, LIWORK MAY BE USED FOR
C WORKSPACE AS DESCRIBED BELOW.
C
C THE RHS VECTOR B IS OVERWRITTEN.
C
C NOTE. F04QAF USES AN ITERATIVE METHOD TO APPROXIMATE THE
C SOLUTION.
C THE NUMBER OF ITERATIONS REQUIRED TO REACH A CERTAIN ACCURACY
C DEPENDS STRONGLY ON THE SCALING OF THE PROBLEM. POOR SCALING OF
C THE ROWS OR COLUMNS OF A SHOULD THEREFORE BE AVOIDED WHERE
C POSSIBLE.
C
C FOR EXAMPLE, IN PROBLEM 1 THE SOLUTION IS UNALTERED BY
C ROW-SCALING. IF A ROW OF A IS VERY SMALL OR LARGE COMPARED TO
C THE OTHER ROWS OF A, THE CORRESPONDING ROW OF ( A B ) SHOULD
C BE SCALED UP OR DOWN.
C
C IN PROBLEMS 1 AND 2, THE SOLUTION X IS EASILY RECOVERED
C FOLLOWING COLUMN-SCALING. IN THE ABSENCE OF BETTER INFORMATION,
C THE NONZERO COLUMNS OF A SHOULD BE SCALED SO THAT THEY ALL HAVE
C THE SAME EUCLIDEAN NORM ( E.G. 1.0 ).
C
C IN PROBLEM 3, THERE IS NO FREEDOM TO RE-SCALE IF DAMP IS
C NONZERO. HOWEVER, THE VALUE OF DAMP SHOULD BE ASSIGNED ONLY
C AFTER ATTENTION HAS BEEN PAID TO THE SCALING OF A.
C
C THE PARAMETER DAMP IS INTENDED TO HELP REGULARIZE
C ILL-CONDITIONED SYSTEMS, BY PREVENTING THE TRUE SOLUTION FROM
C BEING VERY LARGE. ANOTHER AID TO REGULARIZATION IS PROVIDED BY
C THE PARAMETER ACOND, WHICH MAY BE USED TO TERMINATE ITERATIONS
C BEFORE THE COMPUTED SOLUTION BECOMES VERY LARGE.
C
C THE FOLLOWING QUANTITIES ARE USED IN DISCUSSING THE SUBROUTINE
C PARAMETERS...
C
C ABAR = ( A ), BBAR = ( B )
C ( DAMP*I ) ( 0 )
C
C R = B - A*X, RBAR = BBAR - ABAR*X
C
C RNORM = SQRT( NORM( R )**2 + ( DAMP**2 )*( NORM( X )**2 ) )
C = NORM( RBAR )
C
C RELPR = THE RELATIVE PRECISION OF FLOATING-POINT ARITHMETIC
C ON THE MACHINE BEING USED.
C
C F04QAF MINIMIZES THE FUNCTION RNORM WITH RESPECT TO X.
C
C PARAMETERS
C ----------
C
C M INPUT THE NUMBER OF ROWS IN A.
C
C N INPUT THE NUMBER OF COLUMNS IN A.
C
C APROD EXTERNAL SEE ABOVE.
C
C DAMP INPUT THE DAMPING PARAMETER FOR PROBLEM 3 ABOVE.
C ( DAMP SHOULD BE 0.0 FOR PROBLEMS 1 AND 2. )
C IF THE SYSTEM A*X = B IS INCOMPATIBLE, VALUES
C OF DAMP IN THE RANGE 0 TO
C SQRT( RELPR )*NORM( A )
C WILL PROBABLY HAVE A NEGLIGIBLE EFFECT.
C LARGER VALUES OF DAMP WILL TEND TO DECREASE
C THE NORM OF X AND TO REDUCE THE NUMBER OF
C ITERATIONS REQUIRED BY F04QAF.
C
C THE WORK PER ITERATION AND THE STORAGE NEEDED
C BY F04QAF ARE THE SAME FOR ALL VALUES OF DAMP.
C
C LIWORK INPUT THE LENGTH OF THE WORKSPACE ARRAY IWORK.
C LRWORK INPUT THE LENGTH OF THE WORKSPACE ARRAY RWORK.
C IWORK WORKSPACE AN INTEGER ARRAY OF LENGTH LIWORK.
C RWORK WORKSPACE A REAL ARRAY OF LENGTH LRWORK.
C
C NOTE. F04QAF DOES NOT EXPLICITLY USE THE PREVIOUS FOUR
C PARAMETERS, BUT PASSES THEM TO SUBROUTINE APROD FOR
C POSSIBLE USE AS WORKSPACE. IF APROD DOES NOT NEED
C IW OR RW, THE VALUES LENIW = 1 OR LENRW = 1 SHOULD
C BE USED, AND THE ACTUAL PARAMETERS CORRESPONDING TO
C IW OR RW MAY BE ANY CONVENIENT ARRAY OF SUITABLE TYPE.
C
C B( M ) INPUT THE RHS VECTOR. BEWARE THAT B IS
C OVER-WRITTEN BY F04QAF.
C
C X( N ) OUTPUT RETURNS THE COMPUTED SOLUTION X.
C
C SE( N ) OUTPUT RETURNS STANDARD ERROR ESTIMATES FOR THE
C COMPONENTS OF X. FOR EACH I, SE( I ) IS SET
C TO THE VALUE RNORM*SQRT( SIGMA( I, I )/T ),
C WHERE SIGMA( I, I ) IS AN ESTIMATE OF THE ITH
C DIAGONAL OF THE INVERSE OF
C ABAR( TRANSPOSE )*ABAR
C AND T = 1 IF M .LE. N,
C T = M - N IF M .GT. N AND DAMP = 0,
C T = M IF DAMP .NE. 0.
C
C ATOL INPUT AN ESTIMATE OF THE RELATIVE ERROR IN THE DATA
C DEFINING THE MATRIX A. FOR EXAMPLE,
C IF A IS ACCURATE TO ABOUT 6 DIGITS, SET
C ATOL = 1.0E-6 .
C
C BTOL INPUT AN ESTIMATE OF THE RELATIVE ERROR IN THE DATA
C DEFINING THE RHS VECTOR B. FOR EXAMPLE,
C IF B IS ACCURATE TO ABOUT 6 DIGITS, SET
C BTOL = 1.0E-6 .
C
C CONLIM INPUT AN UPPER LIMIT ON COND( ABAR ), THE APPARENT
C CONDITION NUMBER OF THE MATRIX ABAR.
C ITERATIONS WILL BE TERMINATED IF A COMPUTED
C ESTIMATE OF COND( ABAR ) EXCEEDS CONLIM.
C THIS IS INTENDED TO PREVENT CERTAIN SMALL OR
C ZERO SINGULAR VALUES OF A OR ABAR FROM
C COMING INTO EFFECT AND CAUSING UNWANTED GROWTH
C IN THE COMPUTED SOLUTION.
C
C CONLIM AND DAMP MAY BE USED SEPARATELY OR
C TOGETHER TO REGULARIZE ILL-CONDITIONED SYSTEMS.
C
C NORMALLY, CONLIM SHOULD BE IN THE RANGE
C 1000 TO 1/RELPR.
C SUGGESTED VALUE --
C CONLIM = 1/( 100*RELPR ) FOR COMPATIBLE
C SYSTEMS,
C CONLIM = 1/( 10*SQRT( RELPR ) ) FOR LEAST
C SQUARES.
C
C NOTE. IF THE USER IS NOT CONCERNED ABOUT THE PARAMETERS
C ATOL, BTOL AND CONLIM, ANY OR ALL OF THEM MAY BE SET
C TO ZERO. THE EFFECT WILL BE THE SAME AS THE VALUES
C RELPR, RELPR AND 1/RELPR RESPECTIVELY.
C
C ITNLIM INPUT AN UPPER LIMIT ON THE NUMBER OF ITERATIONS.
C SUGGESTED VALUE --
C ITNLIM = N/2 FOR WELL CONDITIONED SYSTEMS,
C ITNLIM = 4*N OTHERWISE.
C
C ITN OUTPUT THE NUMBER OF ITERATIONS TAKEN.
C
C MSGLVL INPUT DETERMINES THE LEVEL OF PRINTING.
C IF MSGLVL .GT. 1 THEN PROGRESS IS MONITERED
C BY PRINTING OUT A SUMMARY LINE PERIODICALLY.
C IF MSGLVL .GT. 0 THEN A SUMMARY IS PRINTED
C AT THE END OF THE ROUTINE.
C OUTPUT IS ON THE UNIT GIVEN BY X04ABF.
C
C INFORM OUTPUT AN INTEGER GIVING THE REASON FOR TERMINATION...
C
C 0 X = 0 IS THE EXACT SOLUTION.
C NO ITERATIONS WERE PERFORMED.
C
C 1 THE EQUATIONS A*X = B ARE PROBABLY
C COMPATIBLE. NORM( A*X - B ) IS SUFFICIENTLY
C SMALL, GIVEN THE VALUES OF ATOL AND BTOL.
C
C 2 THE SYSTEM A*X = B IS PROBABLY NOT
C COMPATIBLE. A LEAST-SQUARES SOLUTION HAS
C BEEN OBTAINED WHICH IS SUFFICIENTLY ACCURATE,
C GIVEN THE VALUE OF ATOL.
C
C 3 THE EQUATIONS A*X = B ARE PROBABLY
C COMPATIBLE. NORM( A*X - B ) IS AS SMALL AS
C SEEMS REASONABLE ON THIS MACHINE.
C
C 4 THE SYSTEM A*X = B IS PROBABLY NOT
C COMPATIBLE. A LEAST-SQUARES SOLUTION HAS
C BEEN OBTAINED WHICH IS AS ACCURATE AS SEEMS
C REASONABLE ON THIS MACHINE.
C
C 5 AN ESTIMATE OF COND( ABAR ) HAS EXCEEDED
C CONLIM. THE SYSTEM A*X = B APPEARS TO BE
C ILL-CONDITIONED. OTHERWISE, THERE COULD BE AN
C AN ERROR IN SUBROUTINE APROD .
C
C 6 COND( ABAR ) SEEMS TO BE SO LARGE THAT THERE
C IS NOT MUCH POINT IN DOING FURTHER ITERATIONS,
C GIVEN THE PRECISION OF THIS MACHINE.
C THERE COULD BE AN ERROR IN SUBROUTINE APROD.
C
C 7 THE ITERATION LIMIT ITNLIM WAS REACHED.
C
C ANORM OUTPUT AN ESTIMATE OF THE FROBENIUS NORM OF ABAR.
C THIS IS THE SQUARE-ROOT OF THE SUM OF SQUARES
C OF THE ELEMENTS OF ABAR.
C IF DAMP IS SMALL AND IF THE COLUMNS OF A
C HAVE ALL BEEN SCALED TO HAVE LENGTH 1.0,
C ANORM SHOULD INCREASE TO ROUGHLY SQRT( N ).
C A RADICALLY DIFFERENT VALUE FOR ANORM MAY
C INDICATE AN ERROR IN SUBROUTINE APROD ( THERE
C MAY BE AN INCONSISTENCY BETWEEN MODES 1 AND
C 2 ).
C
C ACOND OUTPUT AN ESTIMATE OF COND( ABAR ), THE CONDITION
C NUMBER OF ABAR. A VERY HIGH VALUE OF ACOND
C MAY AGAIN INDICATE AN ERROR IN APROD.
C
C RNORM OUTPUT AN ESTIMATE OF THE FINAL VALUE OF NORM( RBAR ),
C THE FUNCTION BEING MINIMIZED ( SEE NOTATION
C ABOVE ). THIS WILL BE SMALL IF A*X = B HAS
C A SOLUTION.
C
C ARNORM OUTPUT AN ESTIMATE OF THE FINAL VALUE OF
C NORM( ABAR( TRANSPOSE )*RBAR ), THE NORM OF
C THE RESIDUAL FOR THE USUAL NORMAL EQUATIONS.
C THIS SHOULD BE SMALL IN ALL CASES. ( ARNORM
C WILL OFTEN BE SMALLER THAN THE TRUE VALUE
C COMPUTED FROM THE OUTPUT VECTOR X. )
C
C XNORM OUTPUT AN ESTIMATE OF THE NORM OF THE FINAL
C SOLUTION VECTOR X.
C
C WORK WORKSPACE ARRAY OF LENGTH AT LEAST 2*N.
C
C
C SUBROUTINES AND FUNCTIONS USED
C ------------------------------
C
C USER APROD
C BLAS COPY , NRM2 , SCAL
C NAG BLAS CNSV , ROTG
C NAG X02ZAZ
C FORTRAN ABS , MOD , SQRT
C
C REFERENCES
C ----------
C
C PAIGE, C.C. AND SAUNDERS, M.A. LSQR: AN ALGORITHM FOR SPARSE
C LINEAR EQUATIONS AND SPARSE LEAST SQUARES.
C ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE 8, 1 ( MARCH 1982 ).
C
C LAWSON, C.L., HANSON, R.J., KINCAID, D.R. AND KROGH, F.T.
C BASIC LINEAR ALGEBRA SUBPROGRAMS FOR FORTRAN USAGE.
C ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE 5, 3 ( SEPT 1979 ),
C 308-323 AND 324-325.
C
C FURTHER COMMENTS
C ----------------
C
C LSQR. VERSION DATED 22 FEBRUARY 1982. MAS.
C F04QAF. THIS VERSION DATED 11 OCT 1983. S.J.HAMMARLING
C
C IN THIS VERSION THE VECTORS V AND W ARE STORED IN THE ARRAY
C WORK AS FOLLOWS.
C
C V( I ) = WORK( I, 1 ), I = 1, 2, ..., N
C W( I ) = WORK( I, 2 ), I = 1, 2, ..., N.
C
C APROD
SUBROUTINE F04YAF(JOB,P,SIGMA,A,NRA,SVD,IRANK,SV,CJ,WORK,IFAIL)
C MARK 11 RELEASE. NAG COPYRIGHT 1983.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C F04YAF RETURNS ELEMENTS OF THE ESTIMATED VARIANCE-COVARIANCE
C MATRIX OF THE SAMPLE REGRESSION COEFFICIENTS FOR THE SOLUTION
C OF A LINEAR REGRESSION PROBLEM.
C
C THE ROUTINE CAN BE USED TO FIND THE ESTIMATED VARIANCES OF THE
C SAMPLE REGRESSION COEFFICIENTS.
C
C FOR A DESCRIPTION OF THE PARAMETERS AND USE OF THIS ROUTINE SEE
C THE NAG LIBRARY MANUAL.
C
C -- WRITTEN ON 29-NOVEMBER-1982. S.J. HAMMARLING.
C
C NAG FORTRAN 66 ROUTINE.
C
SUBROUTINE F04YCF(ICASE,N,X,ESTNRM,WORK,IWORK,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C MARK 15 REVISED. IER-913 (APR 1991).
C
C F04YCF estimates the 1-norm of a square, real matrix A.
C Reverse communication is used for evaluating matrix-vector
C products.
C
C On entry,
C
C N integer. The order of the matrix. N .ge. 1.
C
C IWORK integer(N). Used as workspace.
C
C ICASE integer. ICASE must be set to zero on initial entry.
C
C On intermediate returns:
C
C ICASE = 1 or 2.
C
C X real(N), must be overwritten by
C
C A*X, if ICASE=1,
C transpose(A)*X, if ICASE=2,
C
C and F04YCF must be re-called, with all the other
C parameters unchanged.
C
C On final return,
C
C ICASE = 0.
C
C ESTNRM real. Contains an estimate (a lower bound) for norm(A).
C
C WORK real(N). Contains vector V such that V = A*W, where
C ESTNRM = norm(V)/norm(W) (W is not returned).
C
C Nick Higham, University of Manchester.
C
C Reference:
C N.J. Higham (1987) Fortran Codes for Estimating
C the One-norm of a Real or Complex Matrix, with Applications
C to Condition Estimation, Numerical Analysis Report No. 135,
C University of Manchester, Manchester M13 9PL, England.
C
SUBROUTINE F04ZCF(ICASE,N,X,ESTNRM,WORK,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C MARK 15 REVISED. IER-914 (APR 1991).
C
C F04ZCF estimates the 1-norm of a square, complex matrix A.
C Reverse communication is used for evaluating matrix-vector
C products.
C
C On entry,
C
C N integer. The order of the matrix. N .ge. 1.
C
C ICASE integer. ICASE must be set to zero on initial entry.
C
C On intermediate returns
C
C ICASE = 1 or 2.
C
C X complex(N). X must be overwritten by
C
C A*X, if ICASE=1,
C conjg(A')*X, if ICASE=2,
C
C and F04ZCF must be re-called, with all the other
C parameters unchanged.
C
C On final return,
C
C ICASE = 0.
C
C ESTNRM real. Contains an estimate (a lower bound) for norm(A).
C
C WORK complex(N). Contains vector V such that V = A*W, where
C ESTNRM = norm(V)/norm(W) (W is not returned).
C
C Nick Higham, University of Manchester.
C
C Reference:
C N.J. Higham (1987) Fortran Codes for Estimating
C the One-norm of a Real or Complex Matrix, with Applications
C to Condition Estimation, Numerical Analysis Report No. 135,
C University of Manchester, Manchester M13 9PL, England.
C
SUBROUTINE F05AAF(A,IA,M,N1,N2,S,CC,ICOL,IFAIL)
C MARK 4 RELEASE NAG COPYRIGHT 1974.
C MARK 4.5 REVISED
C MARK 10A REVISED. IER-390 (OCT 1982).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C PERFORMS SCHMIDT ORTHONORMALISATION ON VECTORS N1
C TO N2 OF ARRAY A.
SUBROUTINE F06AAF( A, B, C, S )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06BAF( A, B, C, S )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06BCF( T, C, S )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C MARK 13 REVISED. IER-602 (MAR 1988).
SUBROUTINE F06BEF( JOB, X, Y, Z, C, S )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06BHF( X, Y, Z, C, S )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
DOUBLE PRECISION FUNCTION F06BLF( A, B, FAIL )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
DOUBLE PRECISION FUNCTION F06BMF( SCALE, SSQ )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
DOUBLE PRECISION FUNCTION F06BNF( A, B )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
DOUBLE PRECISION FUNCTION F06BPF( A, B, C )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06CAF( A, B, C, S )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C MARK 14A REVISED. IER-689 (DEC 1989).
C MARK 15A REVISED. IER-916 (APR 1991).
SUBROUTINE F06CBF( A, B, C, S )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06CCF( T, C, S )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C MARK 13 REVISED. IER-603 (MAR 1988).
C MARK 15 REVISED. IER-943 (APR 1991).
SUBROUTINE F06CDF( T, C, S )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C MARK 13 REVISED. IER-604 (MAR 1988).
SUBROUTINE F06CHF( X, Y, Z, C, S )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
COMPLEX*16 FUNCTION F06CLF( A, B, FAIL )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C MARK 13 REVISED. IER-605 (MAR 1988).
SUBROUTINE F06DBF( N, CONST, X, INCX )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06DFF( N, X, INCX, Y, INCY )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
DOUBLE PRECISION FUNCTION F06EAF( N, X, INCX, Y, INCY )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06ECF( N, ALPHA, X, INCX, Y, INCY )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06EDF( N, ALPHA, X, INCX )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06EFF( N, X, INCX, Y, INCY )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06EGF( N, X, INCX, Y, INCY )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
DOUBLE PRECISION FUNCTION F06EJF( N, X, INCX )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
DOUBLE PRECISION FUNCTION F06EKF( N, X, INCX )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06EPF( N, X, INCX, Y, INCY, C, S )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
DOUBLE PRECISION FUNCTION F06ERF(NZ,X,INDX,Y)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C PURPOSE
C
C DDOTI COMPUTES THE VECTOR INNER PRODUCT OF
C A REAL SPARSE VECTOR X
C STORED IN COMPRESSED FORM (X,INDX)
C WITH
C A REAL VECTOR Y IN FULL STORAGE FORM.
C
C ONLY THE ELEMENTS OF Y WHOSE INDICES ARE LISTED IN INDX
C ARE REFERENCED.
C
C ARGUMENTS
C
C INPUT ...
C
C NZ INTEGER NUMBER OF ELEMENTS IN THE COMPRESSED FORM.
C X REAL ARRAY CONTAINING THE VALUES OF THE
C COMPRESSED FORM.
C INDX INTEGER ARRAY CONTAINING THE INDICES OF THE
C COMPRESSED FORM.
C Y REAL ARRAY, ON INPUT, WHICH CONTAINS THE
C VECTOR Y IN FULL STORAGE FORM. ONLY
C THE ELEMENTS CORRESPONDING TO THE
C INDICES IN INDX WILL BE ACCESSED.
C
C OUTPUT ...
C
C DDOTI REAL REAL FUNCTION VALUE EQUAL TO THE
C VECTOR INNER PRODUCT.
C IF NZ .LE. 0 DDOTI IS SET TO ZERO.
C
C SPARSE BASIC LINEAR ALGEBRA SUBPROGRAM
C
C FORTRAN VERSION WRITTEN OCTOBER 1984
C ROGER G GRIMES, BOEING COMPUTER SERVICES
C
SUBROUTINE F06ETF(NZ,A,X,INDX,Y)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C PURPOSE
C
C DAXPYI ADDS A REAL SCALAR MULTIPLE OF
C A REAL SPARSE VECTOR X
C STORED IN COMPRESSED FORM (X,INDX)
C TO
C A REAL VECTOR Y IN FULL STORAGE FORM.
C
C ONLY THE ELEMENTS OF Y WHOSE INDICES ARE LISTED IN INDX
C ARE REFERENCED OR MODIFIED. THE VALUES IN INDX MUST BE
C DISTINCT TO ALLOW CONSISTENT VECTOR OR PARALLEL EXECUTION.
C
C ALTHOUGH DISTINCT INDICES WILL ALLOW VECTOR OR PARALLEL
C EXECUTION, MOST COMPILERS FOR HIGH-PERFORMANCE MACHINES WILL
C BE UNABLE TO GENERATE BEST POSSIBLE CODE WITHOUT SOME
C MODIFICATION, SUCH AS COMPILER DIRECTIVES, TO THIS CODE.
C
C ARGUMENTS
C
C INPUT ...
C
C NZ INTEGER NUMBER OF ELEMENTS IN THE COMPRESSED FORM.
C A REAL SCALAR MULTIPLIER OF X.
C X REAL ARRAY CONTAINING THE VALUES OF THE
C COMPRESSED FORM.
C INDX INTEGER ARRAY CONTAINING THE INDICES OF THE
C COMPRESSED FORM. IT IS ASSUMED THAT
C THE ELEMENTS IN INDX ARE DISTINCT.
C
C UPDATED ...
C
C Y REAL ARRAY, ON INPUT, WHICH CONTAINS THE
C VECTOR Y IN FULL STORAGE FORM. ON
C OUTPUT ONLY THE ELEMENTS CORRESPONDING TO
C THE INDICES IN INDX HAVE BEEN MODIFIED.
C
C SPARSE BASIC LINEAR ALGEBRA SUBPROGRAM
C
C FORTRAN VERSION WRITTEN OCTOBER 1984
C ROGER G GRIMES, BOEING COMPUTER SERVICES
C
SUBROUTINE F06EUF(NZ,Y,X,INDX)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C PURPOSE
C
C DGTHR GATHERS THE SPECIFIED ELEMENTS FROM
C A REAL VECTOR Y IN FULL STORAGE FORM
C INTO
C A REAL VECTOR X IN COMPRESSED FORM (X,INDX).
C
C ONLY THE ELEMENTS OF Y WHOSE INDICES ARE LISTED IN INDX
C ARE REFERENCED.
C
C ARGUMENTS
C
C INPUT ...
C
C NZ INTEGER NUMBER OF ELEMENTS TO BE GATHERED INTO
C COMPRESSED FORM.
C Y REAL ARRAY, ON INPUT, WHICH CONTAINS THE
C VECTOR Y IN FULL STORAGE FORM. ONLY
C THE ELEMENTS CORRESPONDING TO THE INDICES
C IN INDX WILL BE ACCESSED.
C INDX INTEGER ARRAY CONTAINING THE INDICES OF THE
C VALUES TO BE GATHERED INTO COMPRESSED
C FORM.
C
C OUTPUT ...
C
C X REAL ARRAY CONTAINING THE VALUES GATHERED INTO
C THE COMPRESSED FORM.
C
C SPARSE BASIC LINEAR ALGEBRA SUBPROGRAM
C
C FORTRAN VERSION WRITTEN OCTOBER 1984
C ROGER G GRIMES, BOEING COMPUTER SERVICES
C
SUBROUTINE F06EVF(NZ,Y,X,INDX)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C PURPOSE
C
C DGTHRZ GATHERS THE SPECIFIED ELEMENTS FROM
C A REAL VECTOR Y IN FULL STORAGE FORM
C INTO
C A REAL VECTOR X IN COMPRESSED FORM (X,INDX).
C FURTHERMORE THE GATHERED ELEMENTS OF Y ARE SET TO ZERO.
C
C ONLY THE ELEMENTS OF Y WHOSE INDICES ARE LISTED IN INDX
C ARE REFERENCED OR MODIFIED.
C
C ARGUMENTS
C
C INPUT ...
C
C NZ INTEGER NUMBER OF ELEMENTS TO BE GATHERED INTO
C COMPRESSED FORM.
C INDX INTEGER ARRAY CONTAINING THE INDICES OF THE
C VALUES TO BE GATHERED INTO COMPRESSED
C FORM.
C
C UPDATED ...
C
C Y REAL ARRAY, ON INPUT, WHICH CONTAINS THE
C VECTOR Y IN FULL STORAGE FORM. THE
C GATHERED COMPONENTS IN Y ARE SET TO
C ZERO. ONLY THE ELEMENTS CORRESPONDING TO
C THE INDICES IN INDX HAVE BEEN ACCESSED.
C
C OUTPUT ...
C
C X REAL ARRAY CONTAINING THE VALUES GATHERED INTO
C THE COMPRESSED FORM.
C
C SPARSE BASIC LINEAR ALGEBRA SUBPROGRAM
C
C FORTRAN VERSION WRITTEN OCTOBER 1984
C ROGER G GRIMES, BOEING COMPUTER SERVICES
C
SUBROUTINE F06EWF(NZ,X,INDX,Y)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C PURPOSE
C
C DSCTR SCATTERS THE COMPONENTS OF
C A SPARSE VECTOR X STORED IN COMPRESSED FORM (X,INDX)
C INTO
C SPECIFIED COMPONENTS OF A REAL VECTOR Y
C IN FULL STORAGE FORM.
C
C ONLY THE ELEMENTS OF Y WHOSE INDICES ARE LISTED IN INDX
C ARE MODIFIED. THE VALUES IN INDX MUST BE DISTINCT TO
C ALLOW CONSISTENT VECTOR OR PARALLEL EXECUTION.
C
C ALTHOUGH DISTINCT INDICES WILL ALLOW VECTOR OR PARALLEL
C EXECUTION, MOST COMPILERS FOR HIGH-PERFORMANCE MACHINES WILL
C BE UNABLE TO GENERATE BEST POSSIBLE CODE WITHOUT SOME
C MODIFICATION, SUCH AS COMPILER DIRECTIVES, TO THIS CODE.
C
C ARGUMENTS
C
C INPUT ...
C
C NZ INTEGER NUMBER OF ELEMENTS TO BE SCATTERED FROM
C COMPRESSED FORM.
C X REAL ARRAY CONTAINING THE VALUES TO BE
C SCATTERED FROM COMPRESSED FORM INTO FULL
C STORAGE FORM.
C INDX INTEGER ARRAY CONTAINING THE INDICES OF THE VALUES
C TO BE SCATTERED FROM COMPRESSED FORM.
C IT IS ASSUMED THAT THE ELEMENTS IN INDX
C ARE DISTINCT.
C
C OUTPUT ...
C
C Y REAL ARRAY WHOSE ELEMENTS SPECIFIED BY INDX
C HAVE BEEN SET TO THE CORRESPONDING
C ENTRIES OF X. ONLY THE ELEMENTS
C CORRESPONDING TO THE INDICES IN INDX
C HAVE BEEN MODIFIED.
C
C SPARSE BASIC LINEAR ALGEBRA SUBPROGRAM
C
C FORTRAN VERSION WRITTEN OCTOBER 1984
C ROGER G GRIMES, BOEING COMPUTER SERVICES
C
SUBROUTINE F06EXF(NZ,X,INDX,Y,C,S)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C PURPOSE
C
C DROTI APPLIES A GIVENS ROTATION TO
C A SPARSE VECTOR X STORED IN COMPRESSED FORM (X,INDX)
C AND
C ANOTHER VECTOR Y IN FULL STORAGE FORM.
C
C DROTI DOES NOT HANDLE FILL-IN IN X AND THEREFORE, IT IS
C ASSUMED THAT ALL NONZERO COMPONENTS OF Y ARE LISTED IN
C INDX. ONLY THE ELEMENTS OF Y WHOSE INDICES ARE LISTED IN
C INDX ARE REFERENCED OR MODIFIED. THE VALUES IN INDX MUST
C BE DISTINCT TO ALLOW CONSISTENT VECTOR OR PARALLEL EXECUTION.
C
C ALTHOUGH DISTINCT INDICES WILL ALLOW VECTOR OR PARALLEL
C EXECUTION, MOST COMPILERS FOR HIGH-PERFORMANCE MACHINES WILL
C BE UNABLE TO GENERATE BEST POSSIBLE CODE WITHOUT SOME
C MODIFICATION, SUCH AS COMPILER DIRECTIVES, TO THIS CODE.
C
C ARGUMENTS
C
C INPUT ...
C
C NZ INTEGER NUMBER OF ELEMENTS IN THE COMPRESSED FORM.
C INDX INTEGER ARRAY CONTAINING THE INDICIES OF THE
C COMPRESSED FORM. IT IS ASSUMED THAT
C THE ELEMENTS IN INDX ARE DISTINCT.
C C,S REAL THE TWO SCALARS DEFINING THE GIVENS
C ROTATION.
C
C UPDATED ...
C
C X REAL ARRAY CONTAINING THE VALUES OF THE
C SPARSE VECTOR IN COMPRESSED FORM.
C Y REAL ARRAY WHICH CONTAINS THE VECTOR Y
C IN FULL STORAGE FORM. ONLY THE
C ELEMENTS WHOSE INDICIES ARE LISTED IN
C INDX HAVE BEEN REFERENCED OR MODIFIED.
C
C SPARSE BASIC LINEAR ALGEBRA SUBPROGRAM
C
C FORTRAN VERSION WRITTEN OCTOBER 1984
C ROGER G GRIMES, BOEING COMPUTER SERVICES
C
DOUBLE PRECISION FUNCTION F06FAF( N, J, TOLX, X, INCX,
$ TOLY, Y, INCY )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06FBF( N, CONST, X, INCX )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06FCF( N, D, INCD, X, INCX )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06FDF( N, ALPHA, X, INCX, Y, INCY )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06FGF( N, X, INCX )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06FJF( N, X, INCX, SCALE, SUMSQ )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
DOUBLE PRECISION FUNCTION F06FKF( N, W, INCW, X, INCX )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06FLF( N, X, INCX, XMAX, XMIN )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06FPF( N, X, INCX, Y, INCY, C, S )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06FQF( PIVOT, DIRECT, N, ALPHA, X, INCX, C, S )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06FRF( N, ALPHA, X, INCX, TOL, ZETA )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06FSF( N, ALPHA, X, INCX, TOL, Z1 )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06FTF( N, DELTA, Y, INCY, ZETA, Z, INCZ )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06FUF( N, Z, INCZ, Z1, ALPHA, X, INCX )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
COMPLEX*16 FUNCTION F06GAF( N, X, INCX, Y, INCY )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
COMPLEX*16 FUNCTION F06GBF( N, X, INCX, Y, INCY )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06GCF( N, ALPHA, X, INCX, Y, INCY )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06GDF( N, ALPHA, X, INCX )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06GFF( N, X, INCX, Y, INCY )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06GGF( N, X, INCX, Y, INCY )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
COMPLEX*16 FUNCTION F06GRF(NZ,X,INDX,Y)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C PURPOSE
C
C ZDOTUI COMPUTES THE UNCONJUGATED VECTOR INNER PRODUCT OF
C A COMPLEX SPARSE VECTOR X
C STORED IN COMPRESSED FORM (X,INDX)
C WITH
C A COMPLEX VECTOR Y IN FULL STORAGE FORM.
C
C ONLY THE ELEMENTS OF Y WHOSE INDICES ARE LISTED IN INDX
C ARE REFERENCED.
C
C ARGUMENTS
C
C INPUT ...
C
C NZ INTEGER NUMBER OF ELEMENTS IN THE COMPRESSED FORM.
C X COMPLEX ARRAY CONTAINING THE VALUES OF THE
C COMPRESSED FORM.
C INDX INTEGER ARRAY CONTAINING THE INDICES OF THE
C COMPRESSED FORM.
C Y COMPLEX ARRAY, ON INPUT, WHICH CONTAINS THE
C VECTOR Y IN FULL STORAGE FORM. ONLY
C THE ELEMENTS CORRESPONDING TO THE
C INDICES IN INDX WILL BE ACCESSED.
C
C OUTPUT ...
C
C ZDOTUI COMPLEX COMPLEX FUNCTION VALUE EQUAL TO THE
C UNCONJUGATED VECTOR INNER PRODUCT.
C IF NZ .LE. 0 ZDOTCI IS SET TO ZERO.
C
C SPARSE BASIC LINEAR ALGEBRA SUBPROGRAM
C
C FORTRAN VERSION WRITTEN OCTOBER 1984
C ROGER G GRIMES, BOEING COMPUTER SERVICES
C
COMPLEX*16 FUNCTION F06GSF(NZ,X,INDX,Y)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C PURPOSE
C
C ZDOTCI COMPUTES THE CONJUGATED VECTOR INNER PRODUCT OF
C A COMPLEX SPARSE VECTOR X
C STORED IN COMPRESSED FORM (X,INDX)
C WITH
C A COMPLEX VECTOR Y IN FULL STORAGE FORM.
C
C ONLY THE ELEMENTS OF Y WHOSE INDICES ARE LISTED IN INDX
C ARE REFERENCED.
C
C ARGUMENTS
C
C INPUT ...
C
C NZ INTEGER NUMBER OF ELEMENTS IN THE COMPRESSED FORM.
C X COMPLEX ARRAY CONTAINING THE VALUES OF THE
C COMPRESSED FORM.
C INDX INTEGER ARRAY CONTAINING THE INDICES OF THE
C COMPRESSED FORM.
C Y COMPLEX ARRAY, ON INPUT, WHICH CONTAINS THE
C VECTOR Y IN FULL STORAGE FORM. ONLY
C THE ELEMENTS CORRESPONDING TO THE
C INDICES IN INDX WILL BE ACCESSED.
C
C OUTPUT ...
C
C ZDOTCI COMPLEX COMPLEX FUNCTION VALUE EQUAL TO THE
C CONJUGATED VECTOR INNER PRODUCT.
C IF NZ .LE. 0 ZDOTCI IS SET TO ZERO.
C
C SPARSE BASIC LINEAR ALGEBRA SUBPROGRAM
C
C FORTRAN VERSION WRITTEN OCTOBER 1984
C ROGER G GRIMES, BOEING COMPUTER SERVICES
C
SUBROUTINE F06GTF(NZ,A,X,INDX,Y)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C PURPOSE
C
C ZAXPYI ADDS A COMPLEX SCALAR MULTIPLE OF
C A COMPLEX SPARSE VECTOR X
C STORED IN COMPRESSED FORM (X,INDX)
C TO
C A COMPLEX VECTOR Y IN FULL STORAGE FORM.
C
C ONLY THE ELEMENTS OF Y WHOSE INDICES ARE LISTED IN INDX
C ARE REFERENCED OR MODIFIED. THE VALUES IN INDX MUST BE
C DISTINCT TO ALLOW CONSISTENT VECTOR OR PARALLEL EXECUTION.
C
C ALTHOUGH DISTINCT INDICES WILL ALLOW VECTOR OR PARALLEL
C EXECUTION, MOST COMPILERS FOR HIGH-PERFORMANCE MACHINES WILL
C BE UNABLE TO GENERATE BEST POSSIBLE CODE WITHOUT SOME
C MODIFICATION, SUCH AS COMPILER DIRECTIVES, TO THIS CODE.
C
C ARGUMENTS
C
C INPUT ...
C
C NZ INTEGER NUMBER OF ELEMENTS IN THE COMPRESSED FORM.
C A COMPLEX SCALAR MULTIPLIER OF X.
C X COMPLEX ARRAY CONTAINING THE VALUES OF THE
C COMPRESSED FORM.
C INDX INTEGER ARRAY CONTAINING THE INDICES OF THE
C COMPRESSED FORM. IT IS ASSUMED THAT
C THE ELEMENTS IN INDX ARE DISTINCT.
C
C UPDATED ...
C
C Y COMPLEX ARRAY, ON INPUT, WHICH CONTAINS THE
C VECTOR Y IN FULL STORAGE FORM. ON
C OUTPUT ONLY THE ELEMENTS CORRESPONDING TO
C THE INDICES IN INDX HAVE BEEN MODIFIED.
C
C SPARSE BASIC LINEAR ALGEBRA SUBPROGRAM
C
C FORTRAN VERSION WRITTEN OCTOBER 1984
C ROGER G GRIMES, BOEING COMPUTER SERVICES
C
SUBROUTINE F06GUF(NZ,Y,X,INDX)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C PURPOSE
C
C ZGTHR GATHERS THE SPECIFIED ELEMENTS FROM
C A COMPLEX VECTOR Y IN FULL STORAGE FORM
C INTO
C A COMPLEX VECTOR X IN COMPRESSED FORM (X,INDX).
C
C ONLY THE ELEMENTS OF Y WHOSE INDICES ARE LISTED IN INDX
C ARE REFERENCED.
C
C ARGUMENTS
C
C INPUT ...
C
C NZ INTEGER NUMBER OF ELEMENTS TO BE GATHERED INTO
C COMPRESSED FORM.
C Y COMPLEX ARRAY, ON INPUT, WHICH CONTAINS THE
C VECTOR Y IN FULL STORAGE FORM. ONLY
C THE ELEMENTS CORRESPONDING TO THE INDICES
C IN INDX WILL BE ACCESSED.
C INDX INTEGER ARRAY CONTAINING THE INDICES OF THE
C VALUES TO BE GATHERED INTO COMPRESSED
C FORM.
C
C OUTPUT ...
C
C X COMPLEX ARRAY CONTAINING THE VALUES GATHERED INTO
C THE COMPRESSED FORM.
C
C SPARSE BASIC LINEAR ALGEBRA SUBPROGRAM
C
C FORTRAN VERSION WRITTEN OCTOBER 1984
C ROGER G GRIMES, BOEING COMPUTER SERVICES
C
SUBROUTINE F06GVF(NZ,Y,X,INDX)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C PURPOSE
C
C ZGTHRZ GATHERS THE SPECIFIED ELEMENTS FROM
C A COMPLEX VECTOR Y IN FULL STORAGE FORM
C INTO
C A COMPLEX VECTOR X IN COMPRESSED FORM (X,INDX).
C FURTHERMORE THE GATHERED ELEMENTS OF Y ARE SET TO ZERO.
C
C ONLY THE ELEMENTS OF Y WHOSE INDICES ARE LISTED IN INDX
C ARE REFERENCED OR MODIFIED.
C
C ARGUMENTS
C
C INPUT ...
C
C NZ INTEGER NUMBER OF ELEMENTS TO BE GATHERED INTO
C COMPRESSED FORM.
C INDX INTEGER ARRAY CONTAINING THE INDICES OF THE
C VALUES TO BE GATHERED INTO COMPRESSED
C FORM.
C
C UPDATED ...
C
C Y COMPLEX ARRAY, ON INPUT, WHICH CONTAINS THE
C VECTOR Y IN FULL STORAGE FORM. THE
C GATHERED COMPONENTS IN Y ARE SET TO
C ZERO. ONLY THE ELEMENTS CORRESPONDING TO
C THE INDICES IN INDX HAVE BEEN ACCESSED.
C
C OUTPUT ...
C
C X COMPLEX ARRAY CONTAINING THE VALUES GATHERED INTO
C THE COMPRESSED FORM.
C
C SPARSE BASIC LINEAR ALGEBRA SUBPROGRAM
C
C FORTRAN VERSION WRITTEN OCTOBER 1984
C ROGER G GRIMES, BOEING COMPUTER SERVICES
C
SUBROUTINE F06GWF(NZ,X,INDX,Y)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C PURPOSE
C
C ZSCTR SCATTERS THE COMPONENTS OF
C A SPARSE VECTOR X STORED IN COMPRESSED FORM (X,INDX)
C INTO
C SPECIFIED COMPONENTS OF A COMPLEX VECTOR Y
C IN FULL STORAGE FORM.
C
C ONLY THE ELEMENTS OF Y WHOSE INDICES ARE LISTED IN INDX
C ARE MODIFIED. THE VALUES IN INDX MUST BE DISTINCT TO
C ALLOW CONSISTENT VECTOR OR PARALLEL EXECUTION.
C
C ALTHOUGH DISTINCT INDICES WILL ALLOW VECTOR OR PARALLEL
C EXECUTION, MOST COMPILERS FOR HIGH-PERFORMANCE MACHINES WILL
C BE UNABLE TO GENERATE BEST POSSIBLE CODE WITHOUT SOME
C MODIFICATION, SUCH AS COMPILER DIRECTIVES, TO THIS CODE.
C
C ARGUMENTS
C
C INPUT ...
C
C NZ INTEGER NUMBER OF ELEMENTS TO BE SCATTERED FROM
C COMPRESSED FORM.
C X COMPLEX ARRAY CONTAINING THE VALUES TO BE
C SCATTERED FROM COMPRESSED FORM INTO FULL
C STORAGE FORM.
C INDX INTEGER ARRAY CONTAINING THE INDICES OF THE VALUES
C TO BE SCATTERED FROM COMPRESSED FORM.
C IT IS ASSUMED THAT THE ELEMENTS IN INDX
C ARE DISTINCT.
C
C OUTPUT ...
C
C Y COMPLEX ARRAY WHOSE ELEMENTS SPECIFIED BY INDX
C HAVE BEEN SET TO THE CORRESPONDING
C ENTRIES OF X. ONLY THE ELEMENTS
C CORRESPONDING TO THE INDICES IN INDX
C HAVE BEEN MODIFIED.
C
C SPARSE BASIC LINEAR ALGEBRA SUBPROGRAM
C
C FORTRAN VERSION WRITTEN OCTOBER 1984
C ROGER G GRIMES, BOEING COMPUTER SERVICES
C
SUBROUTINE F06HBF( N, CONST, X, INCX )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06HCF( N, D, INCD, X, INCX )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06HDF( N, ALPHA, X, INCX, Y, INCY )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06HGF( N, X, INCX )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06HPF( N, X, INCX, Y, INCY, C, S )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06HQF( PIVOT, DIRECT, N, ALPHA, X, INCX, C, S )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06HRF( N, ALPHA, X, INCX, TOL, THETA )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06HTF( N, DELTA, Y, INCY, THETA, Z, INCZ )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06JDF( N, ALPHA, X, INCX )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
DOUBLE PRECISION FUNCTION F06JJF( N, X, INCX )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
DOUBLE PRECISION FUNCTION F06JKF( N, X, INCX )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
INTEGER FUNCTION F06JLF( N, X, INCX )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
INTEGER FUNCTION F06JMF( N, X, INCX )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06KCF( N, D, INCD, X, INCX )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06KDF( N, ALPHA, X, INCX, Y, INCY )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06KFF( N, X, INCX, Y, INCY )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06KJF( N, X, INCX, SCALE, SUMSQ )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
INTEGER FUNCTION F06KLF( N, X, INCX, TOL )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06KPF( N, X, INCX, Y, INCY, C, S )
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
SUBROUTINE F06PAF( TRANS, M, N, ALPHA, A, LDA, X, INCX,
$ BETA, Y, INCY )
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
SUBROUTINE F06PBF( TRANS, M, N, KL, KU, ALPHA, A, LDA, X, INCX,
$ BETA, Y, INCY )
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
SUBROUTINE F06PCF( UPLO, N, ALPHA, A, LDA, X, INCX,
$ BETA, Y, INCY )
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
SUBROUTINE F06PDF( UPLO, N, K, ALPHA, A, LDA, X, INCX,
$ BETA, Y, INCY )
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
SUBROUTINE F06PEF( UPLO, N, ALPHA, AP, X, INCX, BETA, Y, INCY )
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
SUBROUTINE F06PFF( UPLO, TRANS, DIAG, N, A, LDA, X, INCX )
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
SUBROUTINE F06PGF( UPLO, TRANS, DIAG, N, K, A, LDA, X, INCX )
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
SUBROUTINE F06PHF( UPLO, TRANS, DIAG, N, AP, X, INCX )
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
SUBROUTINE F06PJF( UPLO, TRANS, DIAG, N, A, LDA, X, INCX )
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
SUBROUTINE F06PKF( UPLO, TRANS, DIAG, N, K, A, LDA, X, INCX )
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
SUBROUTINE F06PLF( UPLO, TRANS, DIAG, N, AP, X, INCX )
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
SUBROUTINE F06PMF( M, N, ALPHA, X, INCX, Y, INCY, A, LDA )
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
SUBROUTINE F06PPF( UPLO, N, ALPHA, X, INCX, A, LDA )
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
SUBROUTINE F06PQF( UPLO, N, ALPHA, X, INCX, AP )
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
SUBROUTINE F06PRF( UPLO, N, ALPHA, X, INCX, Y, INCY, A, LDA )
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
SUBROUTINE F06PSF( UPLO, N, ALPHA, X, INCX, Y, INCY, AP )
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
SUBROUTINE F06QFF( MATRIX, M, N, A, LDA, B, LDB )
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE F06QHF( MATRIX, M, N, CONST, DIAG, A, LDA )
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE F06QJF( SIDE, TRANS, N, PERM, K, B, LDB )
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE F06QKF( SIDE, TRANS, N, PERM, K, B, LDB )
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE F06QMF( UPLO, PIVOT, DIRECT, N, K1, K2, C, S, A, LDA )
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE F06QPF( N, ALPHA, X, INCX, Y, INCY, A, LDA, C, S )
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE F06QQF( N, ALPHA, X, INCX, A, LDA, C, S )
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE F06QRF( SIDE, N, K1, K2, C, S, A, LDA )
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE F06QSF( SIDE, N, K1, K2, C, S, A, LDA )
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE F06QTF( SIDE, N, K1, K2, C, S, A, LDA )
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE F06QVF( SIDE, N, K1, K2, C, S, A, LDA )
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE F06QWF( SIDE, N, K1, K2, C, S, A, LDA )
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE F06QXF( SIDE, PIVOT, DIRECT, M, N, K1, K2, C, S, A,
$ LDA )
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
DOUBLE PRECISION FUNCTION F06RAF(NORM,M,N,A,LDA,WORK)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C REAL DLANGE
C ENTRY DLANGE(NORM,M,N,A,LDA,WORK)
C
C Purpose
C =======
C
C DLANGE returns the value of the one norm, or the Frobenius norm, or
C the infinity norm, or the element of largest absolute value of a
C real matrix A.
C
C Description
C ===========
C
C DLANGE returns the value
C
C DLANGE = ( max(abs(A(i,j))), NORM = 'M' or 'm'
C (
C ( norm1(A), NORM = '1', 'O' or 'o'
C (
C ( normI(A), NORM = 'I' or 'i'
C (
C ( normF(A), NORM = 'F', 'f', 'E' or 'e'
C
C where norm1 denotes the one norm of a matrix (maximum column sum),
C normI denotes the infinity norm of a matrix (maximum row sum) and
C normF denotes the Frobenius norm of a matrix (square root of sum of
C squares). Note that max(abs(A(i,j))) is not a matrix norm.
C
C Arguments
C =========
C
C NORM (input) CHARACTER*1
C Specifies the value to be returned in DLANGE as described
C above.
C
C M (input) INTEGER
C The number of rows of the matrix A. M >= 0. When M = 0,
C DLANGE is set to zero.
C
C N (input) INTEGER
C The number of columns of the matrix A. N >= 0. When N = 0,
C DLANGE is set to zero.
C
C A (input) REAL array, dimension (LDA,N)
C The m by n matrix A.
C
C LDA (input) INTEGER
C The leading dimension of the array A. LDA >= max(M,1).
C
C WORK (workspace) REAL array, dimension (LWORK),
C where LWORK >= M when NORM = 'I'; otherwise, WORK is not
C referenced.
C
C
C -- LAPACK auxiliary routine (adapted for NAG Library)
C Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
C Courant Institute, Argonne National Lab, and Rice University
C
DOUBLE PRECISION FUNCTION F06RBF(NORM,N,KL,KU,AB,LDAB,WORK)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C REAL DLANGB
C ENTRY DLANGB(NORM,N,KL,KU,AB,LDAB,WORK)
C
C Purpose
C =======
C
C DLANGB returns the value of the one norm, or the Frobenius norm, or
C the infinity norm, or the element of largest absolute value of an
C n by n band matrix A, with kl sub-diagonals and ku super-diagonals.
C
C Description
C ===========
C
C DLANGB returns the value
C
C DLANGB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
C (
C ( norm1(A), NORM = '1', 'O' or 'o'
C (
C ( normI(A), NORM = 'I' or 'i'
C (
C ( normF(A), NORM = 'F', 'f', 'E' or 'e'
C
C where norm1 denotes the one norm of a matrix (maximum column sum),
C normI denotes the infinity norm of a matrix (maximum row sum) and
C normF denotes the Frobenius norm of a matrix (square root of sum of
C squares). Note that max(abs(A(i,j))) is not a matrix norm.
C
C Arguments
C =========
C
C NORM (input) CHARACTER*1
C Specifies the value to be returned in DLANGB as described
C above.
C
C N (input) INTEGER
C The order of the matrix A. N >= 0. When N = 0, DLANGB is
C set to zero.
C
C KL (input) INTEGER
C The number of sub-diagonals of the matrix A. KL >= 0.
C
C KU (input) INTEGER
C The number of super-diagonals of the matrix A. KU >= 0.
C
C AB (input) REAL array, dimension (LDAB,N)
C The band matrix A, stored in rows 1 to KL+KU+1. The j-th
C column of A is stored in the j-th column of the array AB as
C follows:
C AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).
C
C LDAB (input) INTEGER
C The leading dimension of the array AB. LDAB >= KL+KU+1.
C
C WORK (workspace) REAL array, dimension (LWORK),
C where LWORK >= N when NORM = 'I'; otherwise, WORK is not
C referenced.
C
C
C -- LAPACK auxiliary routine (adapted for NAG Library)
C Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
C Courant Institute, Argonne National Lab, and Rice University
C
DOUBLE PRECISION FUNCTION F06RCF(NORM,UPLO,N,A,LDA,WORK)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C REAL DLANSY
C ENTRY DLANSY(NORM,UPLO,N,A,LDA,WORK)
C
C Purpose
C =======
C
C DLANSY returns the value of the one norm, or the Frobenius norm, or
C the infinity norm, or the element of largest absolute value of a
C real symmetric matrix A.
C
C Description
C ===========
C
C DLANSY returns the value
C
C DLANSY = ( max(abs(A(i,j))), NORM = 'M' or 'm'
C (
C ( norm1(A), NORM = '1', 'O' or 'o'
C (
C ( normI(A), NORM = 'I' or 'i'
C (
C ( normF(A), NORM = 'F', 'f', 'E' or 'e'
C
C where norm1 denotes the one norm of a matrix (maximum column sum),
C normI denotes the infinity norm of a matrix (maximum row sum) and
C normF denotes the Frobenius norm of a matrix (square root of sum of
C squares). Note that max(abs(A(i,j))) is not a matrix norm.
C
C Arguments
C =========
C
C NORM (input) CHARACTER*1
C Specifies the value to be returned in DLANSY as described
C above.
C
C UPLO (input) CHARACTER*1
C Specifies whether the upper or lower triangular part of the
C symmetric matrix A is to be referenced.
C = 'U': Upper triangular part of A is referenced
C = 'L': Lower triangular part of A is referenced
C
C N (input) INTEGER
C The order of the matrix A. N >= 0. When N = 0, DLANSY is
C set to zero.
C
C A (input) REAL array, dimension (LDA,N)
C The symmetric matrix A. If UPLO = 'U', the leading n by n
C upper triangular part of A contains the upper triangular part
C of the matrix A, and the strictly lower triangular part of A
C is not referenced. If UPLO = 'L', the leading n by n lower
C triangular part of A contains the lower triangular part of
C the matrix A, and the strictly upper triangular part of A is
C not referenced.
C
C LDA (input) INTEGER
C The leading dimension of the array A. LDA >= max(N,1).
C
C WORK (workspace) REAL array, dimension (LWORK),
C where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
C WORK is not referenced.
C
C
C -- LAPACK auxiliary routine (adapted for NAG Library)
C Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
C Courant Institute, Argonne National Lab, and Rice University
C
DOUBLE PRECISION FUNCTION F06RDF(NORM,UPLO,N,AP,WORK)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C REAL DLANSP
C ENTRY DLANSP(NORM,UPLO,N,AP,WORK)
C
C Purpose
C =======
C
C DLANSP returns the value of the one norm, or the Frobenius norm, or
C the infinity norm, or the element of largest absolute value of a
C real symmetric matrix A, supplied in packed form.
C
C Description
C ===========
C
C DLANSP returns the value
C
C DLANSP = ( max(abs(A(i,j))), NORM = 'M' or 'm'
C (
C ( norm1(A), NORM = '1', 'O' or 'o'
C (
C ( normI(A), NORM = 'I' or 'i'
C (
C ( normF(A), NORM = 'F', 'f', 'E' or 'e'
C
C where norm1 denotes the one norm of a matrix (maximum column sum),
C normI denotes the infinity norm of a matrix (maximum row sum) and
C normF denotes the Frobenius norm of a matrix (square root of sum of
C squares). Note that max(abs(A(i,j))) is not a matrix norm.
C
C Arguments
C =========
C
C NORM (input) CHARACTER*1
C Specifies the value to be returned in DLANSP as described
C above.
C
C UPLO (input) CHARACTER*1
C Specifies whether the upper or lower triangular part of the
C symmetric matrix A is supplied.
C = 'U': Upper triangular part of A is supplied
C = 'L': Lower triangular part of A is supplied
C
C N (input) INTEGER
C The order of the matrix A. N >= 0. When N = 0, DLANSP is
C set to zero.
C
C AP (input) REAL array, dimension (N*(N+1)/2)
C The upper or lower triangle of the symmetric matrix A, packed
C columnwise in a linear array. The j-th column of A is stored
C in the array AP as follows:
C if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
C if UPLO = 'L', AP(i + (j-1)*(2n-j+2)/2) = A(i,j) for j<=i<=n.
C
C WORK (workspace) REAL array, dimension (LWORK),
C where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
C WORK is not referenced.
C
C
C -- LAPACK auxiliary routine (adapted for NAG Library)
C Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
C Courant Institute, Argonne National Lab, and Rice University
C
DOUBLE PRECISION FUNCTION F06REF(NORM,UPLO,N,K,AB,LDAB,WORK)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C REAL DLANSB
C ENTRY DLANSB(NORM,UPLO,N,K,AB,LDAB,WORK)
C
C Purpose
C =======
C
C DLANSB returns the value of the one norm, or the Frobenius norm, or
C the infinity norm, or the element of largest absolute value of an
C n by n symmetric band matrix A, with k super-diagonals.
C
C Description
C ===========
C
C DLANSB returns the value
C
C DLANSB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
C (
C ( norm1(A), NORM = '1', 'O' or 'o'
C (
C ( normI(A), NORM = 'I' or 'i'
C (
C ( normF(A), NORM = 'F', 'f', 'E' or 'e'
C
C where norm1 denotes the one norm of a matrix (maximum column sum),
C normI denotes the infinity norm of a matrix (maximum row sum) and
C normF denotes the Frobenius norm of a matrix (square root of sum of
C squares). Note that max(abs(A(i,j))) is not a matrix norm.
C
C Arguments
C =========
C
C NORM (input) CHARACTER*1
C Specifies the value to be returned in DLANSB as described
C above.
C
C UPLO (input) CHARACTER*1
C Specifies whether the upper or lower triangular part of the
C band matrix A is supplied.
C = 'U': Upper triangular part is supplied
C = 'L': Lower triangular part is supplied
C
C N (input) INTEGER
C The order of the matrix A. N >= 0. When N = 0, DLANSB is
C set to zero.
C
C K (input) INTEGER
C The number of super-diagonals or sub-diagonals of the
C band matrix A. K >= 0.
C
C AB (input) REAL array, dimension (LDAB,N)
C The upper or lower triangle of the symmetric band matrix A,
C stored in the first K+1 rows of AB. The j-th column of A is
C stored in the j-th column of the array AB as follows:
C if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
C if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+k).
C
C LDAB (input) INTEGER
C The leading dimension of the array AB. LDAB >= K+1.
C
C WORK (workspace) REAL array, dimension (LWORK),
C where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
C WORK is not referenced.
C
C
C -- LAPACK auxiliary routine (adapted for NAG Library)
C Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
C Courant Institute, Argonne National Lab, and Rice University
C
DOUBLE PRECISION FUNCTION F06RJF(NORM,UPLO,DIAG,M,N,A,LDA,WORK)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C REAL DLANTR
C ENTRY DLANTR(NORM,UPLO,DIAG,M,N,A,LDA,WORK)
C
C Purpose
C =======
C
C DLANTR returns the value of the one norm, or the Frobenius norm, or
C the infinity norm, or the element of largest absolute value of a
C trapezoidal or triangular matrix A.
C
C Description
C ===========
C
C DLANTR returns the value
C
C DLANTR = ( max(abs(A(i,j))), NORM = 'M' or 'm'
C (
C ( norm1(A), NORM = '1', 'O' or 'o'
C (
C ( normI(A), NORM = 'I' or 'i'
C (
C ( normF(A), NORM = 'F', 'f', 'E' or 'e'
C
C where norm1 denotes the one norm of a matrix (maximum column sum),
C normI denotes the infinity norm of a matrix (maximum row sum) and
C normF denotes the Frobenius norm of a matrix (square root of sum of
C squares). Note that max(abs(A(i,j))) is not a matrix norm.
C
C Arguments
C =========
C
C NORM (input) CHARACTER*1
C Specifies the value to be returned in DLANTR as described
C above.
C
C UPLO (input) CHARACTER*1
C Specifies whether the matrix A is upper or lower trapezoidal.
C = 'U': Upper trapezoidal
C = 'L': Lower trapezoidal
C Note that A is triangular instead of trapezoidal if M = N.
C
C DIAG (input) CHARACTER*1
C Specifies whether or not the matrix A has unit diagonal.
C = 'N': Non-unit diagonal
C = 'U': Unit diagonal
C
C M (input) INTEGER
C The number of rows of the matrix A. M >= 0, and if
C UPLO = 'U', M <= N. When M = 0, DLANTR is set to zero.
C
C N (input) INTEGER
C The number of columns of the matrix A. N >= 0, and if
C UPLO = 'L', N <= M. When N = 0, DLANTR is set to zero.
C
C A (input) REAL array, dimension (LDA,N)
C The trapezoidal matrix A (A is triangular if M = N).
C If UPLO = 'U', the leading m by n upper trapezoidal part of
C the array A contains the upper trapezoidal matrix, and the
C strictly lower triangular part of A is not referenced.
C If UPLO = 'L', the leading m by n lower trapezoidal part of
C the array A contains the lower trapezoidal matrix, and the
C strictly upper triangular part of A is not referenced. Note
C that when DIAG = 'U', the diagonal elements of A are not
C referenced and are assumed to be one.
C
C LDA (input) INTEGER
C The leading dimension of the array A. LDA >= max(M,1).
C
C WORK (workspace) REAL array, dimension (LWORK),
C where LWORK >= M when NORM = 'I'; otherwise, WORK is not
C referenced.
C
C
C -- LAPACK auxiliary routine (adapted for NAG Library)
C Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
C Courant Institute, Argonne National Lab, and Rice University
C
DOUBLE PRECISION FUNCTION F06RKF(NORM,UPLO,DIAG,N,AP,WORK)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C REAL DLANTP
C ENTRY DLANTP(NORM,UPLO,DIAG,N,AP,WORK)
C
C Purpose
C =======
C
C DLANTP returns the value of the one norm, or the Frobenius norm, or
C the infinity norm, or the element of largest absolute value of a
C triangular matrix A, supplied in packed form.
C
C Description
C ===========
C
C DLANTP returns the value
C
C DLANTP = ( max(abs(A(i,j))), NORM = 'M' or 'm'
C (
C ( norm1(A), NORM = '1', 'O' or 'o'
C (
C ( normI(A), NORM = 'I' or 'i'
C (
C ( normF(A), NORM = 'F', 'f', 'E' or 'e'
C
C where norm1 denotes the one norm of a matrix (maximum column sum),
C normI denotes the infinity norm of a matrix (maximum row sum) and
C normF denotes the Frobenius norm of a matrix (square root of sum of
C squares). Note that max(abs(A(i,j))) is not a matrix norm.
C
C Arguments
C =========
C
C NORM (input) CHARACTER*1
C Specifies the value to be returned in DLANTP as described
C above.
C
C UPLO (input) CHARACTER*1
C Specifies whether the matrix A is upper or lower triangular.
C = 'U': Upper triangular
C = 'L': Lower triangular
C
C DIAG (input) CHARACTER*1
C Specifies whether or not the matrix A is unit triangular.
C = 'N': Non-unit triangular
C = 'U': Unit triangular
C
C N (input) INTEGER
C The order of the matrix A. N >= 0. When N = 0, DLANTP is
C set to zero.
C
C AP (input) REAL array, dimension (N*(N+1)/2)
C The upper or lower triangular matrix A, packed columnwise in
C a linear array. The j-th column of A is stored in the array
C AP as follows:
C if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
C if UPLO = 'L', AP(i + (j-1)*(2n-j+2)/2) = A(i,j) for j<=i<=n.
C Note that when DIAG = 'U', the elements of the array AP
C corresponding to the diagonal elements of the matrix A are
C not referenced, but are assumed to be one.
C
C WORK (workspace) REAL array, dimension (LWORK),
C where LWORK >= N when NORM = 'I'; otherwise, WORK is not
C referenced.
C
C
C -- LAPACK auxiliary routine (adapted for NAG Library)
C Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
C Courant Institute, Argonne National Lab, and Rice University
C
DOUBLE PRECISION FUNCTION F06RLF(NORM,UPLO,DIAG,N,K,AB,LDAB,WORK)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C REAL DLANTB
C ENTRY DLANTB(NORM,UPLO,DIAG,N,K,AB,LDAB,WORK)
C
C Purpose
C =======
C
C DLANTB returns the value of the one norm, or the Frobenius norm, or
C the infinity norm, or the element of largest absolute value of an
C n by n triangular band matrix A, with ( k + 1 ) diagonals.
C
C Description
C ===========
C
C DLANTB returns the value
C
C DLANTB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
C (
C ( norm1(A), NORM = '1', 'O' or 'o'
C (
C ( normI(A), NORM = 'I' or 'i'
C (
C ( normF(A), NORM = 'F', 'f', 'E' or 'e'
C
C where norm1 denotes the one norm of a matrix (maximum column sum),
C normI denotes the infinity norm of a matrix (maximum row sum) and
C normF denotes the Frobenius norm of a matrix (square root of sum of
C squares). Note that max(abs(A(i,j))) is not a matrix norm.
C
C Arguments
C =========
C
C NORM (input) CHARACTER*1
C Specifies the value to be returned in DLANTB as described
C above.
C
C UPLO (input) CHARACTER*1
C Specifies whether the matrix A is upper or lower triangular.
C = 'U': Upper triangular
C = 'L': Lower triangular
C
C DIAG (input) CHARACTER*1
C Specifies whether or not the matrix A is unit triangular.
C = 'N': Non-unit triangular
C = 'U': Unit triangular
C
C N (input) INTEGER
C The order of the matrix A. N >= 0. When N = 0, DLANTB is
C set to zero.
C
C K (input) INTEGER
C The number of super-diagonals of the matrix A if UPLO = 'U',
C or the number of sub-diagonals of the matrix A if UPLO = 'L'.
C K >= 0.
C
C AB (input) REAL array, dimension (LDAB,N)
C The upper or lower triangular band matrix A, stored in the
C first k+1 rows of AB. The j-th column of A is stored
C in the j-th column of the array AB as follows:
C if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
C if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+k).
C Note that when DIAG = 'U', the elements of the array AB
C corresponding to the diagonal elements of the matrix A are
C not referenced, but are assumed to be one.
C
C LDAB (input) INTEGER
C The leading dimension of the array AB. LDAB >= K+1.
C
C WORK (workspace) REAL array, dimension (LWORK),
C where LWORK >= N when NORM = 'I'; otherwise, WORK is not
C referenced.
C
C
C -- LAPACK auxiliary routine (adapted for NAG Library)
C Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
C Courant Institute, Argonne National Lab, and Rice University
C
DOUBLE PRECISION FUNCTION F06RMF(NORM,N,A,LDA,WORK)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C REAL DLANHS
C ENTRY DLANHS(NORM,N,A,LDA,WORK)
C
C Purpose
C =======
C
C DLANHS returns the value of the one norm, or the Frobenius norm, or
C the infinity norm, or the element of largest absolute value of a
C Hessenberg matrix A.
C
C Description
C ===========
C
C DLANHS returns the value
C
C DLANHS = ( max(abs(A(i,j))), NORM = 'M' or 'm'
C (
C ( norm1(A), NORM = '1', 'O' or 'o'
C (
C ( normI(A), NORM = 'I' or 'i'
C (
C ( normF(A), NORM = 'F', 'f', 'E' or 'e'
C
C where norm1 denotes the one norm of a matrix (maximum column sum),
C normI denotes the infinity norm of a matrix (maximum row sum) and
C normF denotes the Frobenius norm of a matrix (square root of sum of
C squares). Note that max(abs(A(i,j))) is not a matrix norm.
C
C Arguments
C =========
C
C NORM (input) CHARACTER*1
C Specifies the value to be returned in DLANHS as described
C above.
C
C N (input) INTEGER
C The order of the matrix A. N >= 0. When N = 0, DLANHS is
C set to zero.
C
C A (input) REAL array, dimension (LDA,N)
C The n by n upper Hessenberg matrix A; the part of A below the
C first sub-diagonal is not referenced.
C
C LDA (input) INTEGER
C The leading dimension of the array A. LDA >= max(N,1).
C
C WORK (workspace) REAL array, dimension (LWORK),
C where LWORK >= N when NORM = 'I'; otherwise, WORK is not
C referenced.
C
C
C -- LAPACK auxiliary routine (adapted for NAG Library)
C Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
C Courant Institute, Argonne National Lab, and Rice University
C
SUBROUTINE F06SAF( TRANS, M, N, ALPHA, A, LDA, X, INCX,
$ BETA, Y, INCY )
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
SUBROUTINE F06SBF( TRANS, M, N, KL, KU, ALPHA, A, LDA, X, INCX,
$ BETA, Y, INCY )
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
SUBROUTINE F06SCF( UPLO, N, ALPHA, A, LDA, X, INCX,
$ BETA, Y, INCY )
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
SUBROUTINE F06SDF( UPLO, N, K, ALPHA, A, LDA, X, INCX,
$ BETA, Y, INCY )
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
SUBROUTINE F06SEF( UPLO, N, ALPHA, AP, X, INCX, BETA, Y, INCY )
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
SUBROUTINE F06SFF( UPLO, TRANS, DIAG, N, A, LDA, X, INCX )
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
SUBROUTINE F06SGF( UPLO, TRANS, DIAG, N, K, A, LDA, X, INCX )
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
SUBROUTINE F06SHF( UPLO, TRANS, DIAG, N, AP, X, INCX )
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
SUBROUTINE F06SJF( UPLO, TRANS, DIAG, N, A, LDA, X, INCX )
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
SUBROUTINE F06SKF( UPLO, TRANS, DIAG, N, K, A, LDA, X, INCX )
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
SUBROUTINE F06SLF( UPLO, TRANS, DIAG, N, AP, X, INCX )
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
SUBROUTINE F06SMF( M, N, ALPHA, X, INCX, Y, INCY, A, LDA )
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
SUBROUTINE F06SNF( M, N, ALPHA, X, INCX, Y, INCY, A, LDA )
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
SUBROUTINE F06SPF( UPLO, N, ALPHA, X, INCX, A, LDA )
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
SUBROUTINE F06SQF( UPLO, N, ALPHA, X, INCX, AP )
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
SUBROUTINE F06SRF( UPLO, N, ALPHA, X, INCX, Y, INCY, A, LDA )
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
SUBROUTINE F06SSF( UPLO, N, ALPHA, X, INCX, Y, INCY, AP )
C MARK 13 RE-ISSUE. NAG COPYRIGHT 1988.
SUBROUTINE F06TFF( MATRIX, M, N, A, LDA, B, LDB )
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE F06THF( MATRIX, M, N, CONST, DIAG, A, LDA )
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE F06TMF( UPLO, PIVOT, DIRECT, N, K1, K2, C, S, A, LDA )
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE F06TPF( N, ALPHA, X, INCX, Y, INCY, A, LDA, C, S )
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE F06TQF( N, ALPHA, X, INCX, A, LDA, C, S )
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE F06TRF( SIDE, N, K1, K2, C, S, A, LDA )
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE F06TSF( SIDE, N, K1, K2, C, S, A, LDA )
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE F06TTF( SIDE, N, K1, K2, C, S, A, LDA )
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE F06TVF( SIDE, N, K1, K2, C, S, A, LDA )
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE F06TWF( SIDE, N, K1, K2, C, S, A, LDA )
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE F06TXF( SIDE, PIVOT, DIRECT, M, N, K1, K2, C, S, A,
$ LDA )
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE F06TYF( SIDE, PIVOT, DIRECT, M, N, K1, K2, C, S, A,
$ LDA )
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
DOUBLE PRECISION FUNCTION F06UAF(NORM,M,N,A,LDA,WORK)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C REAL ZLANGE
C ENTRY ZLANGE(NORM,M,N,A,LDA,WORK)
C
C Purpose
C =======
C
C ZLANGE returns the value of the one norm, or the Frobenius norm, or
C the infinity norm, or the element of largest absolute value of a
C complex matrix A.
C
C Description
C ===========
C
C ZLANGE returns the value
C
C ZLANGE = ( max(abs(A(i,j))), NORM = 'M' or 'm'
C (
C ( norm1(A), NORM = '1', 'O' or 'o'
C (
C ( normI(A), NORM = 'I' or 'i'
C (
C ( normF(A), NORM = 'F', 'f', 'E' or 'e'
C
C where norm1 denotes the one norm of a matrix (maximum column sum),
C normI denotes the infinity norm of a matrix (maximum row sum) and
C normF denotes the Frobenius norm of a matrix (square root of sum of
C squares). Note that max(abs(A(i,j))) is not a matrix norm.
C
C Arguments
C =========
C
C NORM (input) CHARACTER*1
C Specifies the value to be returned in ZLANGE as described
C above.
C
C M (input) INTEGER
C The number of rows of the matrix A. M >= 0. When M = 0,
C ZLANGE is set to zero.
C
C N (input) INTEGER
C The number of columns of the matrix A. N >= 0. When N = 0,
C ZLANGE is set to zero.
C
C A (input) COMPLEX array, dimension (LDA,N)
C The m by n matrix A.
C
C LDA (input) INTEGER
C The leading dimension of the array A. LDA >= max(M,1).
C
C WORK (workspace) REAL array, dimension (LWORK),
C where LWORK >= M when NORM = 'I'; otherwise, WORK is not
C referenced.
C
C
C -- LAPACK auxiliary routine (adapted for NAG Library)
C Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
C Courant Institute, Argonne National Lab, and Rice University
C
DOUBLE PRECISION FUNCTION F06UBF(NORM,N,KL,KU,AB,LDAB,WORK)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C REAL ZLANGB
C ENTRY ZLANGB(NORM,N,KL,KU,AB,LDAB,WORK)
C
C Purpose
C =======
C
C ZLANGB returns the value of the one norm, or the Frobenius norm, or
C the infinity norm, or the element of largest absolute value of an
C n by n band matrix A, with kl sub-diagonals and ku super-diagonals.
C
C Description
C ===========
C
C ZLANGB returns the value
C
C ZLANGB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
C (
C ( norm1(A), NORM = '1', 'O' or 'o'
C (
C ( normI(A), NORM = 'I' or 'i'
C (
C ( normF(A), NORM = 'F', 'f', 'E' or 'e'
C
C where norm1 denotes the one norm of a matrix (maximum column sum),
C normI denotes the infinity norm of a matrix (maximum row sum) and
C normF denotes the Frobenius norm of a matrix (square root of sum of
C squares). Note that max(abs(A(i,j))) is not a matrix norm.
C
C Arguments
C =========
C
C NORM (input) CHARACTER*1
C Specifies the value to be returned in ZLANGB as described
C above.
C
C N (input) INTEGER
C The order of the matrix A. N >= 0. When N = 0, ZLANGB is
C set to zero.
C
C KL (input) INTEGER
C The number of sub-diagonals of the matrix A. KL >= 0.
C
C KU (input) INTEGER
C The number of super-diagonals of the matrix A. KU >= 0.
C
C AB (input) COMPLEX array, dimension (LDAB,N)
C The band matrix A, stored in rows 1 to KL+KU+1. The j-th
C column of A is stored in the j-th column of the array AB as
C follows:
C AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).
C
C LDAB (input) INTEGER
C The leading dimension of the array AB. LDAB >= KL+KU+1.
C
C WORK (workspace) REAL array, dimension (LWORK),
C where LWORK >= N when NORM = 'I'; otherwise, WORK is not
C referenced.
C
C
C -- LAPACK auxiliary routine (adapted for NAG Library)
C Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
C Courant Institute, Argonne National Lab, and Rice University
C
DOUBLE PRECISION FUNCTION F06UCF(NORM,UPLO,N,A,LDA,WORK)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C REAL ZLANHE
C ENTRY ZLANHE(NORM,UPLO,N,A,LDA,WORK)
C
C Purpose
C =======
C
C ZLANHE returns the value of the one norm, or the Frobenius norm, or
C the infinity norm, or the element of largest absolute value of a
C complex hermitian matrix A.
C
C Description
C ===========
C
C ZLANHE returns the value
C
C ZLANHE = ( max(abs(A(i,j))), NORM = 'M' or 'm'
C (
C ( norm1(A), NORM = '1', 'O' or 'o'
C (
C ( normI(A), NORM = 'I' or 'i'
C (
C ( normF(A), NORM = 'F', 'f', 'E' or 'e'
C
C where norm1 denotes the one norm of a matrix (maximum column sum),
C normI denotes the infinity norm of a matrix (maximum row sum) and
C normF denotes the Frobenius norm of a matrix (square root of sum of
C squares). Note that max(abs(A(i,j))) is not a matrix norm.
C
C Arguments
C =========
C
C NORM (input) CHARACTER*1
C Specifies the value to be returned in ZLANHE as described
C above.
C
C UPLO (input) CHARACTER*1
C Specifies whether the upper or lower triangular part of the
C hermitian matrix A is to be referenced.
C = 'U': Upper triangular part of A is referenced
C = 'L': Lower triangular part of A is referenced
C
C N (input) INTEGER
C The order of the matrix A. N >= 0. When N = 0, ZLANHE is
C set to zero.
C
C A (input) COMPLEX array, dimension (LDA,N)
C The hermitian matrix A. If UPLO = 'U', the leading n by n
C upper triangular part of A contains the upper triangular part
C of the matrix A, and the strictly lower triangular part of A
C is not referenced. If UPLO = 'L', the leading n by n lower
C triangular part of A contains the lower triangular part of
C the matrix A, and the strictly upper triangular part of A is
C not referenced. Note that the imaginary parts of the diagonal
C elements need not be set and are assumed to be zero.
C
C LDA (input) INTEGER
C The leading dimension of the array A. LDA >= max(N,1).
C
C WORK (workspace) REAL array, dimension (LWORK),
C where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
C WORK is not referenced.
C
C
C -- LAPACK auxiliary routine (adapted for NAG Library)
C Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
C Courant Institute, Argonne National Lab, and Rice University
C
DOUBLE PRECISION FUNCTION F06UDF(NORM,UPLO,N,AP,WORK)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C REAL ZLANHP
C ENTRY ZLANHP(NORM,UPLO,N,AP,WORK)
C
C Purpose
C =======
C
C ZLANHP returns the value of the one norm, or the Frobenius norm, or
C the infinity norm, or the element of largest absolute value of a
C complex hermitian matrix A, supplied in packed form.
C
C Description
C ===========
C
C ZLANHP returns the value
C
C ZLANHP = ( max(abs(A(i,j))), NORM = 'M' or 'm'
C (
C ( norm1(A), NORM = '1', 'O' or 'o'
C (
C ( normI(A), NORM = 'I' or 'i'
C (
C ( normF(A), NORM = 'F', 'f', 'E' or 'e'
C
C where norm1 denotes the one norm of a matrix (maximum column sum),
C normI denotes the infinity norm of a matrix (maximum row sum) and
C normF denotes the Frobenius norm of a matrix (square root of sum of
C squares). Note that max(abs(A(i,j))) is not a matrix norm.
C
C Arguments
C =========
C
C NORM (input) CHARACTER*1
C Specifies the value to be returned in ZLANHP as described
C above.
C
C UPLO (input) CHARACTER*1
C Specifies whether the upper or lower triangular part of the
C hermitian matrix A is supplied.
C = 'U': Upper triangular part of A is supplied
C = 'L': Lower triangular part of A is supplied
C
C N (input) INTEGER
C The order of the matrix A. N >= 0. When N = 0, ZLANHP is
C set to zero.
C
C AP (input) COMPLEX array, dimension (N*(N+1)/2)
C The upper or lower triangle of the hermitian matrix A, packed
C columnwise in a linear array. The j-th column of A is stored
C in the array AP as follows:
C if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
C if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
C Note that the imaginary parts of the diagonal elements need
C not be set and are assumed to be zero.
C
C WORK (workspace) REAL array, dimension (LWORK),
C where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
C WORK is not referenced.
C
C
C -- LAPACK auxiliary routine (adapted for NAG Library)
C Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
C Courant Institute, Argonne National Lab, and Rice University
C
DOUBLE PRECISION FUNCTION F06UEF(NORM,UPLO,N,K,AB,LDAB,WORK)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C REAL ZLANHB
C ENTRY ZLANHB(NORM,UPLO,N,K,AB,LDAB,WORK)
C
C Purpose
C =======
C
C ZLANHB returns the value of the one norm, or the Frobenius norm, or
C the infinity norm, or the element of largest absolute value of an
C n by n hermitian band matrix A, with k super-diagonals.
C
C Description
C ===========
C
C ZLANHB returns the value
C
C ZLANHB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
C (
C ( norm1(A), NORM = '1', 'O' or 'o'
C (
C ( normI(A), NORM = 'I' or 'i'
C (
C ( normF(A), NORM = 'F', 'f', 'E' or 'e'
C
C where norm1 denotes the one norm of a matrix (maximum column sum),
C normI denotes the infinity norm of a matrix (maximum row sum) and
C normF denotes the Frobenius norm of a matrix (square root of sum of
C squares). Note that max(abs(A(i,j))) is not a matrix norm.
C
C Arguments
C =========
C
C NORM (input) CHARACTER*1
C Specifies the value to be returned in ZLANHB as described
C above.
C
C UPLO (input) CHARACTER*1
C Specifies whether the upper or lower triangular part of the
C band matrix A is supplied.
C = 'U': Upper triangular
C = 'L': Lower triangular
C
C N (input) INTEGER
C The order of the matrix A. N >= 0. When N = 0, ZLANHB is
C set to zero.
C
C K (input) INTEGER
C The number of super-diagonals or sub-diagonals of the
C band matrix A. K >= 0.
C
C AB (input) COMPLEX array, dimension (LDAB,N)
C The upper or lower triangle of the hermitian band matrix A,
C stored in the first K+1 rows of AB. The j-th column of A is
C stored in the j-th column of the array AB as follows:
C if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
C if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+k).
C Note that the imaginary parts of the diagonal elements need
C not be set and are assumed to be zero.
C
C LDAB (input) INTEGER
C The leading dimension of the array AB. LDAB >= K+1.
C
C WORK (workspace) REAL array, dimension (LWORK),
C where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
C WORK is not referenced.
C
C
C -- LAPACK auxiliary routine (adapted for NAG Library)
C Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
C Courant Institute, Argonne National Lab, and Rice University
C
DOUBLE PRECISION FUNCTION F06UFF(NORM,UPLO,N,A,LDA,WORK)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C REAL ZLANSY
C ENTRY ZLANSY(NORM,UPLO,N,A,LDA,WORK)
C
C Purpose
C =======
C
C ZLANSY returns the value of the one norm, or the Frobenius norm, or
C the infinity norm, or the element of largest absolute value of a
C complex symmetric matrix A.
C
C Description
C ===========
C
C ZLANSY returns the value
C
C ZLANSY = ( max(abs(A(i,j))), NORM = 'M' or 'm'
C (
C ( norm1(A), NORM = '1', 'O' or 'o'
C (
C ( normI(A), NORM = 'I' or 'i'
C (
C ( normF(A), NORM = 'F', 'f', 'E' or 'e'
C
C where norm1 denotes the one norm of a matrix (maximum column sum),
C normI denotes the infinity norm of a matrix (maximum row sum) and
C normF denotes the Frobenius norm of a matrix (square root of sum of
C squares). Note that max(abs(A(i,j))) is not a matrix norm.
C
C Arguments
C =========
C
C NORM (input) CHARACTER*1
C Specifies the value to be returned in ZLANSY as described
C above.
C
C UPLO (input) CHARACTER*1
C Specifies whether the upper or lower triangular part of the
C symmetric matrix A is to be referenced.
C = 'U': Upper triangular part of A is referenced
C = 'L': Lower triangular part of A is referenced
C
C N (input) INTEGER
C The order of the matrix A. N >= 0. When N = 0, ZLANSY is
C set to zero.
C
C A (input) COMPLEX array, dimension (LDA,N)
C The symmetric matrix A. If UPLO = 'U', the leading n by n
C upper triangular part of A contains the upper triangular part
C of the matrix A, and the strictly lower triangular part of A
C is not referenced. If UPLO = 'L', the leading n by n lower
C triangular part of A contains the lower triangular part of
C the matrix A, and the strictly upper triangular part of A is
C not referenced.
C
C LDA (input) INTEGER
C The leading dimension of the array A. LDA >= max(N,1).
C
C WORK (workspace) REAL array, dimension (LWORK),
C where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
C WORK is not referenced.
C
C
C -- LAPACK auxiliary routine (adapted for NAG Library)
C Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
C Courant Institute, Argonne National Lab, and Rice University
C
DOUBLE PRECISION FUNCTION F06UGF(NORM,UPLO,N,AP,WORK)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C REAL ZLANSP
C ENTRY ZLANSP(NORM,UPLO,N,AP,WORK)
C
C Purpose
C =======
C
C ZLANSP returns the value of the one norm, or the Frobenius norm, or
C the infinity norm, or the element of largest absolute value of a
C complex symmetric matrix A, supplied in packed form.
C
C Description
C ===========
C
C ZLANSP returns the value
C
C ZLANSP = ( max(abs(A(i,j))), NORM = 'M' or 'm'
C (
C ( norm1(A), NORM = '1', 'O' or 'o'
C (
C ( normI(A), NORM = 'I' or 'i'
C (
C ( normF(A), NORM = 'F', 'f', 'E' or 'e'
C
C where norm1 denotes the one norm of a matrix (maximum column sum),
C normI denotes the infinity norm of a matrix (maximum row sum) and
C normF denotes the Frobenius norm of a matrix (square root of sum of
C squares). Note that max(abs(A(i,j))) is not a matrix norm.
C
C Arguments
C =========
C
C NORM (input) CHARACTER*1
C Specifies the value to be returned in ZLANSP as described
C above.
C
C UPLO (input) CHARACTER*1
C Specifies whether the upper or lower triangular part of the
C symmetric matrix A is supplied.
C = 'U': Upper triangular part of A is supplied
C = 'L': Lower triangular part of A is supplied
C
C N (input) INTEGER
C The order of the matrix A. N >= 0. When N = 0, ZLANSP is
C set to zero.
C
C AP (input) COMPLEX array, dimension (N*(N+1)/2)
C The upper or lower triangle of the symmetric matrix A, packed
C columnwise in a linear array. The j-th column of A is stored
C in the array AP as follows:
C if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
C if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
C
C WORK (workspace) REAL array, dimension (LWORK),
C where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
C WORK is not referenced.
C
C
C -- LAPACK auxiliary routine (adapted for NAG Library)
C Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
C Courant Institute, Argonne National Lab, and Rice University
C
DOUBLE PRECISION FUNCTION F06UHF(NORM,UPLO,N,K,AB,LDAB,WORK)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C REAL ZLANSB
C ENTRY ZLANSB(NORM,UPLO,N,K,AB,LDAB,WORK)
C
C Purpose
C =======
C
C ZLANSB returns the value of the one norm, or the Frobenius norm, or
C the infinity norm, or the element of largest absolute value of an
C n by n symmetric band matrix A, with k super-diagonals.
C
C Description
C ===========
C
C ZLANSB returns the value
C
C ZLANSB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
C (
C ( norm1(A), NORM = '1', 'O' or 'o'
C (
C ( normI(A), NORM = 'I' or 'i'
C (
C ( normF(A), NORM = 'F', 'f', 'E' or 'e'
C
C where norm1 denotes the one norm of a matrix (maximum column sum),
C normI denotes the infinity norm of a matrix (maximum row sum) and
C normF denotes the Frobenius norm of a matrix (square root of sum of
C squares). Note that max(abs(A(i,j))) is not a matrix norm.
C
C Arguments
C =========
C
C NORM (input) CHARACTER*1
C Specifies the value to be returned in ZLANSB as described
C above.
C
C UPLO (input) CHARACTER*1
C Specifies whether the upper or lower triangular part of the
C band matrix A is supplied.
C = 'U': Upper triangular part is supplied
C = 'L': Lower triangular part is supplied
C
C N (input) INTEGER
C The order of the matrix A. N >= 0. When N = 0, ZLANSB is
C set to zero.
C
C K (input) INTEGER
C The number of super-diagonals or sub-diagonals of the
C band matrix A. K >= 0.
C
C AB (input) COMPLEX array, dimension (LDAB,N)
C The upper or lower triangle of the symmetric band matrix A,
C stored in the first K+1 rows of AB. The j-th column of A is
C stored in the j-th column of the array AB as follows:
C if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
C if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+k).
C
C LDAB (input) INTEGER
C The leading dimension of the array AB. LDAB >= K+1.
C
C WORK (workspace) REAL array, dimension (LWORK),
C where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
C WORK is not referenced.
C
C
C -- LAPACK auxiliary routine (adapted for NAG Library)
C Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
C Courant Institute, Argonne National Lab, and Rice University
C
DOUBLE PRECISION FUNCTION F06UJF(NORM,UPLO,DIAG,M,N,A,LDA,WORK)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C REAL ZLANTR
C ENTRY ZLANTR(NORM,UPLO,DIAG,M,N,A,LDA,WORK)
C
C Purpose
C =======
C
C ZLANTR returns the value of the one norm, or the Frobenius norm, or
C the infinity norm, or the element of largest absolute value of a
C trapezoidal or triangular matrix A.
C
C Description
C ===========
C
C ZLANTR returns the value
C
C ZLANTR = ( max(abs(A(i,j))), NORM = 'M' or 'm'
C (
C ( norm1(A), NORM = '1', 'O' or 'o'
C (
C ( normI(A), NORM = 'I' or 'i'
C (
C ( normF(A), NORM = 'F', 'f', 'E' or 'e'
C
C where norm1 denotes the one norm of a matrix (maximum column sum),
C normI denotes the infinity norm of a matrix (maximum row sum) and
C normF denotes the Frobenius norm of a matrix (square root of sum of
C squares). Note that max(abs(A(i,j))) is not a matrix norm.
C
C Arguments
C =========
C
C NORM (input) CHARACTER*1
C Specifies the value to be returned in ZLANTR as described
C above.
C
C UPLO (input) CHARACTER*1
C Specifies whether the matrix A is upper or lower trapezoidal.
C = 'U': Upper trapezoidal
C = 'L': Lower trapezoidal
C Note that A is triangular instead of trapezoidal if M = N.
C
C DIAG (input) CHARACTER*1
C Specifies whether or not the matrix A has unit diagonal.
C = 'N': Non-unit diagonal
C = 'U': Unit diagonal
C
C M (input) INTEGER
C The number of rows of the matrix A. M >= 0, and if
C UPLO = 'U', M <= N. When M = 0, ZLANTR is set to zero.
C
C N (input) INTEGER
C The number of columns of the matrix A. N >= 0, and if
C UPLO = 'L', N <= M. When N = 0, ZLANTR is set to zero.
C
C A (input) COMPLEX array, dimension (LDA,N)
C The trapezoidal matrix A (A is triangular if M = N).
C If UPLO = 'U', the leading m by n upper trapezoidal part of
C the array A contains the upper trapezoidal matrix, and the
C strictly lower triangular part of A is not referenced.
C If UPLO = 'L', the leading m by n lower trapezoidal part of
C the array A contains the lower trapezoidal matrix, and the
C strictly upper triangular part of A is not referenced. Note
C that when DIAG = 'U', the diagonal elements of A are not
C referenced and are assumed to be one.
C
C LDA (input) INTEGER
C The leading dimension of the array A. LDA >= max(M,1).
C
C WORK (workspace) REAL array, dimension (LWORK),
C where LWORK >= M when NORM = 'I'; otherwise, WORK is not
C referenced.
C
C
C -- LAPACK auxiliary routine (adapted for NAG Library)
C Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
C Courant Institute, Argonne National Lab, and Rice University
C
DOUBLE PRECISION FUNCTION F06UKF(NORM,UPLO,DIAG,N,AP,WORK)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C REAL ZLANTP
C ENTRY ZLANTP(NORM,UPLO,DIAG,N,AP,WORK)
C
C Purpose
C =======
C
C ZLANTP returns the value of the one norm, or the Frobenius norm, or
C the infinity norm, or the element of largest absolute value of a
C triangular matrix A, supplied in packed form.
C
C Description
C ===========
C
C ZLANTP returns the value
C
C ZLANTP = ( max(abs(A(i,j))), NORM = 'M' or 'm'
C (
C ( norm1(A), NORM = '1', 'O' or 'o'
C (
C ( normI(A), NORM = 'I' or 'i'
C (
C ( normF(A), NORM = 'F', 'f', 'E' or 'e'
C
C where norm1 denotes the one norm of a matrix (maximum column sum),
C normI denotes the infinity norm of a matrix (maximum row sum) and
C normF denotes the Frobenius norm of a matrix (square root of sum of
C squares). Note that max(abs(A(i,j))) is not a matrix norm.
C
C Arguments
C =========
C
C NORM (input) CHARACTER*1
C Specifies the value to be returned in ZLANTP as described
C above.
C
C UPLO (input) CHARACTER*1
C Specifies whether the matrix A is upper or lower triangular.
C = 'U': Upper triangular
C = 'L': Lower triangular
C
C DIAG (input) CHARACTER*1
C Specifies whether or not the matrix A is unit triangular.
C = 'N': Non-unit triangular
C = 'U': Unit triangular
C
C N (input) INTEGER
C The order of the matrix A. N >= 0. When N = 0, ZLANTP is
C set to zero.
C
C AP (input) COMPLEX array, dimension (N*(N+1)/2)
C The upper or lower triangular matrix A, packed columnwise in
C a linear array. The j-th column of A is stored in the array
C AP as follows:
C if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
C if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
C Note that when DIAG = 'U', the elements of the array AP
C corresponding to the diagonal elements of the matrix A are
C not referenced, but are assumed to be one.
C
C WORK (workspace) REAL array, dimension (LWORK),
C where LWORK >= N when NORM = 'I'; otherwise, WORK is not
C referenced.
C
C
C -- LAPACK auxiliary routine (adapted for NAG Library)
C Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
C Courant Institute, Argonne National Lab, and Rice University
C
DOUBLE PRECISION FUNCTION F06ULF(NORM,UPLO,DIAG,N,K,AB,LDAB,WORK)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C REAL ZLANTB
C ENTRY ZLANTB(NORM,UPLO,DIAG,N,K,AB,LDAB,WORK)
C
C Purpose
C =======
C
C ZLANTB returns the value of the one norm, or the Frobenius norm, or
C the infinity norm, or the element of largest absolute value of an
C n by n triangular band matrix A, with ( k + 1 ) diagonals.
C
C Description
C ===========
C
C ZLANTB returns the value
C
C ZLANTB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
C (
C ( norm1(A), NORM = '1', 'O' or 'o'
C (
C ( normI(A), NORM = 'I' or 'i'
C (
C ( normF(A), NORM = 'F', 'f', 'E' or 'e'
C
C where norm1 denotes the one norm of a matrix (maximum column sum),
C normI denotes the infinity norm of a matrix (maximum row sum) and
C normF denotes the Frobenius norm of a matrix (square root of sum of
C squares). Note that max(abs(A(i,j))) is not a matrix norm.
C
C Arguments
C =========
C
C NORM (input) CHARACTER*1
C Specifies the value to be returned in ZLANTB as described
C above.
C
C UPLO (input) CHARACTER*1
C Specifies whether the matrix A is upper or lower triangular.
C = 'U': Upper triangular
C = 'L': Lower triangular
C
C DIAG (input) CHARACTER*1
C Specifies whether or not the matrix A is unit triangular.
C = 'N': Non-unit triangular
C = 'U': Unit triangular
C
C N (input) INTEGER
C The order of the matrix A. N >= 0. When N = 0, ZLANTB is
C set to zero.
C
C K (input) INTEGER
C The number of super-diagonals of the matrix A if UPLO = 'U',
C or the number of sub-diagonals of the matrix A if UPLO = 'L'.
C K >= 0.
C
C AB (input) COMPLEX array, dimension (LDAB,N)
C The upper or lower triangular band matrix A, stored in the
C first k+1 rows of AB. The j-th column of A is stored
C in the j-th column of the array AB as follows:
C if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
C if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+k).
C Note that when DIAG = 'U', the elements of the array AB
C corresponding to the diagonal elements of the matrix A are
C not referenced, but are assumed to be one.
C
C LDAB (input) INTEGER
C The leading dimension of the array AB. LDAB >= K+1.
C
C WORK (workspace) REAL array, dimension (LWORK),
C where LWORK >= N when NORM = 'I'; otherwise, WORK is not
C referenced.
C
C
C -- LAPACK auxiliary routine (adapted for NAG Library)
C Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
C Courant Institute, Argonne National Lab, and Rice University
C
DOUBLE PRECISION FUNCTION F06UMF(NORM,N,A,LDA,WORK)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C REAL ZLANHS
C ENTRY ZLANHS(NORM,N,A,LDA,WORK)
C
C Purpose
C =======
C
C ZLANHS returns the value of the one norm, or the Frobenius norm, or
C the infinity norm, or the element of largest absolute value of a
C Hessenberg matrix A.
C
C Description
C ===========
C
C ZLANHS returns the value
C
C ZLANHS = ( max(abs(A(i,j))), NORM = 'M' or 'm'
C (
C ( norm1(A), NORM = '1', 'O' or 'o'
C (
C ( normI(A), NORM = 'I' or 'i'
C (
C ( normF(A), NORM = 'F', 'f', 'E' or 'e'
C
C where norm1 denotes the one norm of a matrix (maximum column sum),
C normI denotes the infinity norm of a matrix (maximum row sum) and
C normF denotes the Frobenius norm of a matrix (square root of sum of
C squares). Note that max(abs(A(i,j))) is not a matrix norm.
C
C Arguments
C =========
C
C NORM (input) CHARACTER*1
C Specifies the value to be returned in ZLANHS as described
C above.
C
C N (input) INTEGER
C The order of the matrix A. N >= 0. When N = 0, ZLANHS is
C set to zero.
C
C A (input) COMPLEX array, dimension (LDA,N)
C The n by n upper Hessenberg matrix A; the part of A below the
C first sub-diagonal is not referenced.
C
C LDA (input) INTEGER
C The leading dimension of the array A. LDA >= max(N,1).
C
C WORK (workspace) REAL array, dimension (LWORK),
C where LWORK >= N when NORM = 'I'; otherwise, WORK is not
C referenced.
C
C
C -- LAPACK auxiliary routine (adapted for NAG Library)
C Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
C Courant Institute, Argonne National Lab, and Rice University
C
SUBROUTINE F06VJF( SIDE, TRANS, N, PERM, K, B, LDB )
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE F06VKF( SIDE, TRANS, N, PERM, K, B, LDB )
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE F06VXF( SIDE, PIVOT, DIRECT, M, N, K1, K2, C, S, A,
$ LDA )
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
SUBROUTINE F06YAF(TRANSA,TRANSB,M,N,K,ALPHA,A,LDA,B,LDB,BETA,C,
* LDC)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C Purpose
C =======
C
C DGEMM performs one of the matrix-matrix operations
C
C C := alpha*op( A )*op( B ) + beta*C,
C
C where op( X ) is one of
C
C op( X ) = X or op( X ) = X',
C
C alpha and beta are scalars, and A, B and C are matrices, with op( A )
C an m by k matrix, op( B ) a k by n matrix and C an m by n matrix.
C
C Parameters
C ==========
C
C TRANSA - CHARACTER*1.
C On entry, TRANSA specifies the form of op( A ) to be used in
C the matrix multiplication as follows:
C
C TRANSA = 'N' or 'n', op( A ) = A.
C
C TRANSA = 'T' or 't', op( A ) = A'.
C
C TRANSA = 'C' or 'c', op( A ) = A'.
C
C Unchanged on exit.
C
C TRANSB - CHARACTER*1.
C On entry, TRANSB specifies the form of op( B ) to be used in
C the matrix multiplication as follows:
C
C TRANSB = 'N' or 'n', op( B ) = B.
C
C TRANSB = 'T' or 't', op( B ) = B'.
C
C TRANSB = 'C' or 'c', op( B ) = B'.
C
C Unchanged on exit.
C
C M - INTEGER.
C On entry, M specifies the number of rows of the matrix
C op( A ) and of the matrix C. M must be at least zero.
C Unchanged on exit.
C
C N - INTEGER.
C On entry, N specifies the number of columns of the matrix
C op( B ) and the number of columns of the matrix C. N must be
C at least zero.
C Unchanged on exit.
C
C K - INTEGER.
C On entry, K specifies the number of columns of the matrix
C op( A ) and the number of rows of the matrix op( B ). K must
C be at least zero.
C Unchanged on exit.
C
C ALPHA - REAL .
C On entry, ALPHA specifies the scalar alpha.
C Unchanged on exit.
C
C A - REAL array of DIMENSION ( LDA, ka ), where ka is
C k when TRANSA = 'N' or 'n', and is m otherwise.
C Before entry with TRANSA = 'N' or 'n', the leading m by k
C part of the array A must contain the matrix A, otherwise
C the leading k by m part of the array A must contain the
C matrix A.
C Unchanged on exit.
C
C LDA - INTEGER.
C On entry, LDA specifies the first dimension of A as declared
C in the calling (sub) program. When TRANSA = 'N' or 'n' then
C LDA must be at least max( 1, m ), otherwise LDA must be at
C least max( 1, k ).
C Unchanged on exit.
C
C B - REAL array of DIMENSION ( LDB, kb ), where kb is
C n when TRANSB = 'N' or 'n', and is k otherwise.
C Before entry with TRANSB = 'N' or 'n', the leading k by n
C part of the array B must contain the matrix B, otherwise
C the leading n by k part of the array B must contain the
C matrix B.
C Unchanged on exit.
C
C LDB - INTEGER.
C On entry, LDB specifies the first dimension of B as declared
C in the calling (sub) program. When TRANSB = 'N' or 'n' then
C LDB must be at least max( 1, k ), otherwise LDB must be at
C least max( 1, n ).
C Unchanged on exit.
C
C BETA - REAL .
C On entry, BETA specifies the scalar beta. When BETA is
C supplied as zero then C need not be set on input.
C Unchanged on exit.
C
C C - REAL array of DIMENSION ( LDC, n ).
C Before entry, the leading m by n part of the array C must
C contain the matrix C, except when beta is zero, in which
C case C need not be set on entry.
C On exit, the array C is overwritten by the m by n matrix
C ( alpha*op( A )*op( B ) + beta*C ).
C
C LDC - INTEGER.
C On entry, LDC specifies the first dimension of C as declared
C in the calling (sub) program. LDC must be at least
C max( 1, m ).
C Unchanged on exit.
C
C
C Level 3 Blas routine.
C
C -- Written on 8-February-1989.
C Jack Dongarra, Argonne National Laboratory.
C Iain Duff, AERE Harwell.
C Jeremy Du Croz, Numerical Algorithms Group Ltd.
C Sven Hammarling, Numerical Algorithms Group Ltd.
C
C
SUBROUTINE F06YCF(SIDE,UPLO,M,N,ALPHA,A,LDA,B,LDB,BETA,C,LDC)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C Purpose
C =======
C
C DSYMM performs one of the matrix-matrix operations
C
C C := alpha*A*B + beta*C,
C
C or
C
C C := alpha*B*A + beta*C,
C
C where alpha and beta are scalars, A is a symmetric matrix and B and
C C are m by n matrices.
C
C Parameters
C ==========
C
C SIDE - CHARACTER*1.
C On entry, SIDE specifies whether the symmetric matrix A
C appears on the left or right in the operation as follows:
C
C SIDE = 'L' or 'l' C := alpha*A*B + beta*C,
C
C SIDE = 'R' or 'r' C := alpha*B*A + beta*C,
C
C Unchanged on exit.
C
C UPLO - CHARACTER*1.
C On entry, UPLO specifies whether the upper or lower
C triangular part of the symmetric matrix A is to be
C referenced as follows:
C
C UPLO = 'U' or 'u' Only the upper triangular part of the
C symmetric matrix is to be referenced.
C
C UPLO = 'L' or 'l' Only the lower triangular part of the
C symmetric matrix is to be referenced.
C
C Unchanged on exit.
C
C M - INTEGER.
C On entry, M specifies the number of rows of the matrix C.
C M must be at least zero.
C Unchanged on exit.
C
C N - INTEGER.
C On entry, N specifies the number of columns of the matrix C.
C N must be at least zero.
C Unchanged on exit.
C
C ALPHA - REAL .
C On entry, ALPHA specifies the scalar alpha.
C Unchanged on exit.
C
C A - REAL array of DIMENSION ( LDA, ka ), where ka is
C m when SIDE = 'L' or 'l' and is n otherwise.
C Before entry with SIDE = 'L' or 'l', the m by m part of
C the array A must contain the symmetric matrix, such that
C when UPLO = 'U' or 'u', the leading m by m upper triangular
C part of the array A must contain the upper triangular part
C of the symmetric matrix and the strictly lower triangular
C part of A is not referenced, and when UPLO = 'L' or 'l',
C the leading m by m lower triangular part of the array A
C must contain the lower triangular part of the symmetric
C matrix and the strictly upper triangular part of A is not
C referenced.
C Before entry with SIDE = 'R' or 'r', the n by n part of
C the array A must contain the symmetric matrix, such that
C when UPLO = 'U' or 'u', the leading n by n upper triangular
C part of the array A must contain the upper triangular part
C of the symmetric matrix and the strictly lower triangular
C part of A is not referenced, and when UPLO = 'L' or 'l',
C the leading n by n lower triangular part of the array A
C must contain the lower triangular part of the symmetric
C matrix and the strictly upper triangular part of A is not
C referenced.
C Unchanged on exit.
C
C LDA - INTEGER.
C On entry, LDA specifies the first dimension of A as declared
C in the calling (sub) program. When SIDE = 'L' or 'l' then
C LDA must be at least max( 1, m ), otherwise LDA must be at
C least max( 1, n ).
C Unchanged on exit.
C
C B - REAL array of DIMENSION ( LDB, n ).
C Before entry, the leading m by n part of the array B must
C contain the matrix B.
C Unchanged on exit.
C
C LDB - INTEGER.
C On entry, LDB specifies the first dimension of B as declared
C in the calling (sub) program. LDB must be at least
C max( 1, m ).
C Unchanged on exit.
C
C BETA - REAL .
C On entry, BETA specifies the scalar beta. When BETA is
C supplied as zero then C need not be set on input.
C Unchanged on exit.
C
C C - REAL array of DIMENSION ( LDC, n ).
C Before entry, the leading m by n part of the array C must
C contain the matrix C, except when beta is zero, in which
C case C need not be set on entry.
C On exit, the array C is overwritten by the m by n updated
C matrix.
C
C LDC - INTEGER.
C On entry, LDC specifies the first dimension of C as declared
C in the calling (sub) program. LDC must be at least
C max( 1, m ).
C Unchanged on exit.
C
C
C Level 3 Blas routine.
C
C -- Written on 8-February-1989.
C Jack Dongarra, Argonne National Laboratory.
C Iain Duff, AERE Harwell.
C Jeremy Du Croz, Numerical Algorithms Group Ltd.
C Sven Hammarling, Numerical Algorithms Group Ltd.
C
C
SUBROUTINE F06YFF(SIDE,UPLO,TRANSA,DIAG,M,N,ALPHA,A,LDA,B,LDB)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C Purpose
C =======
C
C DTRMM performs one of the matrix-matrix operations
C
C B := alpha*op( A )*B, or B := alpha*B*op( A ),
C
C where alpha is a scalar, B is an m by n matrix, A is a unit, or
C non-unit, upper or lower triangular matrix and op( A ) is one of
C
C op( A ) = A or op( A ) = A'.
C
C Parameters
C ==========
C
C SIDE - CHARACTER*1.
C On entry, SIDE specifies whether op( A ) multiplies B from
C the left or right as follows:
C
C SIDE = 'L' or 'l' B := alpha*op( A )*B.
C
C SIDE = 'R' or 'r' B := alpha*B*op( A ).
C
C Unchanged on exit.
C
C UPLO - CHARACTER*1.
C On entry, UPLO specifies whether the matrix A is an upper or
C lower triangular matrix as follows:
C
C UPLO = 'U' or 'u' A is an upper triangular matrix.
C
C UPLO = 'L' or 'l' A is a lower triangular matrix.
C
C Unchanged on exit.
C
C TRANSA - CHARACTER*1.
C On entry, TRANSA specifies the form of op( A ) to be used in
C the matrix multiplication as follows:
C
C TRANSA = 'N' or 'n' op( A ) = A.
C
C TRANSA = 'T' or 't' op( A ) = A'.
C
C TRANSA = 'C' or 'c' op( A ) = A'.
C
C Unchanged on exit.
C
C DIAG - CHARACTER*1.
C On entry, DIAG specifies whether or not A is unit triangular
C as follows:
C
C DIAG = 'U' or 'u' A is assumed to be unit triangular.
C
C DIAG = 'N' or 'n' A is not assumed to be unit
C triangular.
C
C Unchanged on exit.
C
C M - INTEGER.
C On entry, M specifies the number of rows of B. M must be at
C least zero.
C Unchanged on exit.
C
C N - INTEGER.
C On entry, N specifies the number of columns of B. N must be
C at least zero.
C Unchanged on exit.
C
C ALPHA - REAL .
C On entry, ALPHA specifies the scalar alpha. When alpha is
C zero then A is not referenced and B need not be set before
C entry.
C Unchanged on exit.
C
C A - REAL array of DIMENSION ( LDA, k ), where k is m
C when SIDE = 'L' or 'l' and is n when SIDE = 'R' or 'r'.
C Before entry with UPLO = 'U' or 'u', the leading k by k
C upper triangular part of the array A must contain the upper
C triangular matrix and the strictly lower triangular part of
C A is not referenced.
C Before entry with UPLO = 'L' or 'l', the leading k by k
C lower triangular part of the array A must contain the lower
C triangular matrix and the strictly upper triangular part of
C A is not referenced.
C Note that when DIAG = 'U' or 'u', the diagonal elements of
C A are not referenced either, but are assumed to be unity.
C Unchanged on exit.
C
C LDA - INTEGER.
C On entry, LDA specifies the first dimension of A as declared
C in the calling (sub) program. When SIDE = 'L' or 'l' then
C LDA must be at least max( 1, m ), when SIDE = 'R' or 'r'
C then LDA must be at least max( 1, n ).
C Unchanged on exit.
C
C B - REAL array of DIMENSION ( LDB, n ).
C Before entry, the leading m by n part of the array B must
C contain the matrix B, and on exit is overwritten by the
C transformed matrix.
C
C LDB - INTEGER.
C On entry, LDB specifies the first dimension of B as declared
C in the calling (sub) program. LDB must be at least
C max( 1, m ).
C Unchanged on exit.
C
C
C Level 3 Blas routine.
C
C -- Written on 8-February-1989.
C Jack Dongarra, Argonne National Laboratory.
C Iain Duff, AERE Harwell.
C Jeremy Du Croz, Numerical Algorithms Group Ltd.
C Sven Hammarling, Numerical Algorithms Group Ltd.
C
C
SUBROUTINE F06YJF(SIDE,UPLO,TRANSA,DIAG,M,N,ALPHA,A,LDA,B,LDB)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C Purpose
C =======
C
C DTRSM solves one of the matrix equations
C
C op( A )*X = alpha*B, or X*op( A ) = alpha*B,
C
C where alpha is a scalar, X and B are m by n matrices, A is a unit, or
C non-unit, upper or lower triangular matrix and op( A ) is one of
C
C op( A ) = A or op( A ) = A'.
C
C The matrix X is overwritten on B.
C
C Parameters
C ==========
C
C SIDE - CHARACTER*1.
C On entry, SIDE specifies whether op( A ) appears on the left
C or right of X as follows:
C
C SIDE = 'L' or 'l' op( A )*X = alpha*B.
C
C SIDE = 'R' or 'r' X*op( A ) = alpha*B.
C
C Unchanged on exit.
C
C UPLO - CHARACTER*1.
C On entry, UPLO specifies whether the matrix A is an upper or
C lower triangular matrix as follows:
C
C UPLO = 'U' or 'u' A is an upper triangular matrix.
C
C UPLO = 'L' or 'l' A is a lower triangular matrix.
C
C Unchanged on exit.
C
C TRANSA - CHARACTER*1.
C On entry, TRANSA specifies the form of op( A ) to be used in
C the matrix multiplication as follows:
C
C TRANSA = 'N' or 'n' op( A ) = A.
C
C TRANSA = 'T' or 't' op( A ) = A'.
C
C TRANSA = 'C' or 'c' op( A ) = A'.
C
C Unchanged on exit.
C
C DIAG - CHARACTER*1.
C On entry, DIAG specifies whether or not A is unit triangular
C as follows:
C
C DIAG = 'U' or 'u' A is assumed to be unit triangular.
C
C DIAG = 'N' or 'n' A is not assumed to be unit
C triangular.
C
C Unchanged on exit.
C
C M - INTEGER.
C On entry, M specifies the number of rows of B. M must be at
C least zero.
C Unchanged on exit.
C
C N - INTEGER.
C On entry, N specifies the number of columns of B. N must be
C at least zero.
C Unchanged on exit.
C
C ALPHA - REAL .
C On entry, ALPHA specifies the scalar alpha. When alpha is
C zero then A is not referenced and B need not be set before
C entry.
C Unchanged on exit.
C
C A - REAL array of DIMENSION ( LDA, k ), where k is m
C when SIDE = 'L' or 'l' and is n when SIDE = 'R' or 'r'.
C Before entry with UPLO = 'U' or 'u', the leading k by k
C upper triangular part of the array A must contain the upper
C triangular matrix and the strictly lower triangular part of
C A is not referenced.
C Before entry with UPLO = 'L' or 'l', the leading k by k
C lower triangular part of the array A must contain the lower
C triangular matrix and the strictly upper triangular part of
C A is not referenced.
C Note that when DIAG = 'U' or 'u', the diagonal elements of
C A are not referenced either, but are assumed to be unity.
C Unchanged on exit.
C
C LDA - INTEGER.
C On entry, LDA specifies the first dimension of A as declared
C in the calling (sub) program. When SIDE = 'L' or 'l' then
C LDA must be at least max( 1, m ), when SIDE = 'R' or 'r'
C then LDA must be at least max( 1, n ).
C Unchanged on exit.
C
C B - REAL array of DIMENSION ( LDB, n ).
C Before entry, the leading m by n part of the array B must
C contain the right-hand side matrix B, and on exit is
C overwritten by the solution matrix X.
C
C LDB - INTEGER.
C On entry, LDB specifies the first dimension of B as declared
C in the calling (sub) program. LDB must be at least
C max( 1, m ).
C Unchanged on exit.
C
C
C Level 3 Blas routine.
C
C
C -- Written on 8-February-1989.
C Jack Dongarra, Argonne National Laboratory.
C Iain Duff, AERE Harwell.
C Jeremy Du Croz, Numerical Algorithms Group Ltd.
C Sven Hammarling, Numerical Algorithms Group Ltd.
C
C
SUBROUTINE F06YPF(UPLO,TRANS,N,K,ALPHA,A,LDA,BETA,C,LDC)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C Purpose
C =======
C
C DSYRK performs one of the symmetric rank k operations
C
C C := alpha*A*A' + beta*C,
C
C or
C
C C := alpha*A'*A + beta*C,
C
C where alpha and beta are scalars, C is an n by n symmetric matrix
C and A is an n by k matrix in the first case and a k by n matrix
C in the second case.
C
C Parameters
C ==========
C
C UPLO - CHARACTER*1.
C On entry, UPLO specifies whether the upper or lower
C triangular part of the array C is to be referenced as
C follows:
C
C UPLO = 'U' or 'u' Only the upper triangular part of C
C is to be referenced.
C
C UPLO = 'L' or 'l' Only the lower triangular part of C
C is to be referenced.
C
C Unchanged on exit.
C
C TRANS - CHARACTER*1.
C On entry, TRANS specifies the operation to be performed as
C follows:
C
C TRANS = 'N' or 'n' C := alpha*A*A' + beta*C.
C
C TRANS = 'T' or 't' C := alpha*A'*A + beta*C.
C
C TRANS = 'C' or 'c' C := alpha*A'*A + beta*C.
C
C Unchanged on exit.
C
C N - INTEGER.
C On entry, N specifies the order of the matrix C. N must be
C at least zero.
C Unchanged on exit.
C
C K - INTEGER.
C On entry with TRANS = 'N' or 'n', K specifies the number
C of columns of the matrix A, and on entry with
C TRANS = 'T' or 't' or 'C' or 'c', K specifies the number
C of rows of the matrix A. K must be at least zero.
C Unchanged on exit.
C
C ALPHA - REAL .
C On entry, ALPHA specifies the scalar alpha.
C Unchanged on exit.
C
C A - REAL array of DIMENSION ( LDA, ka ), where ka is
C k when TRANS = 'N' or 'n', and is n otherwise.
C Before entry with TRANS = 'N' or 'n', the leading n by k
C part of the array A must contain the matrix A, otherwise
C the leading k by n part of the array A must contain the
C matrix A.
C Unchanged on exit.
C
C LDA - INTEGER.
C On entry, LDA specifies the first dimension of A as declared
C in the calling (sub) program. When TRANS = 'N' or 'n'
C then LDA must be at least max( 1, n ), otherwise LDA must
C be at least max( 1, k ).
C Unchanged on exit.
C
C BETA - REAL .
C On entry, BETA specifies the scalar beta.
C Unchanged on exit.
C
C C - REAL array of DIMENSION ( LDC, n ).
C Before entry with UPLO = 'U' or 'u', the leading n by n
C upper triangular part of the array C must contain the upper
C triangular part of the symmetric matrix and the strictly
C lower triangular part of C is not referenced. On exit, the
C upper triangular part of the array C is overwritten by the
C upper triangular part of the updated matrix.
C Before entry with UPLO = 'L' or 'l', the leading n by n
C lower triangular part of the array C must contain the lower
C triangular part of the symmetric matrix and the strictly
C upper triangular part of C is not referenced. On exit, the
C lower triangular part of the array C is overwritten by the
C lower triangular part of the updated matrix.
C
C LDC - INTEGER.
C On entry, LDC specifies the first dimension of C as declared
C in the calling (sub) program. LDC must be at least
C max( 1, n ).
C Unchanged on exit.
C
C
C Level 3 Blas routine.
C
C -- Written on 8-February-1989.
C Jack Dongarra, Argonne National Laboratory.
C Iain Duff, AERE Harwell.
C Jeremy Du Croz, Numerical Algorithms Group Ltd.
C Sven Hammarling, Numerical Algorithms Group Ltd.
C
C
SUBROUTINE F06YRF(UPLO,TRANS,N,K,ALPHA,A,LDA,B,LDB,BETA,C,LDC)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C Purpose
C =======
C
C DSYR2K performs one of the symmetric rank 2k operations
C
C C := alpha*A*B' + alpha*B*A' + beta*C,
C
C or
C
C C := alpha*A'*B + alpha*B'*A + beta*C,
C
C where alpha and beta are scalars, C is an n by n symmetric matrix
C and A and B are n by k matrices in the first case and k by n
C matrices in the second case.
C
C Parameters
C ==========
C
C UPLO - CHARACTER*1.
C On entry, UPLO specifies whether the upper or lower
C triangular part of the array C is to be referenced as
C follows:
C
C UPLO = 'U' or 'u' Only the upper triangular part of C
C is to be referenced.
C
C UPLO = 'L' or 'l' Only the lower triangular part of C
C is to be referenced.
C
C Unchanged on exit.
C
C TRANS - CHARACTER*1.
C On entry, TRANS specifies the operation to be performed as
C follows:
C
C TRANS = 'N' or 'n' C := alpha*A*B' + alpha*B*A' +
C beta*C.
C
C TRANS = 'T' or 't' C := alpha*A'*B + alpha*B'*A +
C beta*C.
C
C TRANS = 'C' or 'c' C := alpha*A'*B + alpha*B'*A +
C beta*C.
C
C Unchanged on exit.
C
C N - INTEGER.
C On entry, N specifies the order of the matrix C. N must be
C at least zero.
C Unchanged on exit.
C
C K - INTEGER.
C On entry with TRANS = 'N' or 'n', K specifies the number
C of columns of the matrices A and B, and on entry with
C TRANS = 'T' or 't' or 'C' or 'c', K specifies the number
C of rows of the matrices A and B. K must be at least zero.
C Unchanged on exit.
C
C ALPHA - REAL .
C On entry, ALPHA specifies the scalar alpha.
C Unchanged on exit.
C
C A - REAL array of DIMENSION ( LDA, ka ), where ka is
C k when TRANS = 'N' or 'n', and is n otherwise.
C Before entry with TRANS = 'N' or 'n', the leading n by k
C part of the array A must contain the matrix A, otherwise
C the leading k by n part of the array A must contain the
C matrix A.
C Unchanged on exit.
C
C LDA - INTEGER.
C On entry, LDA specifies the first dimension of A as declared
C in the calling (sub) program. When TRANS = 'N' or 'n'
C then LDA must be at least max( 1, n ), otherwise LDA must
C be at least max( 1, k ).
C Unchanged on exit.
C
C B - REAL array of DIMENSION ( LDB, kb ), where kb is
C k when TRANS = 'N' or 'n', and is n otherwise.
C Before entry with TRANS = 'N' or 'n', the leading n by k
C part of the array B must contain the matrix B, otherwise
C the leading k by n part of the array B must contain the
C matrix B.
C Unchanged on exit.
C
C LDB - INTEGER.
C On entry, LDB specifies the first dimension of B as declared
C in the calling (sub) program. When TRANS = 'N' or 'n'
C then LDB must be at least max( 1, n ), otherwise LDB must
C be at least max( 1, k ).
C Unchanged on exit.
C
C BETA - REAL .
C On entry, BETA specifies the scalar beta.
C Unchanged on exit.
C
C C - REAL array of DIMENSION ( LDC, n ).
C Before entry with UPLO = 'U' or 'u', the leading n by n
C upper triangular part of the array C must contain the upper
C triangular part of the symmetric matrix and the strictly
C lower triangular part of C is not referenced. On exit, the
C upper triangular part of the array C is overwritten by the
C upper triangular part of the updated matrix.
C Before entry with UPLO = 'L' or 'l', the leading n by n
C lower triangular part of the array C must contain the lower
C triangular part of the symmetric matrix and the strictly
C upper triangular part of C is not referenced. On exit, the
C lower triangular part of the array C is overwritten by the
C lower triangular part of the updated matrix.
C
C LDC - INTEGER.
C On entry, LDC specifies the first dimension of C as declared
C in the calling (sub) program. LDC must be at least
C max( 1, n ).
C Unchanged on exit.
C
C
C Level 3 Blas routine.
C
C
C -- Written on 8-February-1989.
C Jack Dongarra, Argonne National Laboratory.
C Iain Duff, AERE Harwell.
C Jeremy Du Croz, Numerical Algorithms Group Ltd.
C Sven Hammarling, Numerical Algorithms Group Ltd.
C
C
SUBROUTINE F06ZAF(TRANSA,TRANSB,M,N,K,ALPHA,A,LDA,B,LDB,BETA,C,
* LDC)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C Purpose
C =======
C
C ZGEMM performs one of the matrix-matrix operations
C
C C := alpha*op( A )*op( B ) + beta*C,
C
C where op( X ) is one of
C
C op( X ) = X or op( X ) = X' or op( X ) = conjg( X' ),
C
C alpha and beta are scalars, and A, B and C are matrices, with op( A )
C an m by k matrix, op( B ) a k by n matrix and C an m by n matrix.
C
C Parameters
C ==========
C
C TRANSA - CHARACTER*1.
C On entry, TRANSA specifies the form of op( A ) to be used in
C the matrix multiplication as follows:
C
C TRANSA = 'N' or 'n', op( A ) = A.
C
C TRANSA = 'T' or 't', op( A ) = A'.
C
C TRANSA = 'C' or 'c', op( A ) = conjg( A' ).
C
C Unchanged on exit.
C
C TRANSB - CHARACTER*1.
C On entry, TRANSB specifies the form of op( B ) to be used in
C the matrix multiplication as follows:
C
C TRANSB = 'N' or 'n', op( B ) = B.
C
C TRANSB = 'T' or 't', op( B ) = B'.
C
C TRANSB = 'C' or 'c', op( B ) = conjg( B' ).
C
C Unchanged on exit.
C
C M - INTEGER.
C On entry, M specifies the number of rows of the matrix
C op( A ) and of the matrix C. M must be at least zero.
C Unchanged on exit.
C
C N - INTEGER.
C On entry, N specifies the number of columns of the matrix
C op( B ) and the number of columns of the matrix C. N must be
C at least zero.
C Unchanged on exit.
C
C K - INTEGER.
C On entry, K specifies the number of columns of the matrix
C op( A ) and the number of rows of the matrix op( B ). K must
C be at least zero.
C Unchanged on exit.
C
C ALPHA - COMPLEX .
C On entry, ALPHA specifies the scalar alpha.
C Unchanged on exit.
C
C A - COMPLEX array of DIMENSION ( LDA, ka ), where ka is
C k when TRANSA = 'N' or 'n', and is m otherwise.
C Before entry with TRANSA = 'N' or 'n', the leading m by k
C part of the array A must contain the matrix A, otherwise
C the leading k by m part of the array A must contain the
C matrix A.
C Unchanged on exit.
C
C LDA - INTEGER.
C On entry, LDA specifies the first dimension of A as declared
C in the calling (sub) program. When TRANSA = 'N' or 'n' then
C LDA must be at least max( 1, m ), otherwise LDA must be at
C least max( 1, k ).
C Unchanged on exit.
C
C B - COMPLEX array of DIMENSION ( LDB, kb ), where kb is
C n when TRANSB = 'N' or 'n', and is k otherwise.
C Before entry with TRANSB = 'N' or 'n', the leading k by n
C part of the array B must contain the matrix B, otherwise
C the leading n by k part of the array B must contain the
C matrix B.
C Unchanged on exit.
C
C LDB - INTEGER.
C On entry, LDB specifies the first dimension of B as declared
C in the calling (sub) program. When TRANSB = 'N' or 'n' then
C LDB must be at least max( 1, k ), otherwise LDB must be at
C least max( 1, n ).
C Unchanged on exit.
C
C BETA - COMPLEX .
C On entry, BETA specifies the scalar beta. When BETA is
C supplied as zero then C need not be set on input.
C Unchanged on exit.
C
C C - COMPLEX array of DIMENSION ( LDC, n ).
C Before entry, the leading m by n part of the array C must
C contain the matrix C, except when beta is zero, in which
C case C need not be set on entry.
C On exit, the array C is overwritten by the m by n matrix
C ( alpha*op( A )*op( B ) + beta*C ).
C
C LDC - INTEGER.
C On entry, LDC specifies the first dimension of C as declared
C in the calling (sub) program. LDC must be at least
C max( 1, m ).
C Unchanged on exit.
C
C
C Level 3 Blas routine.
C
C -- Written on 8-February-1989.
C Jack Dongarra, Argonne National Laboratory.
C Iain Duff, AERE Harwell.
C Jeremy Du Croz, Numerical Algorithms Group Ltd.
C Sven Hammarling, Numerical Algorithms Group Ltd.
C
C
SUBROUTINE F06ZCF(SIDE,UPLO,M,N,ALPHA,A,LDA,B,LDB,BETA,C,LDC)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C Purpose
C =======
C
C ZHEMM performs one of the matrix-matrix operations
C
C C := alpha*A*B + beta*C,
C
C or
C
C C := alpha*B*A + beta*C,
C
C where alpha and beta are scalars, A is an hermitian matrix and B and
C C are m by n matrices.
C
C Parameters
C ==========
C
C SIDE - CHARACTER*1.
C On entry, SIDE specifies whether the hermitian matrix A
C appears on the left or right in the operation as follows:
C
C SIDE = 'L' or 'l' C := alpha*A*B + beta*C,
C
C SIDE = 'R' or 'r' C := alpha*B*A + beta*C,
C
C Unchanged on exit.
C
C UPLO - CHARACTER*1.
C On entry, UPLO specifies whether the upper or lower
C triangular part of the hermitian matrix A is to be
C referenced as follows:
C
C UPLO = 'U' or 'u' Only the upper triangular part of the
C hermitian matrix is to be referenced.
C
C UPLO = 'L' or 'l' Only the lower triangular part of the
C hermitian matrix is to be referenced.
C
C Unchanged on exit.
C
C M - INTEGER.
C On entry, M specifies the number of rows of the matrix C.
C M must be at least zero.
C Unchanged on exit.
C
C N - INTEGER.
C On entry, N specifies the number of columns of the matrix C.
C N must be at least zero.
C Unchanged on exit.
C
C ALPHA - COMPLEX .
C On entry, ALPHA specifies the scalar alpha.
C Unchanged on exit.
C
C A - COMPLEX array of DIMENSION ( LDA, ka ), where ka is
C m when SIDE = 'L' or 'l' and is n otherwise.
C Before entry with SIDE = 'L' or 'l', the m by m part of
C the array A must contain the hermitian matrix, such that
C when UPLO = 'U' or 'u', the leading m by m upper triangular
C part of the array A must contain the upper triangular part
C of the hermitian matrix and the strictly lower triangular
C part of A is not referenced, and when UPLO = 'L' or 'l',
C the leading m by m lower triangular part of the array A
C must contain the lower triangular part of the hermitian
C matrix and the strictly upper triangular part of A is not
C referenced.
C Before entry with SIDE = 'R' or 'r', the n by n part of
C the array A must contain the hermitian matrix, such that
C when UPLO = 'U' or 'u', the leading n by n upper triangular
C part of the array A must contain the upper triangular part
C of the hermitian matrix and the strictly lower triangular
C part of A is not referenced, and when UPLO = 'L' or 'l',
C the leading n by n lower triangular part of the array A
C must contain the lower triangular part of the hermitian
C matrix and the strictly upper triangular part of A is not
C referenced.
C Note that the imaginary parts of the diagonal elements need
C not be set, they are assumed to be zero.
C Unchanged on exit.
C
C LDA - INTEGER.
C On entry, LDA specifies the first dimension of A as declared
C in the calling (sub) program. When SIDE = 'L' or 'l' then
C LDA must be at least max( 1, m ), otherwise LDA must be at
C least max( 1, n ).
C Unchanged on exit.
C
C B - COMPLEX array of DIMENSION ( LDB, n ).
C Before entry, the leading m by n part of the array B must
C contain the matrix B.
C Unchanged on exit.
C
C LDB - INTEGER.
C On entry, LDB specifies the first dimension of B as declared
C in the calling (sub) program. LDB must be at least
C max( 1, m ).
C Unchanged on exit.
C
C BETA - COMPLEX .
C On entry, BETA specifies the scalar beta. When BETA is
C supplied as zero then C need not be set on input.
C Unchanged on exit.
C
C C - COMPLEX array of DIMENSION ( LDC, n ).
C Before entry, the leading m by n part of the array C must
C contain the matrix C, except when beta is zero, in which
C case C need not be set on entry.
C On exit, the array C is overwritten by the m by n updated
C matrix.
C
C LDC - INTEGER.
C On entry, LDC specifies the first dimension of C as declared
C in the calling (sub) program. LDC must be at least
C max( 1, m ).
C Unchanged on exit.
C
C
C Level 3 Blas routine.
C
C -- Written on 8-February-1989.
C Jack Dongarra, Argonne National Laboratory.
C Iain Duff, AERE Harwell.
C Jeremy Du Croz, Numerical Algorithms Group Ltd.
C Sven Hammarling, Numerical Algorithms Group Ltd.
C
C
SUBROUTINE F06ZFF(SIDE,UPLO,TRANSA,DIAG,M,N,ALPHA,A,LDA,B,LDB)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C Purpose
C =======
C
C ZTRMM performs one of the matrix-matrix operations
C
C B := alpha*op( A )*B, or B := alpha*B*op( A )
C
C where alpha is a scalar, B is an m by n matrix, A is a unit, or
C non-unit, upper or lower triangular matrix and op( A ) is one of
C
C op( A ) = A or op( A ) = A' or op( A ) = conjg( A' ).
C
C Parameters
C ==========
C
C SIDE - CHARACTER*1.
C On entry, SIDE specifies whether op( A ) multiplies B from
C the left or right as follows:
C
C SIDE = 'L' or 'l' B := alpha*op( A )*B.
C
C SIDE = 'R' or 'r' B := alpha*B*op( A ).
C
C Unchanged on exit.
C
C UPLO - CHARACTER*1.
C On entry, UPLO specifies whether the matrix A is an upper or
C lower triangular matrix as follows:
C
C UPLO = 'U' or 'u' A is an upper triangular matrix.
C
C UPLO = 'L' or 'l' A is a lower triangular matrix.
C
C Unchanged on exit.
C
C TRANSA - CHARACTER*1.
C On entry, TRANSA specifies the form of op( A ) to be used in
C the matrix multiplication as follows:
C
C TRANSA = 'N' or 'n' op( A ) = A.
C
C TRANSA = 'T' or 't' op( A ) = A'.
C
C TRANSA = 'C' or 'c' op( A ) = conjg( A' ).
C
C Unchanged on exit.
C
C DIAG - CHARACTER*1.
C On entry, DIAG specifies whether or not A is unit triangular
C as follows:
C
C DIAG = 'U' or 'u' A is assumed to be unit triangular.
C
C DIAG = 'N' or 'n' A is not assumed to be unit
C triangular.
C
C Unchanged on exit.
C
C M - INTEGER.
C On entry, M specifies the number of rows of B. M must be at
C least zero.
C Unchanged on exit.
C
C N - INTEGER.
C On entry, N specifies the number of columns of B. N must be
C at least zero.
C Unchanged on exit.
C
C ALPHA - COMPLEX .
C On entry, ALPHA specifies the scalar alpha. When alpha is
C zero then A is not referenced and B need not be set before
C entry.
C Unchanged on exit.
C
C A - COMPLEX array of DIMENSION ( LDA, k ), where k is m
C when SIDE = 'L' or 'l' and is n when SIDE = 'R' or 'r'.
C Before entry with UPLO = 'U' or 'u', the leading k by k
C upper triangular part of the array A must contain the upper
C triangular matrix and the strictly lower triangular part of
C A is not referenced.
C Before entry with UPLO = 'L' or 'l', the leading k by k
C lower triangular part of the array A must contain the lower
C triangular matrix and the strictly upper triangular part of
C A is not referenced.
C Note that when DIAG = 'U' or 'u', the diagonal elements of
C A are not referenced either, but are assumed to be unity.
C Unchanged on exit.
C
C LDA - INTEGER.
C On entry, LDA specifies the first dimension of A as declared
C in the calling (sub) program. When SIDE = 'L' or 'l' then
C LDA must be at least max( 1, m ), when SIDE = 'R' or 'r'
C then LDA must be at least max( 1, n ).
C Unchanged on exit.
C
C B - COMPLEX array of DIMENSION ( LDB, n ).
C Before entry, the leading m by n part of the array B must
C contain the matrix B, and on exit is overwritten by the
C transformed matrix.
C
C LDB - INTEGER.
C On entry, LDB specifies the first dimension of B as declared
C in the calling (sub) program. LDB must be at least
C max( 1, m ).
C Unchanged on exit.
C
C
C Level 3 Blas routine.
C
C -- Written on 8-February-1989.
C Jack Dongarra, Argonne National Laboratory.
C Iain Duff, AERE Harwell.
C Jeremy Du Croz, Numerical Algorithms Group Ltd.
C Sven Hammarling, Numerical Algorithms Group Ltd.
C
C
SUBROUTINE F06ZJF(SIDE,UPLO,TRANSA,DIAG,M,N,ALPHA,A,LDA,B,LDB)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C Purpose
C =======
C
C ZTRSM solves one of the matrix equations
C
C op( A )*X = alpha*B, or X*op( A ) = alpha*B,
C
C where alpha is a scalar, X and B are m by n matrices, A is a unit, or
C non-unit, upper or lower triangular matrix and op( A ) is one of
C
C op( A ) = A or op( A ) = A' or op( A ) = conjg( A' ).
C
C The matrix X is overwritten on B.
C
C Parameters
C ==========
C
C SIDE - CHARACTER*1.
C On entry, SIDE specifies whether op( A ) appears on the left
C or right of X as follows:
C
C SIDE = 'L' or 'l' op( A )*X = alpha*B.
C
C SIDE = 'R' or 'r' X*op( A ) = alpha*B.
C
C Unchanged on exit.
C
C UPLO - CHARACTER*1.
C On entry, UPLO specifies whether the matrix A is an upper or
C lower triangular matrix as follows:
C
C UPLO = 'U' or 'u' A is an upper triangular matrix.
C
C UPLO = 'L' or 'l' A is a lower triangular matrix.
C
C Unchanged on exit.
C
C TRANSA - CHARACTER*1.
C On entry, TRANSA specifies the form of op( A ) to be used in
C the matrix multiplication as follows:
C
C TRANSA = 'N' or 'n' op( A ) = A.
C
C TRANSA = 'T' or 't' op( A ) = A'.
C
C TRANSA = 'C' or 'c' op( A ) = conjg( A' ).
C
C Unchanged on exit.
C
C DIAG - CHARACTER*1.
C On entry, DIAG specifies whether or not A is unit triangular
C as follows:
C
C DIAG = 'U' or 'u' A is assumed to be unit triangular.
C
C DIAG = 'N' or 'n' A is not assumed to be unit
C triangular.
C
C Unchanged on exit.
C
C M - INTEGER.
C On entry, M specifies the number of rows of B. M must be at
C least zero.
C Unchanged on exit.
C
C N - INTEGER.
C On entry, N specifies the number of columns of B. N must be
C at least zero.
C Unchanged on exit.
C
C ALPHA - COMPLEX .
C On entry, ALPHA specifies the scalar alpha. When alpha is
C zero then A is not referenced and B need not be set before
C entry.
C Unchanged on exit.
C
C A - COMPLEX array of DIMENSION ( LDA, k ), where k is m
C when SIDE = 'L' or 'l' and is n when SIDE = 'R' or 'r'.
C Before entry with UPLO = 'U' or 'u', the leading k by k
C upper triangular part of the array A must contain the upper
C triangular matrix and the strictly lower triangular part of
C A is not referenced.
C Before entry with UPLO = 'L' or 'l', the leading k by k
C lower triangular part of the array A must contain the lower
C triangular matrix and the strictly upper triangular part of
C A is not referenced.
C Note that when DIAG = 'U' or 'u', the diagonal elements of
C A are not referenced either, but are assumed to be unity.
C Unchanged on exit.
C
C LDA - INTEGER.
C On entry, LDA specifies the first dimension of A as declared
C in the calling (sub) program. When SIDE = 'L' or 'l' then
C LDA must be at least max( 1, m ), when SIDE = 'R' or 'r'
C then LDA must be at least max( 1, n ).
C Unchanged on exit.
C
C B - COMPLEX array of DIMENSION ( LDB, n ).
C Before entry, the leading m by n part of the array B must
C contain the right-hand side matrix B, and on exit is
C overwritten by the solution matrix X.
C
C LDB - INTEGER.
C On entry, LDB specifies the first dimension of B as declared
C in the calling (sub) program. LDB must be at least
C max( 1, m ).
C Unchanged on exit.
C
C
C Level 3 Blas routine.
C
C -- Written on 8-February-1989.
C Jack Dongarra, Argonne National Laboratory.
C Iain Duff, AERE Harwell.
C Jeremy Du Croz, Numerical Algorithms Group Ltd.
C Sven Hammarling, Numerical Algorithms Group Ltd.
C
C
SUBROUTINE F06ZPF(UPLO,TRANS,N,K,ALPHA,A,LDA,BETA,C,LDC)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C Purpose
C =======
C
C ZHERK performs one of the hermitian rank k operations
C
C C := alpha*A*conjg( A' ) + beta*C,
C
C or
C
C C := alpha*conjg( A' )*A + beta*C,
C
C where alpha and beta are real scalars, C is an n by n hermitian
C matrix and A is an n by k matrix in the first case and a k by n
C matrix in the second case.
C
C Parameters
C ==========
C
C UPLO - CHARACTER*1.
C On entry, UPLO specifies whether the upper or lower
C triangular part of the array C is to be referenced as
C follows:
C
C UPLO = 'U' or 'u' Only the upper triangular part of C
C is to be referenced.
C
C UPLO = 'L' or 'l' Only the lower triangular part of C
C is to be referenced.
C
C Unchanged on exit.
C
C TRANS - CHARACTER*1.
C On entry, TRANS specifies the operation to be performed as
C follows:
C
C TRANS = 'N' or 'n' C := alpha*A*conjg( A' ) + beta*C.
C
C TRANS = 'C' or 'c' C := alpha*conjg( A' )*A + beta*C.
C
C Unchanged on exit.
C
C N - INTEGER.
C On entry, N specifies the order of the matrix C. N must be
C at least zero.
C Unchanged on exit.
C
C K - INTEGER.
C On entry with TRANS = 'N' or 'n', K specifies the number
C of columns of the matrix A, and on entry with
C TRANS = 'C' or 'c', K specifies the number of rows of the
C matrix A. K must be at least zero.
C Unchanged on exit.
C
C ALPHA - REAL .
C On entry, ALPHA specifies the scalar alpha.
C Unchanged on exit.
C
C A - COMPLEX array of DIMENSION ( LDA, ka ), where ka is
C k when TRANS = 'N' or 'n', and is n otherwise.
C Before entry with TRANS = 'N' or 'n', the leading n by k
C part of the array A must contain the matrix A, otherwise
C the leading k by n part of the array A must contain the
C matrix A.
C Unchanged on exit.
C
C LDA - INTEGER.
C On entry, LDA specifies the first dimension of A as declared
C in the calling (sub) program. When TRANS = 'N' or 'n'
C then LDA must be at least max( 1, n ), otherwise LDA must
C be at least max( 1, k ).
C Unchanged on exit.
C
C BETA - REAL .
C On entry, BETA specifies the scalar beta.
C Unchanged on exit.
C
C C - COMPLEX array of DIMENSION ( LDC, n ).
C Before entry with UPLO = 'U' or 'u', the leading n by n
C upper triangular part of the array C must contain the upper
C triangular part of the hermitian matrix and the strictly
C lower triangular part of C is not referenced. On exit, the
C upper triangular part of the array C is overwritten by the
C upper triangular part of the updated matrix.
C Before entry with UPLO = 'L' or 'l', the leading n by n
C lower triangular part of the array C must contain the lower
C triangular part of the hermitian matrix and the strictly
C upper triangular part of C is not referenced. On exit, the
C lower triangular part of the array C is overwritten by the
C lower triangular part of the updated matrix.
C Note that the imaginary parts of the diagonal elements need
C not be set, they are assumed to be zero, and on exit they
C are set to zero.
C
C LDC - INTEGER.
C On entry, LDC specifies the first dimension of C as declared
C in the calling (sub) program. LDC must be at least
C max( 1, n ).
C Unchanged on exit.
C
C
C Level 3 Blas routine.
C
C -- Written on 8-February-1989.
C Jack Dongarra, Argonne National Laboratory.
C Iain Duff, AERE Harwell.
C Jeremy Du Croz, Numerical Algorithms Group Ltd.
C Sven Hammarling, Numerical Algorithms Group Ltd.
C
C
SUBROUTINE F06ZRF(UPLO,TRANS,N,K,ALPHA,A,LDA,B,LDB,BETA,C,LDC)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C Purpose
C =======
C
C ZHER2K performs one of the hermitian rank 2k operations
C
C C := alpha*A*conjg( B' ) + conjg( alpha )*B*conjg( A' ) + beta*C,
C
C or
C
C C := alpha*conjg( A' )*B + conjg( alpha )*conjg( B' )*A + beta*C,
C
C where alpha and beta are scalars with beta real, C is an n by n
C hermitian matrix and A and B are n by k matrices in the first case
C and k by n matrices in the second case.
C
C Parameters
C ==========
C
C UPLO - CHARACTER*1.
C On entry, UPLO specifies whether the upper or lower
C triangular part of the array C is to be referenced as
C follows:
C
C UPLO = 'U' or 'u' Only the upper triangular part of C
C is to be referenced.
C
C UPLO = 'L' or 'l' Only the lower triangular part of C
C is to be referenced.
C
C Unchanged on exit.
C
C TRANS - CHARACTER*1.
C On entry, TRANS specifies the operation to be performed as
C follows:
C
C TRANS = 'N' or 'n' C := alpha*A*conjg( B' ) +
C conjg( alpha )*B*conjg( A' ) +
C beta*C.
C
C TRANS = 'C' or 'c' C := alpha*conjg( A' )*B +
C conjg( alpha )*conjg( B' )*A +
C beta*C.
C
C Unchanged on exit.
C
C N - INTEGER.
C On entry, N specifies the order of the matrix C. N must be
C at least zero.
C Unchanged on exit.
C
C K - INTEGER.
C On entry with TRANS = 'N' or 'n', K specifies the number
C of columns of the matrices A and B, and on entry with
C TRANS = 'C' or 'c', K specifies the number of rows of the
C matrices A and B. K must be at least zero.
C Unchanged on exit.
C
C ALPHA - COMPLEX .
C On entry, ALPHA specifies the scalar alpha.
C Unchanged on exit.
C
C A - COMPLEX array of DIMENSION ( LDA, ka ), where ka is
C k when TRANS = 'N' or 'n', and is n otherwise.
C Before entry with TRANS = 'N' or 'n', the leading n by k
C part of the array A must contain the matrix A, otherwise
C the leading k by n part of the array A must contain the
C matrix A.
C Unchanged on exit.
C
C LDA - INTEGER.
C On entry, LDA specifies the first dimension of A as declared
C in the calling (sub) program. When TRANS = 'N' or 'n'
C then LDA must be at least max( 1, n ), otherwise LDA must
C be at least max( 1, k ).
C Unchanged on exit.
C
C B - COMPLEX array of DIMENSION ( LDB, kb ), where kb is
C k when TRANS = 'N' or 'n', and is n otherwise.
C Before entry with TRANS = 'N' or 'n', the leading n by k
C part of the array B must contain the matrix B, otherwise
C the leading k by n part of the array B must contain the
C matrix B.
C Unchanged on exit.
C
C LDB - INTEGER.
C On entry, LDB specifies the first dimension of B as declared
C in the calling (sub) program. When TRANS = 'N' or 'n'
C then LDB must be at least max( 1, n ), otherwise LDB must
C be at least max( 1, k ).
C Unchanged on exit.
C
C BETA - REAL .
C On entry, BETA specifies the scalar beta.
C Unchanged on exit.
C
C C - COMPLEX array of DIMENSION ( LDC, n ).
C Before entry with UPLO = 'U' or 'u', the leading n by n
C upper triangular part of the array C must contain the upper
C triangular part of the hermitian matrix and the strictly
C lower triangular part of C is not referenced. On exit, the
C upper triangular part of the array C is overwritten by the
C upper triangular part of the updated matrix.
C Before entry with UPLO = 'L' or 'l', the leading n by n
C lower triangular part of the array C must contain the lower
C triangular part of the hermitian matrix and the strictly
C upper triangular part of C is not referenced. On exit, the
C lower triangular part of the array C is overwritten by the
C lower triangular part of the updated matrix.
C Note that the imaginary parts of the diagonal elements need
C not be set, they are assumed to be zero, and on exit they
C are set to zero.
C
C LDC - INTEGER.
C On entry, LDC specifies the first dimension of C as declared
C in the calling (sub) program. LDC must be at least
C max( 1, n ).
C Unchanged on exit.
C
C
C Level 3 Blas routine.
C
C -- Written on 8-February-1989.
C Jack Dongarra, Argonne National Laboratory.
C Iain Duff, AERE Harwell.
C Jeremy Du Croz, Numerical Algorithms Group Ltd.
C Sven Hammarling, Numerical Algorithms Group Ltd.
C
C
SUBROUTINE F06ZTF(SIDE,UPLO,M,N,ALPHA,A,LDA,B,LDB,BETA,C,LDC)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C Purpose
C =======
C
C ZSYMM performs one of the matrix-matrix operations
C
C C := alpha*A*B + beta*C,
C
C or
C
C C := alpha*B*A + beta*C,
C
C where alpha and beta are scalars, A is a symmetric matrix and B and
C C are m by n matrices.
C
C Parameters
C ==========
C
C SIDE - CHARACTER*1.
C On entry, SIDE specifies whether the symmetric matrix A
C appears on the left or right in the operation as follows:
C
C SIDE = 'L' or 'l' C := alpha*A*B + beta*C,
C
C SIDE = 'R' or 'r' C := alpha*B*A + beta*C,
C
C Unchanged on exit.
C
C UPLO - CHARACTER*1.
C On entry, UPLO specifies whether the upper or lower
C triangular part of the symmetric matrix A is to be
C referenced as follows:
C
C UPLO = 'U' or 'u' Only the upper triangular part of the
C symmetric matrix is to be referenced.
C
C UPLO = 'L' or 'l' Only the lower triangular part of the
C symmetric matrix is to be referenced.
C
C Unchanged on exit.
C
C M - INTEGER.
C On entry, M specifies the number of rows of the matrix C.
C M must be at least zero.
C Unchanged on exit.
C
C N - INTEGER.
C On entry, N specifies the number of columns of the matrix C.
C N must be at least zero.
C Unchanged on exit.
C
C ALPHA - COMPLEX .
C On entry, ALPHA specifies the scalar alpha.
C Unchanged on exit.
C
C A - COMPLEX array of DIMENSION ( LDA, ka ), where ka is
C m when SIDE = 'L' or 'l' and is n otherwise.
C Before entry with SIDE = 'L' or 'l', the m by m part of
C the array A must contain the symmetric matrix, such that
C when UPLO = 'U' or 'u', the leading m by m upper triangular
C part of the array A must contain the upper triangular part
C of the symmetric matrix and the strictly lower triangular
C part of A is not referenced, and when UPLO = 'L' or 'l',
C the leading m by m lower triangular part of the array A
C must contain the lower triangular part of the symmetric
C matrix and the strictly upper triangular part of A is not
C referenced.
C Before entry with SIDE = 'R' or 'r', the n by n part of
C the array A must contain the symmetric matrix, such that
C when UPLO = 'U' or 'u', the leading n by n upper triangular
C part of the array A must contain the upper triangular part
C of the symmetric matrix and the strictly lower triangular
C part of A is not referenced, and when UPLO = 'L' or 'l',
C the leading n by n lower triangular part of the array A
C must contain the lower triangular part of the symmetric
C matrix and the strictly upper triangular part of A is not
C referenced.
C Unchanged on exit.
C
C LDA - INTEGER.
C On entry, LDA specifies the first dimension of A as declared
C in the calling (sub) program. When SIDE = 'L' or 'l' then
C LDA must be at least max( 1, m ), otherwise LDA must be at
C least max( 1, n ).
C Unchanged on exit.
C
C B - COMPLEX array of DIMENSION ( LDB, n ).
C Before entry, the leading m by n part of the array B must
C contain the matrix B.
C Unchanged on exit.
C
C LDB - INTEGER.
C On entry, LDB specifies the first dimension of B as declared
C in the calling (sub) program. LDB must be at least
C max( 1, m ).
C Unchanged on exit.
C
C BETA - COMPLEX .
C On entry, BETA specifies the scalar beta. When BETA is
C supplied as zero then C need not be set on input.
C Unchanged on exit.
C
C C - COMPLEX array of DIMENSION ( LDC, n ).
C Before entry, the leading m by n part of the array C must
C contain the matrix C, except when beta is zero, in which
C case C need not be set on entry.
C On exit, the array C is overwritten by the m by n updated
C matrix.
C
C LDC - INTEGER.
C On entry, LDC specifies the first dimension of C as declared
C in the calling (sub) program. LDC must be at least
C max( 1, m ).
C Unchanged on exit.
C
C
C Level 3 Blas routine.
C
C -- Written on 8-February-1989.
C Jack Dongarra, Argonne National Laboratory.
C Iain Duff, AERE Harwell.
C Jeremy Du Croz, Numerical Algorithms Group Ltd.
C Sven Hammarling, Numerical Algorithms Group Ltd.
C
C
SUBROUTINE F06ZUF(UPLO,TRANS,N,K,ALPHA,A,LDA,BETA,C,LDC)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C Purpose
C =======
C
C ZSYRK performs one of the symmetric rank k operations
C
C C := alpha*A*A' + beta*C,
C
C or
C
C C := alpha*A'*A + beta*C,
C
C where alpha and beta are scalars, C is an n by n symmetric matrix
C and A is an n by k matrix in the first case and a k by n matrix
C in the second case.
C
C Parameters
C ==========
C
C UPLO - CHARACTER*1.
C On entry, UPLO specifies whether the upper or lower
C triangular part of the array C is to be referenced as
C follows:
C
C UPLO = 'U' or 'u' Only the upper triangular part of C
C is to be referenced.
C
C UPLO = 'L' or 'l' Only the lower triangular part of C
C is to be referenced.
C
C Unchanged on exit.
C
C TRANS - CHARACTER*1.
C On entry, TRANS specifies the operation to be performed as
C follows:
C
C TRANS = 'N' or 'n' C := alpha*A*A' + beta*C.
C
C TRANS = 'T' or 't' C := alpha*A'*A + beta*C.
C
C Unchanged on exit.
C
C N - INTEGER.
C On entry, N specifies the order of the matrix C. N must be
C at least zero.
C Unchanged on exit.
C
C K - INTEGER.
C On entry with TRANS = 'N' or 'n', K specifies the number
C of columns of the matrix A, and on entry with
C TRANS = 'T' or 't', K specifies the number of rows of the
C matrix A. K must be at least zero.
C Unchanged on exit.
C
C ALPHA - COMPLEX .
C On entry, ALPHA specifies the scalar alpha.
C Unchanged on exit.
C
C A - COMPLEX array of DIMENSION ( LDA, ka ), where ka is
C k when TRANS = 'N' or 'n', and is n otherwise.
C Before entry with TRANS = 'N' or 'n', the leading n by k
C part of the array A must contain the matrix A, otherwise
C the leading k by n part of the array A must contain the
C matrix A.
C Unchanged on exit.
C
C LDA - INTEGER.
C On entry, LDA specifies the first dimension of A as declared
C in the calling (sub) program. When TRANS = 'N' or 'n'
C then LDA must be at least max( 1, n ), otherwise LDA must
C be at least max( 1, k ).
C Unchanged on exit.
C
C BETA - COMPLEX .
C On entry, BETA specifies the scalar beta.
C Unchanged on exit.
C
C C - COMPLEX array of DIMENSION ( LDC, n ).
C Before entry with UPLO = 'U' or 'u', the leading n by n
C upper triangular part of the array C must contain the upper
C triangular part of the symmetric matrix and the strictly
C lower triangular part of C is not referenced. On exit, the
C upper triangular part of the array C is overwritten by the
C upper triangular part of the updated matrix.
C Before entry with UPLO = 'L' or 'l', the leading n by n
C lower triangular part of the array C must contain the lower
C triangular part of the symmetric matrix and the strictly
C upper triangular part of C is not referenced. On exit, the
C lower triangular part of the array C is overwritten by the
C lower triangular part of the updated matrix.
C
C LDC - INTEGER.
C On entry, LDC specifies the first dimension of C as declared
C in the calling (sub) program. LDC must be at least
C max( 1, n ).
C Unchanged on exit.
C
C
C Level 3 Blas routine.
C
C -- Written on 8-February-1989.
C Jack Dongarra, Argonne National Laboratory.
C Iain Duff, AERE Harwell.
C Jeremy Du Croz, Numerical Algorithms Group Ltd.
C Sven Hammarling, Numerical Algorithms Group Ltd.
C
C
SUBROUTINE F06ZWF(UPLO,TRANS,N,K,ALPHA,A,LDA,B,LDB,BETA,C,LDC)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C Purpose
C =======
C
C ZSYR2K performs one of the symmetric rank 2k operations
C
C C := alpha*A*B' + alpha*B*A' + beta*C,
C
C or
C
C C := alpha*A'*B + alpha*B'*A + beta*C,
C
C where alpha and beta are scalars, C is an n by n symmetric matrix
C and A and B are n by k matrices in the first case and k by n
C matrices in the second case.
C
C Parameters
C ==========
C
C UPLO - CHARACTER*1.
C On entry, UPLO specifies whether the upper or lower
C triangular part of the array C is to be referenced as
C follows:
C
C UPLO = 'U' or 'u' Only the upper triangular part of C
C is to be referenced.
C
C UPLO = 'L' or 'l' Only the lower triangular part of C
C is to be referenced.
C
C Unchanged on exit.
C
C TRANS - CHARACTER*1.
C On entry, TRANS specifies the operation to be performed as
C follows:
C
C TRANS = 'N' or 'n' C := alpha*A*B' + alpha*B*A' +
C beta*C.
C
C TRANS = 'T' or 't' C := alpha*A'*B + alpha*B'*A +
C beta*C.
C
C Unchanged on exit.
C
C N - INTEGER.
C On entry, N specifies the order of the matrix C. N must be
C at least zero.
C Unchanged on exit.
C
C K - INTEGER.
C On entry with TRANS = 'N' or 'n', K specifies the number
C of columns of the matrices A and B, and on entry with
C TRANS = 'T' or 't', K specifies the number of rows of the
C matrices A and B. K must be at least zero.
C Unchanged on exit.
C
C ALPHA - COMPLEX .
C On entry, ALPHA specifies the scalar alpha.
C Unchanged on exit.
C
C A - COMPLEX array of DIMENSION ( LDA, ka ), where ka is
C k when TRANS = 'N' or 'n', and is n otherwise.
C Before entry with TRANS = 'N' or 'n', the leading n by k
C part of the array A must contain the matrix A, otherwise
C the leading k by n part of the array A must contain the
C matrix A.
C Unchanged on exit.
C
C LDA - INTEGER.
C On entry, LDA specifies the first dimension of A as declared
C in the calling (sub) program. When TRANS = 'N' or 'n'
C then LDA must be at least max( 1, n ), otherwise LDA must
C be at least max( 1, k ).
C Unchanged on exit.
C
C B - COMPLEX array of DIMENSION ( LDB, kb ), where kb is
C k when TRANS = 'N' or 'n', and is n otherwise.
C Before entry with TRANS = 'N' or 'n', the leading n by k
C part of the array B must contain the matrix B, otherwise
C the leading k by n part of the array B must contain the
C matrix B.
C Unchanged on exit.
C
C LDB - INTEGER.
C On entry, LDB specifies the first dimension of B as declared
C in the calling (sub) program. When TRANS = 'N' or 'n'
C then LDB must be at least max( 1, n ), otherwise LDB must
C be at least max( 1, k ).
C Unchanged on exit.
C
C BETA - COMPLEX .
C On entry, BETA specifies the scalar beta.
C Unchanged on exit.
C
C C - COMPLEX array of DIMENSION ( LDC, n ).
C Before entry with UPLO = 'U' or 'u', the leading n by n
C upper triangular part of the array C must contain the upper
C triangular part of the symmetric matrix and the strictly
C lower triangular part of C is not referenced. On exit, the
C upper triangular part of the array C is overwritten by the
C upper triangular part of the updated matrix.
C Before entry with UPLO = 'L' or 'l', the leading n by n
C lower triangular part of the array C must contain the lower
C triangular part of the symmetric matrix and the strictly
C upper triangular part of C is not referenced. On exit, the
C lower triangular part of the array C is overwritten by the
C lower triangular part of the updated matrix.
C
C LDC - INTEGER.
C On entry, LDC specifies the first dimension of C as declared
C in the calling (sub) program. LDC must be at least
C max( 1, n ).
C Unchanged on exit.
C
C
C Level 3 Blas routine.
C
C -- Written on 8-February-1989.
C Jack Dongarra, Argonne National Laboratory.
C Iain Duff, AERE Harwell.
C Jeremy Du Croz, Numerical Algorithms Group Ltd.
C Sven Hammarling, Numerical Algorithms Group Ltd.
C
C
SUBROUTINE F07ADF(M,N,A,LDA,IPIV,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C MARK 16 REVISED. IER-1009 (JUN 1993).
SUBROUTINE F07AEF(TRANS,N,NRHS,A,LDA,IPIV,B,LDB,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C MARK 16 REVISED. IER-1013 (JUN 1993).
SUBROUTINE F07AGF(NORM,N,A,LDA,ANORM,RCOND,WORK,IWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07AHF(TRANS,N,NRHS,A,LDA,AF,LDAF,IPIV,B,LDB,X,LDX,
* FERR,BERR,WORK,IWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07AJF(N,A,LDA,IPIV,WORK,LWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07ARF(M,N,A,LDA,IPIV,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C MARK 16 REVISED. IER-1015 (JUN 1993).
C
SUBROUTINE F07ASF(TRANS,N,NRHS,A,LDA,IPIV,B,LDB,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C MARK 16 REVISED. IER-1019 (JUN 1993).
SUBROUTINE F07AUF(NORM,N,A,LDA,ANORM,RCOND,WORK,RWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07AVF(TRANS,N,NRHS,A,LDA,AF,LDAF,IPIV,B,LDB,X,LDX,
* FERR,BERR,WORK,RWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07AWF(N,A,LDA,IPIV,WORK,LWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07BDF(M,N,KL,KU,AB,LDAB,IPIV,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07BEF(TRANS,N,KL,KU,NRHS,AB,LDAB,IPIV,B,LDB,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07BGF(NORM,N,KL,KU,AB,LDAB,IPIV,ANORM,RCOND,WORK,
* IWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07BHF(TRANS,N,KL,KU,NRHS,AB,LDAB,AFB,LDAFB,IPIV,B,LDB,
* X,LDX,FERR,BERR,WORK,IWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07BRF(M,N,KL,KU,AB,LDAB,IPIV,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07BSF(TRANS,N,KL,KU,NRHS,AB,LDAB,IPIV,B,LDB,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07BUF(NORM,N,KL,KU,AB,LDAB,IPIV,ANORM,RCOND,WORK,
* RWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07BVF(TRANS,N,KL,KU,NRHS,AB,LDAB,AFB,LDAFB,IPIV,B,LDB,
* X,LDX,FERR,BERR,WORK,RWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07FDF(UPLO,N,A,LDA,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07FEF(UPLO,N,NRHS,A,LDA,B,LDB,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C MARK 16 REVISED. IER-1021 (JUN 1993).
SUBROUTINE F07FGF(UPLO,N,A,LDA,ANORM,RCOND,WORK,IWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07FHF(UPLO,N,NRHS,A,LDA,AF,LDAF,B,LDB,X,LDX,FERR,BERR,
* WORK,IWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07FJF(UPLO,N,A,LDA,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07FRF(UPLO,N,A,LDA,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07FSF(UPLO,N,NRHS,A,LDA,B,LDB,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C MARK 16 REVISED. IER-1022 (JUN 1993).
SUBROUTINE F07FUF(UPLO,N,A,LDA,ANORM,RCOND,WORK,RWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07FVF(UPLO,N,NRHS,A,LDA,AF,LDAF,B,LDB,X,LDX,FERR,BERR,
* WORK,RWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07FWF(UPLO,N,A,LDA,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07GDF(UPLO,N,AP,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07GEF(UPLO,N,NRHS,AP,B,LDB,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07GGF(UPLO,N,AP,ANORM,RCOND,WORK,IWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07GHF(UPLO,N,NRHS,AP,AFP,B,LDB,X,LDX,FERR,BERR,WORK,
* IWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07GJF(UPLO,N,AP,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07GRF(UPLO,N,AP,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07GSF(UPLO,N,NRHS,AP,B,LDB,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07GUF(UPLO,N,AP,ANORM,RCOND,WORK,RWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07GVF(UPLO,N,NRHS,AP,AFP,B,LDB,X,LDX,FERR,BERR,WORK,
* RWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07GWF(UPLO,N,AP,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07HDF(UPLO,N,KD,AB,LDAB,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07HEF(UPLO,N,KD,NRHS,AB,LDAB,B,LDB,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07HGF(UPLO,N,KD,AB,LDAB,ANORM,RCOND,WORK,IWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07HHF(UPLO,N,KD,NRHS,AB,LDAB,AFB,LDAFB,B,LDB,X,LDX,
* FERR,BERR,WORK,IWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07HRF(UPLO,N,KD,AB,LDAB,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07HSF(UPLO,N,KD,NRHS,AB,LDAB,B,LDB,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07HUF(UPLO,N,KD,AB,LDAB,ANORM,RCOND,WORK,RWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07HVF(UPLO,N,KD,NRHS,AB,LDAB,AFB,LDAFB,B,LDB,X,LDX,
* FERR,BERR,WORK,RWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07MDF(UPLO,N,A,LDA,IPIV,WORK,LWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07MEF(UPLO,N,NRHS,A,LDA,IPIV,B,LDB,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07MGF(UPLO,N,A,LDA,IPIV,ANORM,RCOND,WORK,IWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07MHF(UPLO,N,NRHS,A,LDA,AF,LDAF,IPIV,B,LDB,X,LDX,FERR,
* BERR,WORK,IWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07MJF(UPLO,N,A,LDA,IPIV,WORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07MRF(UPLO,N,A,LDA,IPIV,WORK,LWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07MSF(UPLO,N,NRHS,A,LDA,IPIV,B,LDB,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07MUF(UPLO,N,A,LDA,IPIV,ANORM,RCOND,WORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07MVF(UPLO,N,NRHS,A,LDA,AF,LDAF,IPIV,B,LDB,X,LDX,FERR,
* BERR,WORK,RWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07MWF(UPLO,N,A,LDA,IPIV,WORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07NRF(UPLO,N,A,LDA,IPIV,WORK,LWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07NSF(UPLO,N,NRHS,A,LDA,IPIV,B,LDB,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07NUF(UPLO,N,A,LDA,IPIV,ANORM,RCOND,WORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07NVF(UPLO,N,NRHS,A,LDA,AF,LDAF,IPIV,B,LDB,X,LDX,FERR,
* BERR,WORK,RWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07NWF(UPLO,N,A,LDA,IPIV,WORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07PDF(UPLO,N,AP,IPIV,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07PEF(UPLO,N,NRHS,AP,IPIV,B,LDB,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07PGF(UPLO,N,AP,IPIV,ANORM,RCOND,WORK,IWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07PHF(UPLO,N,NRHS,AP,AFP,IPIV,B,LDB,X,LDX,FERR,BERR,
* WORK,IWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07PJF(UPLO,N,AP,IPIV,WORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07PRF(UPLO,N,AP,IPIV,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07PSF(UPLO,N,NRHS,AP,IPIV,B,LDB,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07PUF(UPLO,N,AP,IPIV,ANORM,RCOND,WORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07PVF(UPLO,N,NRHS,AP,AFP,IPIV,B,LDB,X,LDX,FERR,BERR,
* WORK,RWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07PWF(UPLO,N,AP,IPIV,WORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07QRF(UPLO,N,AP,IPIV,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07QSF(UPLO,N,NRHS,AP,IPIV,B,LDB,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07QUF(UPLO,N,AP,IPIV,ANORM,RCOND,WORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07QVF(UPLO,N,NRHS,AP,AFP,IPIV,B,LDB,X,LDX,FERR,BERR,
* WORK,RWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07QWF(UPLO,N,AP,IPIV,WORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07TEF(UPLO,TRANS,DIAG,N,NRHS,A,LDA,B,LDB,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C MARK 16 REVISED. IER-1023 (JUN 1993).
SUBROUTINE F07TGF(NORM,UPLO,DIAG,N,A,LDA,RCOND,WORK,IWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07THF(UPLO,TRANS,DIAG,N,NRHS,A,LDA,B,LDB,X,LDX,FERR,
* BERR,WORK,IWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07TJF(UPLO,DIAG,N,A,LDA,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07TSF(UPLO,TRANS,DIAG,N,NRHS,A,LDA,B,LDB,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C MARK 16 REVISED. IER-1025 (JUN 1993).
SUBROUTINE F07TUF(NORM,UPLO,DIAG,N,A,LDA,RCOND,WORK,RWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07TVF(UPLO,TRANS,DIAG,N,NRHS,A,LDA,B,LDB,X,LDX,FERR,
* BERR,WORK,RWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07TWF(UPLO,DIAG,N,A,LDA,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07UEF(UPLO,TRANS,DIAG,N,NRHS,AP,B,LDB,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07UGF(NORM,UPLO,DIAG,N,AP,RCOND,WORK,IWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07UHF(UPLO,TRANS,DIAG,N,NRHS,AP,B,LDB,X,LDX,FERR,BERR,
* WORK,IWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07UJF(UPLO,DIAG,N,AP,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07USF(UPLO,TRANS,DIAG,N,NRHS,AP,B,LDB,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07UUF(NORM,UPLO,DIAG,N,AP,RCOND,WORK,RWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07UVF(UPLO,TRANS,DIAG,N,NRHS,AP,B,LDB,X,LDX,FERR,BERR,
* WORK,RWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07UWF(UPLO,DIAG,N,AP,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07VEF(UPLO,TRANS,DIAG,N,KD,NRHS,AB,LDAB,B,LDB,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07VGF(NORM,UPLO,DIAG,N,KD,AB,LDAB,RCOND,WORK,IWORK,
* INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07VHF(UPLO,TRANS,DIAG,N,KD,NRHS,AB,LDAB,B,LDB,X,LDX,
* FERR,BERR,WORK,IWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07VSF(UPLO,TRANS,DIAG,N,KD,NRHS,AB,LDAB,B,LDB,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07VUF(NORM,UPLO,DIAG,N,KD,AB,LDAB,RCOND,WORK,RWORK,
* INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F07VVF(UPLO,TRANS,DIAG,N,KD,NRHS,AB,LDAB,B,LDB,X,LDX,
* FERR,BERR,WORK,RWORK,INFO)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
SUBROUTINE F08AEF(M,N,A,LDA,TAU,WORK,LWORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08AFF(M,N,K,A,LDA,TAU,WORK,LWORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08AGF(SIDE,TRANS,M,N,K,A,LDA,TAU,C,LDC,WORK,LWORK,
* INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08AHF(M,N,A,LDA,TAU,WORK,LWORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08AJF(M,N,K,A,LDA,TAU,WORK,LWORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08AKF(SIDE,TRANS,M,N,K,A,LDA,TAU,C,LDC,WORK,LWORK,
* INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08ASF(M,N,A,LDA,TAU,WORK,LWORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08ATF(M,N,K,A,LDA,TAU,WORK,LWORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08AUF(SIDE,TRANS,M,N,K,A,LDA,TAU,C,LDC,WORK,LWORK,
* INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08AVF(M,N,A,LDA,TAU,WORK,LWORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08AWF(M,N,K,A,LDA,TAU,WORK,LWORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08AXF(SIDE,TRANS,M,N,K,A,LDA,TAU,C,LDC,WORK,LWORK,
* INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08BEF(M,N,A,LDA,JPVT,TAU,WORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08BSF(M,N,A,LDA,JPVT,TAU,WORK,RWORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08FEF(UPLO,N,A,LDA,D,E,TAU,WORK,LWORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08FFF(UPLO,N,A,LDA,TAU,WORK,LWORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08FGF(SIDE,UPLO,TRANS,M,N,A,LDA,TAU,C,LDC,WORK,LWORK,
* INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08FSF(UPLO,N,A,LDA,D,E,TAU,WORK,LWORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08FTF(UPLO,N,A,LDA,TAU,WORK,LWORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08FUF(SIDE,UPLO,TRANS,M,N,A,LDA,TAU,C,LDC,WORK,LWORK,
* INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08GEF(UPLO,N,AP,D,E,TAU,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08GFF(UPLO,N,AP,TAU,Q,LDQ,WORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08GGF(SIDE,UPLO,TRANS,M,N,AP,TAU,C,LDC,WORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08GSF(UPLO,N,AP,D,E,TAU,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08GTF(UPLO,N,AP,TAU,Q,LDQ,WORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08GUF(SIDE,UPLO,TRANS,M,N,AP,TAU,C,LDC,WORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08HEF(VECT,UPLO,N,KD,AB,LDAB,D,E,Q,LDQ,WORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08HSF(VECT,UPLO,N,KD,AB,LDAB,D,E,Q,LDQ,WORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08JEF(COMPZ,N,D,E,Z,LDZ,WORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08JFF(N,D,E,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08JGF(COMPZ,N,D,E,Z,LDZ,WORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08JJF(RANGE,ORDER,N,VL,VU,IL,IU,ABSTOL,D,E,M,NSPLIT,W,
* IBLOCK,ISPLIT,WORK,IWORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08JKF(N,D,E,M,W,IBLOCK,ISPLIT,Z,LDZ,WORK,IWORK,IFAIL,
* INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08JSF(COMPZ,N,D,E,Z,LDZ,WORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08JUF(COMPZ,N,D,E,Z,LDZ,WORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08JXF(N,D,E,M,W,IBLOCK,ISPLIT,Z,LDZ,WORK,IWORK,IFAIL,
* INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08KEF(M,N,A,LDA,D,E,TAUQ,TAUP,WORK,LWORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08KFF(VECT,M,N,K,A,LDA,TAU,WORK,LWORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08KGF(VECT,SIDE,TRANS,M,N,K,A,LDA,TAU,C,LDC,WORK,
* LWORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08KSF(M,N,A,LDA,D,E,TAUQ,TAUP,WORK,LWORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08KTF(VECT,M,N,K,A,LDA,TAU,WORK,LWORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08KUF(VECT,SIDE,TRANS,M,N,K,A,LDA,TAU,C,LDC,WORK,
* LWORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08MEF(UPLO,N,NCVT,NRU,NCC,D,E,VT,LDVT,U,LDU,C,LDC,
* WORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08MSF(UPLO,N,NCVT,NRU,NCC,D,E,VT,LDVT,U,LDU,C,LDC,
* WORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08NEF(N,ILO,IHI,A,LDA,TAU,WORK,LWORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08NFF(N,ILO,IHI,A,LDA,TAU,WORK,LWORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08NGF(SIDE,TRANS,M,N,ILO,IHI,A,LDA,TAU,C,LDC,WORK,
* LWORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08NHF(JOB,N,A,LDA,ILO,IHI,SCALE,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08NJF(JOB,SIDE,N,ILO,IHI,SCALE,M,V,LDV,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08NSF(N,ILO,IHI,A,LDA,TAU,WORK,LWORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08NTF(N,ILO,IHI,A,LDA,TAU,WORK,LWORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08NUF(SIDE,TRANS,M,N,ILO,IHI,A,LDA,TAU,C,LDC,WORK,
* LWORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08NVF(JOB,N,A,LDA,ILO,IHI,SCALE,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08NWF(JOB,SIDE,N,ILO,IHI,SCALE,M,V,LDV,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08PEF(JOB,COMPZ,N,ILO,IHI,H,LDH,WR,WI,Z,LDZ,WORK,
* LWORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08PKF(JOB,EIGSRC,INITV,SELECT,N,H,LDH,WR,WI,VL,LDVL,
* VR,LDVR,MM,M,WORK,IFAILL,IFAILR,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08PSF(JOB,COMPZ,N,ILO,IHI,H,LDH,W,Z,LDZ,WORK,LWORK,
* INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08PXF(JOB,EIGSRC,INITV,SELECT,N,H,LDH,W,VL,LDVL,VR,
* LDVR,MM,M,WORK,RWORK,IFAILL,IFAILR,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08QFF(COMPQ,N,T,LDT,Q,LDQ,IFST,ILST,WORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08QGF(JOB,COMPQ,SELECT,N,T,LDT,Q,LDQ,WR,WI,M,S,SEP,
* WORK,LWORK,IWORK,LIWORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08QHF(TRANA,TRANB,ISGN,M,N,A,LDA,B,LDB,C,LDC,SCALE,
* INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08QKF(JOB,HOWMNY,SELECT,N,T,LDT,VL,LDVL,VR,LDVR,MM,M,
* WORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08QLF(JOB,HOWMNY,SELECT,N,T,LDT,VL,LDVL,VR,LDVR,S,SEP,
* MM,M,WORK,LDWORK,IWORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08QTF(COMPQ,N,T,LDT,Q,LDQ,IFST,ILST,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08QUF(JOB,COMPQ,SELECT,N,T,LDT,Q,LDQ,W,M,S,SEP,WORK,
* LWORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08QVF(TRANA,TRANB,ISGN,M,N,A,LDA,B,LDB,C,LDC,SCALE,
* INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08QXF(JOB,HOWMNY,SELECT,N,T,LDT,VL,LDVL,VR,LDVR,MM,M,
* WORK,RWORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08QYF(JOB,HOWMNY,SELECT,N,T,LDT,VL,LDVL,VR,LDVR,S,SEP,
* MM,M,WORK,LDWORK,RWORK,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08SEF(ITYPE,UPLO,N,A,LDA,B,LDB,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08SSF(ITYPE,UPLO,N,A,LDA,B,LDB,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08TEF(ITYPE,UPLO,N,AP,BP,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE F08TSF(ITYPE,UPLO,N,AP,BP,INFO)
C MARK 16 RELEASE. NAG COPYRIGHT 1992.
SUBROUTINE G01AAF(N,X,IWT,WT,XMEAN,S2,S3,S4,XMIN,XMAX,WSUM,IFAIL)
C MARK 4 RELEASE NAG COPYRIGHT 1974.
C MARK 4.5 REVISED
C MARK 10D REVISED. IER-430 (FEB 1984)
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C
C G01AAF - THE ROUTINE RETURNS THE MEAN, THE STANDARD
C DEVIATION, THE COEFFICIENTS OF SKEWNESS AND
C KURTOSIS, AND THE LARGEST AND SMALLEST VALUES
C FROM THE VALUES IN THE ARRAY X - OPTIONALLY
C USING WEIGHTS IN THE ARRAY WT
C
C USES NAG LIBRARY ROUTINE P01AAF
C
SUBROUTINE G01ABF(N,X1,X2,IWT,WT,RES,IFAIL)
C MARK 4 RELEASE NAG COPYRIGHT 1974.
C MARK 4.5 REVISED
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C G01ABF - THE ROUTINE COMPUTES MEANS, STD DEVNS, MINIMA,
C MAXIMA, CORRECTED SUMS OF SQUARES AND PRODUCTS,
C AND THE PRODUCT MOMENT CORRELATION COEFFICIENT
C FOR TWO VARIABLES IN THE ARRAYS X1 AND X2 -
C OPTIONALLY USING WEIGHTS IN THE ARRAY WT
C
C USES NAG LIBRARY ROUTINE P01AAF
C
SUBROUTINE G01ADF(K,X,IFREQ,XMEAN,S2,S3,S4,N,IFAIL)
C MARK 4 RELEASE NAG COPYRIGHT 1974.
C MARK 4.5 REVISED
C MARK 5C REVISED IER-82
C MARK 9 REVISED. IER-331 (SEP 1981).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. IER-619 (APR 1988).
C
C G01ADF - THE ROUTINE COMPUTES MEAN, STD DEVN, AND COEFFTS
C OF SKEWNESS AND KURTOSIS FOR GROUPED DATA
C
C USES NAG LIBRARY ROUTINE P01AAF
C
SUBROUTINE G01AEF(N,K2,X,ICLASS,CINT,IFREQ,XMIN,XMAX,IFAIL)
C MARK 4 RELEASE NAG COPYRIGHT 1974.
C MARK 4.5 REVISED IER-49/42
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 12A REVISED. IER-510 (AUG 1986).
C
C G01AEF - THE ROUTINE CONSTRUCTS A FREQUENCY DISTRIBUTION
C FOR VALUES IN AN ARRAY X WITH EITHER ROUTINE
C CALCULATED OR USER SUPPLIED CLASS INTERVALS
C
C USES NAG LIBRARY ROUTINE P01AAF
C
SUBROUTINE G01AFF(INOB,IPRED,M,N,NOBS,NUM,PRED,CHIS,P,NPOS,NDF,M1,
* N1,IFAIL)
C MARK 4 RELEASE NAG COPYRIGHT 1974.
C MARK 4.5 REVISED LER-14B
C MARK 6B REVISED IER-120 (MAR 1978)
C MARK 8E REVISED. IER-278 (JAN 1981).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C G01AFF - THE ROUTINE PERFORMS AN ANALYSIS OF A TWO-WAY
C CONTINGENCY TABLE. YATES CORRECTION FOR 2*2 TABLE
C OR FISHERS EXACT TEST WHEN TOTAL FREQUENCY LE MINTAB
C
C USES NAG LIBRARY ROUTINES P01AAF, S14AAF
C
SUBROUTINE G01AGF(X,Y,NOBS,ISORT,NSTEPX,NSTEPY,IFAIL)
C MARK 14 RE-ISSUE. NAG COPYRIGHT 1989.
C G01AGF PLOTS TWO ARRAYS AGAINST ONE ANOTHER
C ON A CHARACTER PRINTING DEVICE WITH A CHOSEN NUMBER
C OF CHARACTER POSITIONS IN EACH DIRECTION
SUBROUTINE G01AHF(X,NOBS,NSTEPX,NSTEPY,ISTAND,IWORK,WORK,LWORK,
* XSORT,XBAR,XSTD,IFAIL)
C MARK 14 RE-ISSUE. NAG COPYRIGHT 1989.
C G01AHF PLOTS AN ARRAY OF RESIDUALS AGAINST THE NORMAL SCORES
C FOR A SAMPLE OF THE SAME SIZE, ON A CHARACTER PRINTING DEVICE
C WITH A CHOSEN NUMBER OF CHARACTER POSITIONS IN EACH DIRECTION
SUBROUTINE G01AJF(X,N,NSTEPX,NSTEPY,ITYPE,ISPACE,XMIN,XMAX,XSTEP,
* N1,MULTY,IFAIL)
C MARK 10 RELEASE. NAG COPYRIGHT 1982
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C G01AJF PRINTS A HISTOGRAM OF A VECTOR OF REAL DATA,
C WITH A CHOSEN NUMBER OF CHARACTER POSITIONS IN EACH DIRECTION.
C
SUBROUTINE G01ALF(N,X,IWRK,RES,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
SUBROUTINE G01ARF(RANGE,PRT,N,Y,NSTEPX,NSTEPY,UNIT,PLOT,LDP,LINES,
* SORTY,IWORK,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C Produce a Stem-and-Leaf display of the data in Y()
C
C IWORK() is an integer work array. SORTY() is a real work array.
C
C UNIT is the leaf unit used. Is converted to a power of ten.
C NSTEPX is the maximum width of the plot horizontally
C NSTEPY is the maximum number of lines to feature the display.
C LDP is the leading dimension of the plotting array.
C EPSI is the machine-related epsilon.
C IMAXIN is the maximum permitted integer value
C
SUBROUTINE G01ASF(PRT,M,N,X,LDX,NSTEPX,NSTEPY,PLOT,LDP,WORK,IWORK,
* IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
SUBROUTINE G01BJF(N,P,K,PLEK,PGTK,PEQK,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C
C Returns the lower tail, upper tail and point probabilities
C associated with a Binomial distribution.
C
C Let X denote a random variable having a binomial distribution with
C parameters N and P. The routine computes for given N, P, K:
C
C PLEK = Prob (X .LE. K)
C PGTK = Prob (X .GT. K)
C PEQK = Prob (X .EQ. K)
C
C Reference: L. Knusel, Computation of the Chi-square and Poisson
C distribution, SIAM J. Sci. Statist. Comput. 7, pp 1022-1036, 1986.
C
SUBROUTINE G01BKF(RLAMDA,K,PLEK,PGTK,PEQK,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C
C Returns the lower tail, upper tail and point probabilities
C associated with a Poisson distribution.
C
C Let Z denote a random variable having a Poisson distribution with
C parameters N and P. The routine computes for given RLAMDA and K:
C
C PLEK = Prob (Z .LE. K)
C PGTK = Prob (Z .GT. K)
C PEQK = Prob (Z .EQ. K)
C
C Reference: L. Knusel, Computation of the Chi-square and Poisson
C distribution, SIAM J. Sci. Statist. Comput. 7, pp 1022-1036, 1986.
C
SUBROUTINE G01BLF(N,L,M,K,PLEK,PGTK,PEQK,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C
C Returns the lower tail, upper tail and point probabilities
C associated with a hypergeometric distribution.
C
C Let X denote a random variable having a hypergeometric
C distribution with parameters N, L and M. The routine computes for
C given N, L, M and K:
C
C PLEK = Prob (X .LE. K)
C PGTK = Prob (X .GT. K)
C PEQK = Prob (X .EQ. K)
C
C Reference: L. Knusel, Computation of the Chisquare and Poisson
C distribution, SIAM J. Sci. Stat. Comput. 7, pp 1022-1036, 1986.
C
SUBROUTINE G01DAF(N,PP,ETOL,ERREST,WORK,IW,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C G01DAF COMPUTES A SET OF NORMAL SCORES OF SIZE N
C PARAMETERS -
C N THE NUMBER OF NORMAL SCORES TO BE COMPUTED
C PP THE SCORES OBTAINED ARE STORED IN PP(1) TO PP(N)
C ETOL AN ATTEMPT IS MADE TO COMPUTE THE SCORES WITH
C AN ESTIMATED ACCURACY OF ETOL OR BETTER
C FOR EXAMPLE ETOL MAY BE GIVEN THE VALUE 0.000001
C ERREST THE COMPUTED ESTIMATE OF THE MAXIMUM ERROR
C WORK WORK ARRAY OF DIMENSION IW ( AT LEAST 3*N/2 )
C IW THE DIMENSION OF WORK
C IFAIL GIVEN THE FOLLOWING VALUES
C 0 SUCCESS
C 1 IF N IS LESS THAN 1
C 2 IF ETOL IS LESS THAN OR EQUAL TO ZERO
C 3 IF THE ACCURACY REQUESTED THROUGH ETOL
C COULD NOT BE OBTAINED, PP AND ERREST
C REFER TO THE BEST SOLUTION OBTAINED
C 4 IF THE VALUE OF IW IS TOO SMALL
SUBROUTINE G01DBF(N,PP,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C AS APPL. STATIST. ALGORITHM AS 177.3 (1982), VOL. 31
C
C COMPUTES AN APPROXIMATION TO THE EXPECTED VALUES OF NORMAL ORDER
C STATISTICS FOR A SAMPLE OF SIZE N.
C
SUBROUTINE G01DCF(N,EX1,EX2,SUMM2,VAPVEC,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C MARK 13 REVISED. IER-639 (APR 1988).
C
C AS APPL. STATIST. ALGORITHM AS 128 (1978), VOL. 27
C DAVIS C.S. AND STEPHENS M.A.
C
C COMPUTES AND NORMALISES THE DAVID-JOHNSON APPROXIMATION
C FOR THE COVARIANCE MATRIX OF NORMAL ORDER STATISTICS.
C
C (ANP/AJS)
C
SUBROUTINE G01DDF(X,N,CALWTS,A,W,PW,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C MODIFICATION OF ALGORITHM AS 181 APPL. STATIST. (1982)
C
C CALCULATES SHAPIRO AND WILK W STATISTIC AND ITS SIG. LEVEL
C
SUBROUTINE G01DHF(SCORES,TIES,N,X,R,IWRK,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C
C The character variables SCORES is used to indicate
C what type of scores are required
C
C ... SCORES ...
C R --- RANKS
C B --- BLOM VERSION
C T --- TUKEY VERSION
C V --- VAN DER WAERDEN
C N --- NORMAL
C S --- SAVAGE
C
C The character variables TIES is used to indicate
C how ties should be handled
C
C ... TIES ...
C A --- AVERAGE
C L --- LOWEST
C H --- HIGHEST
C R --- RANDOM
C I --- IGNORE
C
DOUBLE PRECISION FUNCTION G01EAF(TAIL,X,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
DOUBLE PRECISION FUNCTION G01EBF(TAIL,T,DF,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C G01EBF -- RETURNS THE PROBABILITY ASSOCIATED WITH EITHER
C (OR BOTH) TAIL(S) OF THE STUDENTS T-DISTRIBUTION WITH DF DEGREES
C OF FREEDOM THROUGH THE FUNCTION NAME.
C IF TAIL = 'U' or 'u' THE UPPER TAIL PROBABILITY IS RETURNED,
C IF TAIL = 'S' or 's' THE TWO TAIL (SIGNIFICANCE LEVEL)
C PROBABILITY IS RETURNED,
C IF TAIL = 'C' or 'c' THE TWO TAIL (CONFIDENCE INTERVAL)
C PROBABILITY IS RETURNED,
C IF TAIL = 'L' or 'l' THE LOWER TAIL PROBABILITY IS RETURNED.
C
C USES NAG LIBRARY ROUTINES P01AAF, S15ABF, G01EEF
C
DOUBLE PRECISION FUNCTION G01ECF(TAIL,X,DF,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C Computes the lower tail probability of X for a
C CHI-squared distribution with DF degrees of freedom.
C
DOUBLE PRECISION FUNCTION G01EDF(TAIL,F,DF1,DF2,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C Computes the lower/upper tail probability of F for the
C F-distribution with DF1 and DF2 degrees of freedom.
C
SUBROUTINE G01EEF(X,A,B,TOL,P,Q,PDF,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C MARK 16A REVISED. IER-1031 (JUN 1993).
C
C COMPUTES INCOMPLETE BETA FUNCTIONS P(a,b,x) and Q(a,b,x),
C and also the PROBABILITY DENSITY FUNCTION.
C
C BASED ON :-
C ALGORITHM AS 63 APPL. STATIST. (1973) VOL.22, P.409
C
DOUBLE PRECISION FUNCTION G01EFF(TAIL,G,A,B,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
DOUBLE PRECISION FUNCTION G01EMF(Q,V,IR,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C
C COMPUTES THE PROBABILITY INTEGRAL FROM 'ZERO' TO 'Q'.
C OF THE STUDENTIZED RANGE DISTRIBUTION
C
SUBROUTINE G01EPF(N,IP,D,PDL,PDU,WORK,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C
C Calculates upper and lower bounds for the significance of a
C Durbin-Watson statistic.
C
DOUBLE PRECISION FUNCTION G01ERF(T,VK,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
C --------------------------------------------------------
C G01ERF returns the left tail area of the Von Mises
C distribution, equal to the incomplete modified Bessel
C function, of the first kind and zero-th order.
C
C Parameters :
C
C T = Angle in radians, treated as deviation from zero
C (mean) reduced modulo 2PI to the range (-PI,+PI).
C
C VK = concentration parameter, Kappa, VK.GE.0.0.
C
C IFAIL = Error parameter
C
C IFAIL = 1 : On input value of VK was .lt. 0.0.
C
C --------------------------------------------------------
DOUBLE PRECISION FUNCTION G01EYF(N,D,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
DOUBLE PRECISION FUNCTION G01EZF(N1,N2,D,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
DOUBLE PRECISION FUNCTION G01FAF(TAIL,P,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
DOUBLE PRECISION FUNCTION G01FBF(TAIL,P,DF,IFAIL)
C MARK 15 RE-ISSUE. NAG COPYRIGHT 1991.
C
C G01FBF RETURNS THE DEVIATE ASSOCIATED
C WITH THE LOWER, UPPER OR TWO TAIL PROBABILITY P FROM
C THE T DISTRIBUTION WITH DF DEGREES OF FREEDOM.
C
C IF TAIL = 'U' or 'u' THE UPPER TAIL PROB IS RETURNED
C TAIL = 'S' or 's' THE (SIG. LEVEL) TWO TAIL PROB IS RETURNED
C TAIL = 'C' or 'c' THE (CONF. INT.) TWO TAIL PROB IS RETURNED
C TAIL = 'L' or 'l' THE LOWER TAIL PROB IS RETURNED.
C
C BASED ON CACM ALGORITHM 396,CACM,VOL.13,NO.10,1970
C
DOUBLE PRECISION FUNCTION G01FCF(P,DF,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C COMPUTES PERCENTAGE POINTS OF THE CHISQUARE DISTRIBUTION.
C CALLS G01FFF WITH SUITABLE VALUES OF THE PARAMETERS
C
DOUBLE PRECISION FUNCTION G01FDF(P,DF1,DF2,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C MARK 15 REVISED. IER-932 (APR 1991).
C
C Computes the deviate associated with the lower
C tail probability P, of a F-distribution with DF1 and
C DF2 degrees of freedom.
C
DOUBLE PRECISION FUNCTION G01FEF(P,A,B,TOL,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C MARK 15 REVISED. IER-918 (APR 1991).
C
C G01FEF RETURNS THE DEVIATE ASSOCIATED
C WITH THE LOWER TAIL PROBABILITY P FROM
C THE BETA DISTRIBUTION OF THE FIRST KIND
C WITH PARAMETERS A AND B.
C
C BASED ON ALGORITHM AS 109 APPL. STATIST. (1977), VOL.26, NO.1
C
DOUBLE PRECISION FUNCTION G01FFF(P,A,B,TOL,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C MARK 15 REVISED. IER-933 (APR 1991).
C
C Based on:
C Algorithm as 91 Appl. Statist. (1975) Vol.24, P.385.
C Computes the deviate associated with the lower tail
C probability P, of the Gamma distribution with shape
C parameter A and scale parameter B.
C
DOUBLE PRECISION FUNCTION G01FMF(P,V,IR,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C
C COMPUTES THE QUANTILE FOR A GIVEN PROBABIITY.
C
DOUBLE PRECISION FUNCTION G01GBF(T,DF,DELTA,TOL,MAXIT,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C G01GBF -- RETURNS THE PROBABILITY ASSOCIATED WITH THE LOWER
C TAIL OF THE NON-CENTRAL STUDENTS T-DISTRIBUTION
C WITH DF DEGREES OF FREEDOM THROUGH THE FUNCTION NAME
C
DOUBLE PRECISION FUNCTION G01GCF(X,DF,RLAMDA,TOL,MAXIT,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C MARK 16A REVISED. IER-1032 (JUN 1993).
C
C Computes the lower tail probability of X for the non-central
C CHI-Squared distribution with degrees of freedom DF.
C
DOUBLE PRECISION FUNCTION G01GDF(F,DF1,DF2,RLAMDA,TOL,MAXIT,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C Computes the lower tail probability of F for the non-central
C F distribution with DF1 and DF2 degrees of freedom.
C
DOUBLE PRECISION FUNCTION G01GEF(X,A,B,RLAMDA,TOL,MAXIT,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C Based on algorithm AS226 APPL. STATIST. (1987) VOL. 36, NO. 2
C
C Returns the cumulative probability of X for the non - central
C Beta Distribution with parameters A, B and noncentrality RLAMDA.
C
DOUBLE PRECISION FUNCTION G01HAF(X,Y,RHO,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C MARK 15B REVISED. IER-953 (NOV 1991).
C
C This function computes the bivariate normal probability;
C PROB(X.LE.x, Y.LE.y)
C where X and Y are standardised with correlation RHO.
C The algorithm given by D. R. Divgi is used with a 15 term series.
C Ref : Divgi, d.r.,(1979), Calculation of univariate and bivariate
C normal probability functions. The Annals of Statistics,
C 7,4,903-910
C
DOUBLE PRECISION FUNCTION G01HBF(TAIL,N,A,B,XMU,SIG,LDSIG,TOL,WK,
* LWK,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C
C Computes the multivariate normal probabilities.
C
SUBROUTINE G01JCF(A,MULT,RLAMDA,N,C,P,PDF,TOL,MAXIT,WRK,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
SUBROUTINE G01JDF(METHOD,N,RLAM,D,C,PROB,WORK,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C
C Calculates the lower tail probability for a linear combination
C of (central) chi-square variables.
C
DOUBLE PRECISION FUNCTION G01MBF(X)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C
C G01MBF returns the reciprocal of Mills ratio
C i.e. it returns the value of Z(X)/Q(X).
C
SUBROUTINE G01NAF(MOM,MEAN,N,A,LDA,EMU,SIGMA,LDSIG,L,RKUM,RMOM,WK,
* IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
C
C Calculates cumulants and moments of a
C quadratic form in normal variables
C Based on routine CUM by Jan Magnus and Bahram Pesaran
C
SUBROUTINE G01NBF(CASE,MEAN,N,A,LDA,B,LDB,C,LDC,ELA,EMU,SIGMA,
* LDSIG,L1,L2,LMAX,RMOM,ABSERR,EPS,WK,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
C
C Works out L1-th to L2-th moments relating to
C ratios of quadratic forms in normal variables.
C Based on routine QRMOM by Jan Magnus and Bahran Pesaran
C
C Maximum number of moments set by ISDIM (and ISPAR) is 12.
C
SUBROUTINE G02BAF(N,M,X,IX,XBAR,STD,SSP,ISSP,R,IR,IFAIL)
C MARK 4 RELEASE NAG COPYRIGHT 1974.
C MARK 4.5 REVISED
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C NAG SUBROUTINE G02BAF
C WRITTEN 16. 7.73 BY PAUL GRIFFITHS (OXFORD UNIVERSITY)
C (REVAMP OF G02AAF ALLOWING MORE FLEXIBLE ARRAY DIMENSIONS)
C
C COMPUTES MEANS, STANDARD DEVIATIONS, SUMS OF SQUARES AND
C CROSS-PRODUCTS OF DEVIATIONS FROM MEANS, AND PEARSON PRODUCT-
C MOMENT CORRELATION COEFFICIENTS FOR A SET OF DATA IN THE
C ARRAY X.
C
C USES NAG ERROR ROUTINE P01AAF
C
C
C ABOVE DATA STATEMENT MAY BE MACHINE DEPENDENT -- DEPENDS ON
C NUMBER OF CHARACTERS WHICH CAN BE STORED IN A REAL VARIABLE
C
SUBROUTINE G02BBF(N,M,X,IX,MISS,XMISS,XBAR,STD,SSP,ISSP,R,IR,
* NCASES,IFAIL)
C MARK 4 RELEASE NAG COPYRIGHT 1974.
C MARK 4.5 REVISED
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C NAG SUBROUTINE G02BBF
C WRITTEN 16. 7.73 BY PAUL GRIFFITHS (OXFORD UNIVERSITY)
C
C COMPUTES MEANS, STANDARD DEVIATIONS, SUMS OF SQUARES AND
C CROSS-PRODUCTS OF DEVIATIONS FROM MEANS, AND PEARSON PRODUCT-
C MOMENT CORRELATION COEFFICIENTS FOR A SET OF DATA IN THE
C ARRAY X, OMITTING COMPLETELY ANY CASES WITH A MISSING
C OBSERVATION FOR ANY VARIABLE
C
C USES NAG ERROR ROUTINE P01AAF
C NAG LIBRARY ROUTINE X02BEF
C
C
C ABOVE DATA STATEMENT MAY BE MACHINE DEPENDENT -- DEPENDS ON
C NUMBER OF CHARACTERS WHICH MAY BE STORED IN A REAL VARIABLE
C
SUBROUTINE G02BCF(N,M,X,IX,MISS,XMISS,XBAR,STD,SSP,ISSP,R,IR,
* NCASES,COUNT,IC,IFAIL)
C MARK 4 RELEASE NAG COPYRIGHT 1974.
C MARK 4.5 REVISED
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C NAG SUBROUTINE G02BCF
C WRITTEN 16. 7.73 BY PAUL GRIFFITHS (OXFORD UNIVERSITY)
C
C COMPUTES MEANS, STANDARD DEVIATIONS, SUMS OF SQUARES AND
C CROSS-PRODUCTS OF DEVIATIONS FROM MEANS, AND PEARSON PRODUCT-
C MOMENT CORRELATION COEFFICIENTS FOR A SET OF DATA IN THE
C ARRAY X, OMITTING CASES WITH MISSING VALUES FROM ONLY THOSE
C CALCULATIONS INVOLVING THE VARIABLES FOR WHICH THE VALUES
C ARE MISSING
C
C USES NAG ERROR ROUTINE P01AAF
C NAG LIBRARY ROUTINE X02BEF
C
C
C ABOVE DATA STATEMENT MAY BE MACHINE DEPENDENT -- DEPENDS ON
C NUMBER OF CHARACTERS WHICH MAY BE STORED IN A REAL VARIABLE
C
SUBROUTINE G02BDF(N,M,X,IX,XBAR,STD,SSPZ,ISSPZ,RZ,IRZ,IFAIL)
C MARK 4 RELEASE NAG COPYRIGHT 1974.
C MARK 4.5 REVISED
C MARK 10 REVISED. IER-380 (JUN 1982).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C NAG SUBROUTINE G02BDF
C WRITTEN 17. 7.73 BY PAUL GRIFFITHS (OXFORD UNIVERSITY)
C
C COMPUTES MEANS AND STANDARD DEVIATIONS OF VARIABLES SUMS OF
C SQUARES AND CROSS-PRODUCTS ABOUT ZERO AND CORRELATION-LIKE
C COEFFICIENTS FOR A SET OF DATA IN THE ARRAY X.
C
C USES NAG ERROR ROUTINE P01AAF
C
C
C ABOVE DATA STATEMENT MAY BE MACHINE-DEPENDENT -- DEPENDS ON
C NUMBER OF CHARACTERS WHICH CAN BE STORED IN A REAL VARIABLE
C
SUBROUTINE G02BEF(N,M,X,IX,MISS,XMISS,XBAR,STD,SSPZ,ISSPZ,RZ,IRZ,
* NCASES,IFAIL)
C MARK 4 RELEASE NAG COPYRIGHT 1974.
C MARK 4.5 REVISED
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C NAG SUBROUTINE G02BEF
C WRITTEN 17. 7.73 BY PAUL GRIFFITHS (OXFORD UNIVERSITY)
C
C COMPUTES MEANS AND STANDARD DEVIATIONS OF VARIABLES, SUMS OF
C SQUARES AND CROSS-PRODUCTS ABOUT ZERO AND CORRELATION-LIKE
C COEFFICIENTS FOR A SET OF DATA IN THE ARRAY X, OMITTING
C COMPLETELY ANY CASES WITH A MISSING OBSERVATION FOR ANY
C VARIABLE
C
C USES NAG ERROR ROUTINE P01AAF
C NAG LIBRARY ROUTINE X02BEF
C
C
C ABOVE DATA STATEMENT MAY BE MACHINE-DEPENDENT -- DEPENDS ON
C NUMBER OF CHARACTERS WHICH CAN BE STORED IN A REAL VARIABLE
C
SUBROUTINE G02BFF(N,M,X,IX,MISS,XMISS,XBAR,STD,SSPZ,ISSPZ,RZ,IRZ,
* NCASES,COUNT,IC,IFAIL)
C MARK 4 RELEASE NAG COPYRIGHT 1974.
C MARK 4.5 REVISED
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C NAG SUBROUTINE G02BFF
C WRITTEN 17. 7.73 BY PAUL GRIFFITHS (OXFORD UNIVERSITY)
C
C COMPUTES MEANS AND STANDARD DEVIATIONS OF VARIABLES, SUMS OF
C SQUARES AND CROSS-PRODUCTS ABOUT ZERO AND CORRELATION-LIKE
C COEFFICIENTS FOR A SET OF DATA IN THE ARRAY X, OMITTING CASES
C WITH MISSING VALUES FROM ONLY THOSE CALCULATIONS INVOLVING
C THE VARIABLES FOR WHICH THE VALUES ARE MISSING.
C
C USES NAG ERROR ROUTINE P01AAF
C NAG LIBRARY ROUTINE X02BEF
C
C
C ABOVE DATA STATEMENT MAY BE MACHINE-DEPENDENT -- DEPENDS ON
C NUMBER OF CHARACTERS WHICH CAN BE STORED IN A REAL VARIABLE
C
SUBROUTINE G02BGF(N,M,X,IX,NVARS,KVAR,XBAR,STD,SSP,ISSP,R,IR,
* IFAIL)
C MARK 4 RELEASE NAG COPYRIGHT 1974.
C MARK 4.5 REVISED
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C NAG SUBROUTINE G02BGF
C WRITTEN 17. 7.73 BY PAUL GRIFFITHS (OXFORD UNIVERSITY)
C
C COMPUTES MEANS, STANDARD DEVIATIONS, SUMS OF SQUARES AND
C CROSS-PRODUCTS OF DEVIATIONS FROM MEANS, AND PEARSON PRODUCT-
C MOMENT CORRELATION COEFFICIENTS FOR A SET OF DATA IN
C SPECIFIED COLUNNS OF THE ARRAY X
C
C USES NAG ERROR ROUTINE P01AAF
C
C
C ABOVE DATA STATEMENT MAY BE MACHINE DEPENDENT -- DEPENDS ON
C NUMBER OF CHARACTERS WHICH MAY BE STORED IN A REAL VARIABLE
C
SUBROUTINE G02BHF(N,M,X,IX,MISS,XMISS,MISTYP,NVARS,KVAR,XBAR,STD,
* SSP,ISSP,R,IR,NCASES,IFAIL)
C MARK 4 RELEASE NAG COPYRIGHT 1974.
C MARK 4.5 REVISED IER-53/46
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C NAG SUBROUTINE G02BHF
C WRITTEN 17. 8.73 BY PAUL GRIFFITHS (OXFORD UNIVERSITY)
C
C COMPUTES MEANS, STANDARD DEVIATIONS, SUMS OF SQUARES AND
C CROSS-PRODUCTS OF DEVIATIONS FROM MEANS, AND PEARSON PRODUCT-
C MOMENT CORRELATION COEFFICIENTS FOR A SET OF DATA IN
C SPECIFIED COLUMNS OF THE ARRAY X, OMITTING COMPLETELY ANY
C CASES WITH A MISSING OBSERVATION FOR ANY VARIABLE (OVER ALL M
C VARIABLES IF MISTYP=1, OR OVER ONLY THE NVARS VARIABLES IN
C THE SELECTED SUBSET IF MISTYP=0
C
C USES NAG ERROR ROUTINE P01AAF
C NAG LIBRARY ROUTINE X02BEF
C
C
C ABOVE DATA STATEMENT MAY BE MACHINE DEPENDENT -- DEPENDS ON
C NUMBER OF CHARACTERS WHICH MAY BE STORED IN A REAL VARIABLE
C
SUBROUTINE G02BJF(N,M,X,IX,MISS,XMISS,NVARS,KVAR,XBAR,STD,SSP,
* ISSP,R,IR,NCASES,COUNT,IC,IFAIL)
C MARK 4 RELEASE NAG COPYRIGHT 1974.
C MARK 4.5 REVISED
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C NAG SUBROUTINE G02BJF
C WRITTEN 17. 8.73 BY PAUL GRIFFITHS (OXFORD UNIVERSITY)
C
C COMPUTES MEANS, STANDARD DEVIATIONS, SUMS OF SQUARES AND
C CROSS-PRODUCTS OF DEVIATIONS FROM MEANS, AND PEARSON PRODUCT-
C MOMENT CORRELATION COEFFICIENTS FOR A SET OF DATA IN
C SPECIFIED COLUMNS OF THE ARRAY X, OMITTING CASES WITH MISSING
C VALUES FROM ONLY THOSE CALCULATIONS INVOLVING THE VARIABLES
C FOR WHICH THE VALUES ARE MISSING.
C
C USES NAG ERROR ROUTINE P01AAF
C NAG LIBRARY ROUTINE X02BEF
C
C
C ABOVE DATA STATEMENT MAY BE MACHINE DEPENDENT -- DEPENDS ON
C NUMBER OF CHARACTERS WHICH MAY BE STORED IN A REAL VARIABLE
C
SUBROUTINE G02BKF(N,M,X,IX,NVARS,KVAR,XBAR,STD,SSPZ,ISSPZ,RZ,IRZ,
* IFAIL)
C MARK 4 RELEASE NAG COPYRIGHT 1974.
C MARK 4.5 REVISED
C MARK 10 REVISED. IER-381 (JUN 1982).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C NAG SUBROUTINE G02BKF
C WRITTEN 17. 8.73 BY PAUL GRIFFITHS (OXFORD UNIVERSITY)
C
C COMPUTES MEANS AND STANDARD DEVIATIONS OF VARIABLES, SUMS OF
C SQUARES AND CROSS-PRODUCTS ABOUT ZERO AND CORRELATION-LIKE
C COEFFICIENTS FOR A SET OF DATA IN SPECIFIED COLUNNS OF THE
C ARRAY X
C
C USES NAG ERROR ROUTINE P01AAF
C
C
C ABOVE DATA STATEMENT MAY BE MACHINE-DEPENDENT -- DEPENDS ON
C NUMBER OF CHARACTERS WHICH MAY BE STORED IN A REAL VARIABLE
C
SUBROUTINE G02BLF(N,M,X,IX,MISS,XMISS,MISTYP,NVARS,KVAR,XBAR,STD,
* SSPZ,ISSPZ,RZ,IRZ,NCASES,IFAIL)
C MARK 4 RELEASE NAG COPYRIGHT 1974.
C MARK 4.5 REVISED
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C NAG SUBROUTINE G02BLF
C WRITTEN 17.8.73 BY PAUL GRIFFITHS (OXFORD UNIVERSITY)
C
C COMPUTES MEANS AND STANDARD DEVIATIONS OF VARIABLES, SUMS OF
C SQUARES AND CROSS-PRODUCTS ABOUT ZERO AND CORRELATION-LIKE
C COEFFICIENTS FOR A SET OF DATA IN SPECIFIED COLUMNS OF THE
C ARRAY X, OMITTING COMPLETELY ANY CASES WITH A MISSING
C OBSERVATION FOR ANY VARIABLE (OVER ALL M VARIABLES IF
C MISTYP =1, OR OVER ONLY THE NVARS VARIABLES IN THE SELECTED
C SUBSET IF MISTYP=0)
C
C USES NAG ERROR ROUTINE P01AAF
C NAG LIBRARY ROUTINE X02BEF
C
C
C ABOVE DATA STATEMENT MAY BE MACHINE-DEPENDENT -- DEPENDS ON
C NUMBER OF CHARACTERS WHICH CAN BE STORED IN A REAL VARIABLE
C
SUBROUTINE G02BMF(N,M,X,IX,MISS,XMISS,NVARS,KVAR,XBAR,STD,SSPZ,
* ISSPZ,RZ,IRZ,NCASES,COUNT,IC,IFAIL)
C MARK 4 RELEASE NAG COPYRIGHT 1974.
C MARK 4.5 REVISED
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C NAG SUBROUTINE G02BMF
C WRITTEN 17. 8. 73 BY PAUL GRIFFITHS (OXFORD UNIVERSITY)
C
C COMPUTES MEANS AND STANDARD DEVIATIONS OF VARIABLES, SUMS OF
C SQUARES AND CROSS-PRODUCTS ABOUT ZERO AND CORRELATION-LIKE
C COEFFICIENTS FOR A SET OF DATA IN SPECIFIED COLUMNS OF THE
C ARRAY X, OMITTING CASES WITH MISSING VALUES FROM ONLY THOSE
C CALCULATIONS INVOLVING THE VARIABLES FOR WHICH THE VALUES ARE
C MISSING
C
C USES NAG ERROR ROUTINE P01AAF
C NAG LIBRARY ROUTINE X02BEF
C
C
C ABOVE DATA STATEMENT MAY BE MACHINE-DEPENDENT -- DEPENDS ON
C NUMBER OF CHARACTERS WHICH CAN BE STORED IN A REAL VARIABLE
C
SUBROUTINE G02BNF(N,M,X,IX,ITYPE,RR,IRR,KWORKA,KWORKB,WORK1,WORK2,
* IFAIL)
C MARK 4 RELEASE NAG COPYRIGHT 1974.
C MARK 4.5 REVISED
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C NAG SUBROUTINE G02BNF
C WRITTEN 27. 7.73 BY PAUL GRIFFITHS (OXFORD UNIVERSITY)
C
C COMPUTES KENDALL AND/OR SPEARMAN RANK CORRELATION
C COEFFICIENTS
C FOR A SET OF DATA IN THE ARRAY X -- THE ARRAY X IS
C OVERWRITTEN
C BY THE ROUTINE.
C
C USES NAG ERROR ROUTINE P01AAF
C AND AUXILIARY ROUTINES G02BNZ
C G02BNX
C
C
C ABOVE DATA STATEMENT MAY BE MACHINE-DEPENDENT -- DEPENDS ON
C NUMBER OF CHARACTERS WHICH CAN BE STORED IN A REAL VARIABLE
C
SUBROUTINE G02BPF(N,M,X,IX,MISS,XMISS,ITYPE,RR,IRR,NCASES,INCASE,
* KWORKA,KWORKB,KWORKC,WORK1,WORK2,IFAIL)
C MARK 4 RELEASE NAG COPYRIGHT 1974.
C MARK 4.5 REVISED
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C NAG SUBROUTINE G02BPF
C WRITTEN 30. 7.73 BY PAUL GRIFFITHS (OXFORD UNIVERSITY)
C
C COMPUTES KENDALL AND/OR SPEARMAN RANK CORRELATION
C COEFFICIENTS
C FOR A SET OF DATA IN THE ARRAY X, OMITTING COMPLETELY ANY
C CASES WITH A MISSING OBSERVATION FOR ANY VARIABLE -- THE
C ARRAY X IS OVERWRITTEN BY THE ROUTINE
C
C
C USES NAG ERROR ROUTINE P01AAF
C NAG LIBRARY ROUTINE X02BEF
C AND AUXILIARY ROUTINES G02BNZ
C G02BNX
C
C
C ABOVE DATA STATEMENT MAY BE MACHINE-DEPENDENT -- DEPENDS ON
C NUMBER OF CHARACTERS WHICH CAN BE STORED IN A REAL VARIABLE
C
SUBROUTINE G02BQF(N,M,X,IX,ITYPE,RR,IRR,KWORKA,KWORKB,WORK1,WORK2,
* IFAIL)
C MARK 4 RELEASE NAG COPYRIGHT 1974.
C MARK 4.5 REVISED
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C NAG SUBROUTINE G02BQF
C WRITTEN 30. 7.73 BY PAUL GRIFFITHS (OXFORD UNIVERSITY)
C
C COMPUTES KENDALL AND/OR SPEARMAN RANK CORRELATION
C COEFFICIENTS
C FOR A SET OF DATA IN THE ARRAY X -- THE ARRAY X IS PRESERVED.
C
C USES NAG ERROR ROUTINE P01AAF
C AND AUXILIARY ROUTINES G02BNZ
C G02BNY
C
C
C ABOVE DATA STATEMENT MAY BE MACHINE-DEPENDENT -- DEPENDS ON
C NUMBER OF CHARACTERS WHICH CAN BE STORED IN A REAL VARIABLE
C
SUBROUTINE G02BRF(N,M,X,IX,MISS,XMISS,ITYPE,RR,IRR,NCASES,INCASE,
* KWORKA,KWORKB,KWORKC,WORK1,WORK2,IFAIL)
C MARK 4 RELEASE NAG COPYRIGHT 1974.
C MARK 4.5 REVISED
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C NAG SUBROUTINE G02BRF
C WRITTEN 15. 8.73 BY PAUL GRIFFITHS (OXFORD UNIVERSITY)
C
C COMPUTES KENDALL AND/OR SPEARMAN RANK CORRELATION
C COEFFICIENTS
C FOR A SET OF DATA IN THE ARRAY X, OMITTING COMPLETELY ANY
C CASES WITH A MISSING OBSERVATION FOR ANY VARIABLE -- THE
C ARRAY X IS PRESERVED.
C
C USES NAG ERROR ROUTINE P01AAF
C NAG LIBRARY ROUTINE X02BEF
C AND AUXILIARY ROUTINES G02BNZ
C G02BNY
C G02BRZ
C
C
C ABOVE DATA STATEMENT MAY BE MACHINE DEPENDENT -- DEPENDS ON
C NUMBER OF CHARACTERS WHICH CAN BE STORED IN A REAL VARIABLE
C
SUBROUTINE G02BSF(N,M,X,IX,MISS,XMISS,ITYPE,RR,IRR,NCASES,COUNT,
* IC,KWORKA,KWORKB,KWORKC,KWORKD,WORK1,WORK2,
* IFAIL)
C MARK 4 RELEASE NAG COPYRIGHT 1974.
C MARK 4.5 REVISED
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C NAG SUBROUTINE G02BSF
C WRITTEN 16. 8.73 BY PAUL GRIFFITHS (OXFORD UNIVERSITY)
C
C COMPUTES KENDALL AND/OR SPEARMAN RANK CORRELATION
C COEFFICIENTS
C FOR A SET OF DATA IN THE ARRAY X, OMITTING CASES WITH MISSING
C VALUES FROM ONLY THOSE CALCULATIONS INVOLVING THE VARIABLES
C FOR WHICH THE VALUES ARE MISSING--THE ARRAY X IS PRESERVED.
C
C USES NAG ERROR ROUTINE P01AAF
C NAG LIBRARY ROUTINE X02BEF
C AND AUXILIARY ROUTINES G02BNZ
C G02BNY
C G02BRZ
C
C
C ABOVE DATA STATEMENT MAY BE MACHINE-DEPENDENT -- DEPENDS ON
C NUMBER OF CHARACTERS WHICH CAN BE STORED IN A REAL VARIABLE
C
SUBROUTINE G02BTF(MEAN,M,WT,X,INCX,SW,XBAR,C,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C G02BTF RETURNS THE SUM OF MEAN SQUARED DEVIATIONS
C AND THE SUM OF SQUARES OF A DATA SET.
C THESE VALUES ARE CALCULATED USING A STABLE UPDATING METHOD
C WHICH REVISES THE EXISTING VALUES IN A SINGLE PASS.
C
C BASED ON ALGORITHM AS (WV2) COMM. ACM., (1979), VOL.22, NO.9
C
C USES NAG LIBRARY ROUTINE P01ABF
C
SUBROUTINE G02BUF(MEAN,WEIGHT,N,M,X,LDX,WT,SW,WMEAN,C,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C G02BUF CALCULATES :
C WEIGHTED SUMS OF SQUARES MATRIX
C OR UNWEIGHTED SUMS OF SQUARES MATRIX
C OR WEIGHTED SUMS OF SQUARES ABOUT ZERO MATRIX
C OR UNWEIGHTED SUMS OF SQUARES ABOUT ZERO MATRIX
C
C THESE VALUES ARE CALCULATED USING A STABLE UPDATING METHOD
C WHICH REVISES THE EXISTING VALUES IN A SINGLE PASS.
C
C BASED ON ALGORITHM AS (WV2) COMM. ACM., (1979), VOL.22, NO.9
C
C USES NAG LIBRARY ROUTINE P01ABF
C
SUBROUTINE G02BWF(M,R,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C CALCULATES CORRELATION MATRIX FROM SUM OF
C SQUARES AND CROSS-PRODUCTS OF DEVIATIONS
C STORED IN PACKED FORM UPPER-TRIANGULAR BY COLUMNS
C
SUBROUTINE G02BXF(WEIGHT,N,M,X,LDX,WT,XBAR,STD,V,LDV,R,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
SUBROUTINE G02CAF(N,X,Y,RESULT,IFAIL)
C MARK 4 RELEASE NAG COPYRIGHT 1974.
C MARK 4.5 REVISED
C MARK 5C REVISED
C MARK 8 REVISED. IER-229 (APR 1980).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C
C NAG SUBROUTINE G02CAF
C WRITTEN 6.10.73 BY PAUL GRIFFITHS (OXFORD UNIVERSITY)
C
C PERFORMS A SIMPLE LINEAR REGRESSION WITH DEPENDENT VARIABLE Y
C AND INDEPENDENT VARIABLE X.
C
C USES NAG ERROR ROUTINE P01AAF
C NAG LIBRARY ROUTINES X02AKF, X02ALF
C
C
C ABOVE DATA STATEMENT MAY BE MACHINE-DEPENDENT -- DEPENDS ON
C NUMBER OF CHARACTERS WHICH CAN BE STORED IN A REAL VARIABLE
C
SUBROUTINE G02CBF(N,X,Y,RESULT,IFAIL)
C MARK 4 RELEASE NAG COPYRIGHT 1974.
C MARK 4.5 REVISED
C MARK 5C REVISED
C MARK 8 REVISED. IER-229 (APR 1980).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C
C NAG SUBROUTINE G02CBF
C WRITTEN 6.10.73 BY PAUL GRIFFITHS (OXFORD UNIVERSITY)
C
C PERFORMS A SIMPLE LINEAR REGRESSION WITH NO CONSTANT WITH
C DEPENDENT VARIABLE Y AND INDEPENDENT VARIABLE X.
C
C USES NAG ERROR ROUTINE P01AAF
C NAG LIBRARY ROUTINES X02AKF, X02ALF
C
C
C ABOVE DATA STATEMENT MAY BE MACHINE-DEPENDENT -- DEPENDS ON
C NUMBER OF CHARACTERS WHICH CAN BE STORED IN A REAL VARIABLE
C
SUBROUTINE G02CCF(N,X,Y,XMISS,YMISS,RESULT,IFAIL)
C MARK 4 RELEASE NAG COPYRIGHT 1974.
C MARK 4.5 REVISED
C MARK 5C REVISED
C MARK 8 REVISED. IER-229 (APR 1980).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C
C NAG SUBROUTINE G02CCF
C WRITTEN 6.10.73 BY PAUL GRIFFITHS (OXFORD UNIVERSITY)
C
C PERFORMS A SIMPLE LINEAR REGRESSION WITH DEPENDENT VARIABLE Y
C AND INDEPENDENT VARIABLE X, OMITTING CASES INVOLVING MISSING
C VALUES.
C
C USES NAG ERROR ROUTINE P01AAF
C NAG LIBRARY ROUTINES X02AKF, X02ALF
C X02BEF
C
C
C ABOVE DATA STATEMENT MAY BE MACHINE-DEPENDENT -- DEPENDS ON
C NUMBER OF CHARACTERS WHICH CAN BE STORED IN A REAL VARIABLE
C
SUBROUTINE G02CDF(N,X,Y,XMISS,YMISS,RESULT,IFAIL)
C MARK 4 RELEASE NAG COPYRIGHT 1974.
C MARK 4.5 REVISED
C MARK 5C REVISED
C MARK 8 REVISED. IER-229 (APR 1980).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C
C NAG SUBROUTINE G02CDF
C WRITTEN 6.10.73 BY PAUL GRIFFITHS (OXFORD UNIVERSITY)
C
C PERFORMS A SIMPLE LINEAR REGRESSION WITH NO CONSTANT WITH
C DEPENDENT VARIABLE Y AND INDEPENDENT VARIABLE X, OMITTING
C CASES INVOLVING MISSING VALUES.
C
C USES NAG ERROR ROUTINE P01AAF
C NAG LIBRARY ROUTINES X02AKF, X02ALF
C X02BEF
C
C
C ABOVE DATA STATEMENT MAY BE MACHINE-DEPENDENT -- DEPENDS ON
C NUMBER OF CHARACTERS WHICH CAN BE STORED IN A REAL VARIABLE
C
SUBROUTINE G02CEF(NVARS,XBAR,STD,SSP,ISSP,R,IR,NVARS2,KORDER,
* XBAR2,STD2,SSP2,ISSP2,R2,IR2,IFAIL)
C MARK 4 RELEASE NAG COPYRIGHT 1974.
C MARK 4.5 REVISED
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C NAG SUBROUTINE G02CEF
C WRITTEN 6.10.73 BY PAUL GRIFFITHS (OXFORD UNIVERSITY)
C
C TAKES SELECTED ELEMENTS FROM TWO VECTORS (TYPICALLY VECTORS
C OF
C MEANS AND STANDARD DEVIATIONS) TO FORM TWO SMALLER VECTORS,
C AND SELECTED ROWS AND COLUMNS FROM TWO MATRICES (TYPICALLY
C EITHER MATRICES OF SUMS OF SQUARES AND CROSS-PRODUCTS OF
C DEVIATIONS FROM MEANS AND PEARSON PRODUCT-MOMENT CORRELATION
C COEFFICIENTS, OR MATRICES OF SUMS OF SQUARES AND
C CROSS-PRODUCTS
C ABOUT ZERO AND CORRELATION-LIKE COEFFICIENTS) TO FORM TWO
C SMALLER MATRICES, ALLOWING RE-ORDERING OF ELEMENTS IN THE
C PROCESS.
C
C USES NAG ERROR ROUTINE P01AAF
C
C
C ABOVE DATA STATEMENT MAY BE MACHINE-DEPENDENT -- DEPENDS ON
C NUMBER OF CHARACTERS WHICH CAN BE STORED IN A REAL VARIABLE
C
SUBROUTINE G02CFF(NVARS,KORDER,XBAR,STD,SSP,ISSP,R,IR,KWORK,IFAIL)
C MARK 4 RELEASE NAG COPYRIGHT 1974.
C MARK 4.5 REVISED
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C NAG SUBROUTINE G02CFF
C WRITTEN 6.10.73 BY PAUL GRIFFITHS (OXFORD UNIVERSITY)
C
C RE-ORDERS THE ELEMENTS IN TWO VECTORS (TYPICALLY VECTORS OF
C MEANS AND STANDARD DEVIATIONS), AND THE ROWS AND COLUMNS IN
C TWO MATRICES (TYPICALLY EITHER MATRICES OF SUMS OF SQUARES
C AND
C CROSS-PRODUCTS OF DEVIATIONS FROM MEANS AND PEARSON PRODUCT-
C MOMENT CORRELATION COEFFICIENTS, OR MATRICES OF SUMS OF
C SQUARES
C AND CROSS-PRODUCTS ABOUT ZERO AND CORRELATION-LIKE
C COEFFICIENTS).
C
C USES NAG ERROR ROUTINE P01AAF
C
C
C ABOVE DATA STATEMENT MAY BE MACHINE-DEPENDENT -- DEPENDS ON
C NUMBER OF CHARACTERS WHICH CAN BE STORED IN A REAL VARIABLE
C
SUBROUTINE G02CGF(NCASES,NVARS,NIND,XBAR,SSP,ISSP,R,IR,RESULT,
* COEFF,ICOEFF,CONST,RINV,IRINV,C,IC,WKZ,IWKZ,
* IFAIL)
C MARK 4 RELEASE NAG COPYRIGHT 1974.
C MARK 4.5 REVISED
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C NAG SUBROUTINE G02CGF
C WRITTEN 7.10.73 BY PAUL GRIFFITHS (OXFORD UNIVERSITY)
C
C PERFORMS A MULTIPLE LINEAR REGRESSION ON THE SET OF VARIABLES
C WHOSE MEANS, SUMS OF SQUARES AND CROSS-PRODUCTS OF DEVIATIONS
C FROM MEANS, AND PEARSON PRODUCT-MOMENT CORRELATION
C COEFFICIENTS
C ARE GIVEN, WHEN THE CORRELATION MATRIX R IS
C POSITIVE-DEFINITE.
C
C USES NAG ERROR ROUTINE P01AAF
C NAG LIBRARY ROUTINE F04ABF
C AND AUXILIARY ROUTINES G02CGZ
C G02CGY
C
C
C ABOVE DATA STATEMENT MAY BE MACHINE-DEPENDENT -- DEPENDS ON
C NUMBER OF CHARACTERS WHICH CAN BE STORED IN A REAL VARIABLE
C
SUBROUTINE G02CHF(NCASES,NVARS,NIND,SSPZ,ISSPZ,RZ,IRZ,RESULT,
* COEFF,ICOEFF,RZINV,IRZINV,CZ,ICZ,WKZ,IWKZ,IFAIL)
C MARK 4 RELEASE NAG COPYRIGHT 1974.
C MARK 4.5 REVISED
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C NAG SUBROUTINE G02CHF
C WRITTEN 7.10.73 BY PAUL GRIFFITHS (OXFORD UNIVERSITY)
C
C PERFORMS A MULTIPLE LINEAR REGRESSION WITH NO CONSTANT ON THE
C SET OF VARIABLES WHOSE SUMS OF SQUARES AND CROSS-PRODUCTS
C ABOUT ZERO AND CORRELATION-LIKE COEFFICIENTS ARE GIVEN, WHEN
C THE CORRELATION-LIKE MATRIX RZ IS POSITIVE-DEFINITE.
C
C USES NAG ERROR ROUTINE P01AAF
C NAG LIBRARY ROUTINE F04ABF
C AND AUXILIARY ROUTINE G02CGZ
C
C
C ABOVE DATA STATEMENT MAY BE MACHINE-DEPENDENT -- DEPENDS ON
C NUMBER OF CHARACTERS WHICH CAN BE STORED IN A REAL VARIABLE
C
SUBROUTINE G02DAF(MEAN,WEIGHT,N,X,LDX,M,ISX,IP,Y,WT,RSS,IDF,B,SE,
* COV,RES,H,Q,LDQ,SVD,IRANK,P,TOL,WK,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C MARK 15B REVISED. IER-954 (NOV 1991).
C COMPUTES LINEAR REGRESSION
C
C The non-full rank case is covered.
C A QR/SVD approach is used.
C
C
SUBROUTINE G02DCF(UPDATE,MEAN,WEIGHT,M,ISX,Q,LDQ,IP,X,IX,Y,WT,RSS,
* WK,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C MARK 15 REVISED. IER-919 (APR 1991).
C
C ADD (UPDATE='A') OR DROPS (UPDATE='D') AN OBSERVATION
C FROM A LINEAR REGRESSION RETURNING THE UPDATED RSS AND NEW R
C MATRIX AND Q'Y MATRIX STORED IN Q (SEE G02DAF)
C
SUBROUTINE G02DDF(N,IP,Q,LDQ,RSS,IDF,B,SE,COV,SVD,IRANK,P,TOL,WK,
* IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C COMPUTES REGRESSION PARAMETERS ETC FROM R MATRIX AND Q'Y VECTOR
C THESE ARE STORED IN Q
C
SUBROUTINE G02DEF(WEIGHT,N,IP,Q,LDQ,P,WT,X,RSS,TOL,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C G02DEF adds a new variable to a regression. The R and Q'Y matrices
C are updated.
C
SUBROUTINE G02DFF(IP,Q,LDQ,INDX,RSS,WK,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C REMOVES THE INDX TH X VARIABLE FROM THE REGRESSION.
C ONLY R AND Q'Y ARE UPDATED.
C
SUBROUTINE G02DGF(WEIGHT,N,WT,RSS,IP,IRANK,COV,Q,LDQ,SVD,P,Y,B,SE,
* RES,WK,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C CALCULATES A REGRESSION FOR NEW Y VARIABLE
C G02DGF SHOULD BE CALLED AFTER G02DAF
C NOTE: THE FIRST COLUMN OF Q IS OVERWRITTEN
C
SUBROUTINE G02DKF(IP,ICONST,P,C,LDC,B,RSS,IDF,SE,COV,WK,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C CALCULATES LEAST SQUARES ESTIMATES FOR GIVEN SET OF LINEAR
C CONSTRAINTS GIVEN THE SVD SOLUTION FROM G02DAF
C
SUBROUTINE G02DNF(IP,IRANK,B,COV,P,F,EST,STAT,SESTAT,T,TOL,WK,
* IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C GIVES ESTIMATE AND SE OF ESTIMABLE FUNCTION
C
SUBROUTINE G02EAF(MEAN,WEIGHT,N,M,X,LDX,NAME,ISX,Y,WT,NMOD,MODEL,
* LDM,RSS,NTERMS,MRANK,WK,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C CALCULATES ALL POSSIBLE REGRESION USING QR METHOD
C
SUBROUTINE G02ECF(MEAN,N,SIGSQ,TSS,NMOD,NTERMS,RSS,RSQ,CP,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C CALCULATES CP STATISTIC AND R-SQUARED VALUES
C FROM RESIDUAL SUM OF SQUARES
C
SUBROUTINE G02EEF(ISTEP,MEAN,WEIGHT,N,M,X,LDX,NAME,ISX,MAXIP,Y,WT,
* FIN,ADDVAR,NEWVAR,CHRSS,F,MODEL,NTERM,RSS,IDF,
* IFR,FREE,EXSS,Q,LDQ,P,WK,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C FITS A REGRESSION MODEL BY FORWARD SELECTION
C
SUBROUTINE G02FAF(N,IP,NRES,RES,H,RMS,SRES,LDS,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C MARK 14C REVISED. IER-887 (NOV 1990).
C
C CALCULATES STANDARDIZED RESIDUALS FROM RESIDUALS
C AND DIAGONAL OF HAT MATRIX
C
C
SUBROUTINE G02FCF(N,IP,RES,D,PDL,PDU,WORK,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C
SUBROUTINE G02GAF(LINK,MEAN,OFFSET,WEIGHT,N,X,LDX,M,ISX,IP,Y,WT,S,
* A,RSS,IDF,B,IRANK,SE,COV,V,LDV,TOL,MAXIT,IPRINT,
* EPS,WK,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C GLM FOR NORMAL DISTN
C
C
C V(I,1) = ETA, LINEAR PREDICTOR
C V(I,2) = FV, FITTED VALUES
C V(I,3) = VAR, ESTIMATED VARIANCE OF Y(I)
C V(I,4) = WWT, FINAL VALUE OF WORKING WEIGHT
C V(I,5) = R, STANDARDIZED RESIDUALS (PRIOR Y)
C V(I,6) = H, LEVERAGE
C V(I,7) = OFFSET
C V(I,8+) = R (IF QR USED) P' (IF SVD USED)
C
SUBROUTINE G02GBF(LINK,MEAN,OFFSET,WEIGHT,N,X,LDX,M,ISX,IP,Y,T,WT,
* DEV,IDF,B,IRANK,SE,COV,V,LDV,TOL,MAXIT,IPRINT,
* EPS,WK,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C GLM FOR BINOMIAL DATA
C
C
C V(I,1) = ETA, LINEAR PREDICTOR
C V(I,2) = FV, FITTED VALUES
C V(I,3) = VAR, ESTIMATED VARIANCE OF Y(I)
C V(I,4) = WWT, FINAL VALUE OF WORKING WEIGHT
C V(I,5) = R, STANDARDIZED RESIDUALS
C V(I,6) = H, LEVERAGE
C V(I,7) = OFFSET
C V(I,8+) = R (IF QR USED) P' (IF SVD USED)
C
SUBROUTINE G02GCF(LINK,MEAN,OFFSET,WEIGHT,N,X,LDX,M,ISX,IP,Y,WT,A,
* DEV,IDF,B,IRANK,SE,COV,V,LDV,TOL,MAXIT,IPRINT,
* EPS,WK,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C GENERALIZED LINEAR MODEL WITH POISSON ERRORS
C
C V(I,1) = ETA, LINEAR PREDICTOR
C V(I,2) = FV, FITTED VALUES
C V(I,3) = VAR, ESTIMATED VARIANCE OF Y(I)
C V(I,4) = WWT, FINAL VALUE OF WORKING WEIGHT
C V(I,5) = R, STANDARDIZED RESIDUALS
C V(I,6) = H, LEVERAGE
C V(I,7) = OFFSET
C V(I,8+) = R (IF QR USED) P' (IF SVD USED)
C
SUBROUTINE G02GDF(LINK,MEAN,OFFSET,WEIGHT,N,X,LDX,M,ISX,IP,Y,WT,S,
* A,DEV,IDF,B,IRANK,SE,COV,V,LDV,TOL,MAXIT,IPRINT,
* EPS,WK,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C V(I,1) = ETA, LINEAR PREDICTOR
C V(I,2) = FV, FITTED VALUES
C V(I,3) = VAR, ESTIMATED VARIANCE OF Y(I)
C V(I,4) = WWT, FINAL VALUE OF WORKING WEIGHT
C V(I,5) = R, STANDARDIZED RESIDUALS (PRIOR Y)
C V(I,6) = H, LEVERAGE
C V(I,7) = OFFSET
C V(I,8+) = R (IF QR USED) P' (IF SVD USED)
C
SUBROUTINE G02GKF(IP,ICONST,V,LDV,C,LDC,B,S,SE,COV,WK,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C CALCULATES MAXIMUM LIKELIHOOD ESTIMATES FOR GIVEN SET OF LINEAR
C CONSTRAINTS GIVEN THE SVD SOLUTION FROM G02DAF
C
SUBROUTINE G02GNF(IP,IRANK,B,COV,V,LDV,F,EST,STAT,SESTAT,Z,TOL,WK,
* IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C GIVES MAXIMUM LIKELIHOOD ESTIMATE AND SE OF ESTIMABLE FUNCTION
C
SUBROUTINE G02HAF(INDW,IPSI,ISIGMA,INDC,N,M,X,IX,Y,CPSI,H1,H2,H3,
* CUCV,DCHI,THETA,SIGMA,C,IC,RS,WGT,TOL,MAXIT,
* NITMON,WORK,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C
C MASTER ROUTINE FOR ROBUST REGRESSION USING ROBETH ROUTINES
C
SUBROUTINE G02HBF(UCV,N,M,X,IX,A,Z,BL,BD,TOL,MAXIT,NITMON,NIT,WK,
* IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C
C ITERATIVE ALGORITHM FOR THE COMPUTATION OF THE MATRIX A
C (STANDARDIZED CASE, A LOWER TRIANGULAR)
C WHERE INV(A'A) IS A ROBUST EXTIMATE OF X'X
C BASED ON ROUTINES IN ROBETH BY A. MARAZZI
C
SUBROUTINE G02HDF(CHI,PSI,PSIP0,BETA,INDW,ISIGMA,N,M,X,IX,Y,WGT,
* THETA,K,SIGMA,RS,TOL,EPS,MAXIT,NITMON,NIT,WK,
* IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C
C PURPOSE
C -------
C IWLS-ALGORITHM FOR ROBUST AND BOUNDED INFLUENCE LINEAR REGRESSION
C BASED ON ROUTINES IN ROBETH BY A MARAZZI
C
SUBROUTINE G02HFF(PSI,PSP,INDW,INDC,SIGMA,N,M,X,IX,RS,WGT,C,IC,WK,
* IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C MARK 16 REVISED. IER-1034 (JUN 1993).
C
C CALCULATION OF ROBUST ASYMPTOTIC VARIANCE-COVARIANCE MATRIX
C FOR ROBUST REGRESSION
C FOR HUBER TYPE REGRESSION C=FACT*INV(X'X)
C FOR MALLOWS OR SCHWEPPE TYPE REGRESSION
C C=FACT*INV(S1)*S2*INV(S1)
C WHERE S1=(1/N)*(X'DX) AND S2=(1/N)*(X'EX)
C
C BASED ON ROUTINES FROM ROBETH BY A. MARAZZI
C
SUBROUTINE G02HKF(N,M,X,LDX,EPS,COV,THETA,MAXIT,NITMON,TOL,NIT,WK,
* IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C MARK 15A REVISED. IER-921 (APR 1991).
C
C MASTER ROUTINE FOR ROBUST COVARIANCE
C BASED ON ROUTINES IN ROBETH BY A. MARAZZI
C
SUBROUTINE G02HLF(UCV,USERP,INDM,N,M,X,LDX,COV,A,WT,THETA,BL,BD,
* MAXIT,NITMON,TOL,NIT,WK,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C ITERATIVE ALGORITHM FOR THE COMPUTATION OF THE MATRIX A
C AND ROBUST ESIMATE OF COVARIANCE
C HUBER'S (NEWTON-RAPHSON) METHOD
C BASED ON ROUTINES IN ROBETH BY A. MARAZZI
C
C
C PARAMETER CHECK AND INITIALIZATION
C
SUBROUTINE G02HMF(UCV,USERP,INDM,N,M,X,LDX,COV,A,WT,THETA,BL,BD,
* MAXIT,NITMON,TOL,NIT,WK,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C ITERATIVE ALGORITHM FOR THE COMPUTATION OF THE MATRIX A
C AND ROBUST ESIMATE OF COVARIANCE
C STAHEL'S METHOD
C BASED ON ROUTINES IN ROBETH BY A. MARAZZI
C
C
C PARAMETER CHECK AND INITIALIZATION
C
SUBROUTINE G03AAF(MATRIX,STD,WEIGHT,N,M,X,LDX,ISX,S,WT,NVAR,E,LDE,
* P,LDP,V,LDV,WK,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C MARK 16A REVISED. IER-1035 (JUN 1993).
C
C PRINCIPAL COMPONENT ANALYSIS
C
SUBROUTINE G03ACF(WEIGHT,N,M,X,LDX,ISX,NX,ING,NG,WT,NIG,CVM,LDCVM,
* E,LDE,NCV,CVX,LDCVX,TOL,IRANKX,WK,IWK,IFAIL)
SUBROUTINE G03ADF(WEIGHT,N,M,Z,LDZ,ISZ,NX,NY,WT,E,LDE,NCV,CVX,
* LDCVX,MCV,CVY,LDCVY,TOL,WK,IWK,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
SUBROUTINE G03BAF(STAND,G,NVAR,K,FL,LDF,FLR,R,LDR,ACC,MAXIT,ITER,
* WK,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C MARK 16 REVISED. IER-1036 (JUN 1993).
C
C Performs orthogonal rotations.
C
C The NVAR x K matrix in FL is rotated to give matrix in FLR using
C the rotations in R.
C If G = 1 varimax rotation is used
C If G = 0 quartimax rotation is used
C If STAND = 'S' or 's' the rotations are computed from the
C row-standardized matrix.
C
SUBROUTINE G03BCF(STAND,PSCALE,N,M,X,LDX,Y,LDY,YHAT,R,LDR,ALPHA,
* RSS,RES,WK,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C MARK 16 REVISED. IER-1112 (JUL 1993).
C
C Computes Procrustes Rotations ( returned in R) for target matrix Y
C for data in X and fitted values in YHAT
C
C STAND indicates initial scaling/translation
C STAND = 'N' no translation or scaling
C STAND = 'Z' translation to origin
C STAND = 'C' translation to Y centre
C STAND = 'U' unit scaling, no translation
C STAND = 'M' matched scaling and translation to Y
C STAND = 'S' unit scaling and translation to zero
C
C PSCALE indicates if post-rotation scaling is required
C (Unscaled/Scaled)
C
C ALPHA is the least-squares scaling coefficient
C
C RSS is the residual sum of squares
C
C RES is the distance between observed and fitted points
C
SUBROUTINE G03CAF(MATRIX,WEIGHT,N,M,X,LDX,NVAR,ISX,NFAC,WT,E,STAT,
* COM,PSI,RES,FL,LDFL,IOP,IWK,WK,LWK,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C
C Computes maximum likelihood estimates of factor loadings
C The marginal likelihood is maximized using E04LBF
C
SUBROUTINE G03CCF(METHOD,ROTATE,NVAR,NFAC,FL,LDFL,PSI,E,R,LDR,FS,
* LDFS,WK,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C
C Computes factor score coefficients in FS from factor
C unrotated loadings in FL.
C Either Bartlett's method (METHOD='B') or the Regression
C method (METHOD='R') is used.
C Optionally rotations can be supplied in R
C
SUBROUTINE G03DAF(WEIGHT,N,M,X,LDX,ISX,NVAR,ING,NG,WT,NIG,GMEAN,
* LDG,DET,GC,STAT,DF,SIG,WK,IWK,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C
C Computes test for equality of within group covariance matrices
C
SUBROUTINE G03DBF(EQUAL,MODE,NVAR,NG,GMEAN,LDG,GC,NOBS,M,ISX,X,
* LDX,D,LDD,WK,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C
C computes generalized distance from point X to group centroids
C
C for use after G03DAF
C
SUBROUTINE G03DCF(TYPE,EQUAL,PRIORS,NVAR,NG,NIG,GMEAN,LDG,GC,DET,
* NOBS,M,ISX,X,LDX,PRIOR,P,LDP,IAG,ATIQ,ATI,WK,
* IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C
C Allocates observations to groups according to posterior
C probabilities.
C
C for use after G03DAF
C
SUBROUTINE G03EAF(UPDATE,DIST,SCALE,N,M,X,LDX,ISX,S,D,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
C
C Computes a distance matrix, D.
C
C Three distances can be computed
C DIST = A - absolute distances
C DIST = E - Euclidean distances
C DIST = S - Euclidean squared distance
C
C SCALE indicates the scaling of the variables to be used
C SCALE = S - standard deviation
C SCALE = R - range
C SCALE = G - scaling Given in S
C SCALE = U - unscaled
C
C The UPDATE option allows an existing matrix to be updated
C
SUBROUTINE G03ECF(METHOD,N,D,ILC,IUC,CD,IORD,DORD,IWK,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
C
C Performs Hierarchical Cluster Analysis using distance matrix D.
C
C METHOD indicates which type is performed
C METHOD = 1 - single link
C METHOD = 2 - complete link
C METHOD = 3 - group average
C METHOD = 4 - centroid
C METHOD = 5 - median
C METHOD = 6 - minimum variance
C
C ILC, IUC and CD give information about the clustering process
C
C IORD and DORD give information for printing the dendrogram
C
SUBROUTINE G03EFF(WEIGHT,N,M,X,LDX,ISX,NVAR,K,CMEANS,LDC,WT,INC,
* NIC,CSS,CSW,MAXIT,IWK,WK,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
C
C K-Means clustering
C
SUBROUTINE G03EHF(ORIENT,N,DORD,DMIN,DSTEP,NSYM,C,LENC,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
C
C Prints dendogram to character array
C
C Uses results from G03ECF
C
SUBROUTINE G03EJF(N,CD,IORD,DORD,K,DLEVEL,IC,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
C
C Computes an indicator of group allocation from results of G03ECF
C
C Either the number of clusters required, K, or the distance at
C which clusters are to taken, DLEVEL, can be entered.
C
SUBROUTINE G03ZAF(N,M,X,LDX,NVAR,ISX,S,E,Z,LDZ,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C
C Computes standardized values (z-scores) for data matrix X.
C
SUBROUTINE G04AGF(Y,N,K,LSUB,NOBS,L,NGP,GBAR,SGBAR,GM,SS,IDF,F,FP,
* IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 16 REVISED. IER-1115 (JUL 1993).
C G04AGF PERFORMS AN ANALYSIS OF VARIANCE FOR A
C TWO-WAY HIERARCHICAL CLASSIFICATION WITH
C SUBGROUPS OF POSSIBLY UNEQUAL SIZE , AND ALSO
C COMPUTES THE TREATMENT GROUP AND
C SUBGROUP MEANS.
C CHECK PARAMETERS
SUBROUTINE G04BBF(N,Y,IBLOCK,NT,IT,GMEAN,BMEAN,TMEAN,TABLE,LDT,C,
* LDC,IREP,R,EF,TOL,IRDF,WK,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
C
C Analysis for a block design with one factor.
C Special cases are:
C completely randomised (one-way) design,
C randomised complete block,
C
SUBROUTINE G04CAF(N,Y,NFAC,LFAC,NBLOCK,INTER,IRDF,MTERM,TABLE,
* ITOTAL,TMEAN,MAXT,E,IMEAN,SEMEAN,BMEAN,R,IWK,
* IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
C
C Computes ANOVA for general complete factorial design, optionally
C in blocks.
C
DOUBLE PRECISION FUNCTION G05CAF(X)
C MARK 14 RE-ISSUE. NAG COPYRIGHT 1989.
C
C Returns a pseudo-random number uniformly distributed between
C A and B.
C
C Pseudo-random numbers are generated by the auxiliary routine
C G05CAY, 63 at a time, and stored in the array RV in common block
C CG05CA. G05CAF copies one number from the array RV into X,
C calling G05CAY to replenish RV when necessary.
C
C This revised version of G05CAF has been introduced for
C compatibility with the new routines G05FAF, G05FBF and G05FDF,
C introduced at Mark 14.
C
C Jeremy Du Croz, NAG Ltd, June 1989.
C
SUBROUTINE G05CBF(I)
C MARK 14 RE-ISSUE. NAG COPYRIGHT 1989.
C
C initializes the internal variables used by the generator
C routines to produce a repeatable sequence
C
C G05CBF initializes the notional internal variable N in G05CAY
C to 2*ABS(I)+1 and then calls G05CAY to fill the buffer with the
C next LV pseudo-random numbers.
C
C G05CBF also re-initializes the varibles NORMAL, GAMMA and VNORML
C which are used by G05DDF, G05DGF and G05FDF respectively.
C
C ******************** ADVICE FOR IMPLEMENTORS *********************
C
C This version of G05CBF must be used in conjunction with the
C new auxiliary routine G05CAY which has been introduced at Mark 14.
C
C These notes are intended to guide implementors through the text
C changes necessary to implement the basic random number generator
C routines G05CAY, G05CAZ, G05CBF, G05CCF, G05CFZ, G05CGZ. Please
C follow these guidelines, and consult NAG Central Office if in any
C doubt or difficulty. Please send a listing of your final text for
C these routines to Central Office.
C
C 1. Prepare code for G05CAY following guidelines supplied there.
C
C 2. Read "DETAILS-NOTE-1" below.
C
C 3. Activate all lines beginning CAnn, where nn is the value of
C ILIM used in G05CAY.
C
C ******************************************************************
C
C ************************ DETAILS-NOTE-1 **************************
C
C G05CBF must be implemented consistently with G05CAY.
C
C If G05CAY has been implemented simply by selecting suitable
C variant code according to the value of ILIM, then a consistent
C implementation of G05CBF may be obtained by using the variant
C code supplied in comments beginning CAnn where the digits nn
C are the value of ILIM.
C
C If G05CAY has been implemented in machine code, it will still
C be possible on many machines to implement G05CBF in Fortran
C and this will be satisfactory since it is not important for
C G05CBF to be particularly efficient. Essentially the code for
C G05CBF depends only on how the internal variable N is stored in
C the array B in the common block /AG05CA/ and the code given
C below should be applicable provided that N is stored in
C accordance with a particular value of ILIM as defined in the
C text of G05CAY.
C
C ******************************************************************
C
SUBROUTINE G05CCF
C MARK 14 RE-ISSUE. NAG COPYRIGHT 1989.
C MARK 14B REVISED. IER-833 (MAR 1990).
C
C initializes the internal variables used by the generator
C routines to produce a non-repeatable sequence
C
C G05CCF initializes the notional internal variable N in G05CAY
C to some unpredictable value and then calls G05CAY to fill the
C buffer with the next LV pseudo-random numbers.
C
C G05CCF also re-initializes the varibles NORMAL, GAMMA and VNORML
C which are used by G05DDF, G05DGF and G05FDF respectively.
C
C ******************** ADVICE FOR IMPLEMENTORS *********************
C
C This version of G05CCF must be used in conjunction with the
C new auxiliary routine G05CAY which has been introduced at Mark 14.
C
C These notes are intended to guide implementors through the text
C changes necessary to implement the basic random number generator
C routines G05CAY, G05CAZ, G05CBF, G05CCF, G05CFZ, G05CGZ. Please
C follow these guidelines, and consult NAG Central Office if in any
C doubt or difficulty. Please send a listing of your final text for
C these routines to Central Office.
C
C 1. Prepare code for G05CAY following guidelines supplied there.
C
C 2. Read "DETAILS-NOTE-1" below.
C
C 3. Activate all lines beginning CAnn, where nn is the value of
C ILIM used in G05CAY.
C
C ******************************************************************
C
C ************************ DETAILS-NOTE-1 **************************
C
C G05CCF must be implemented consistently with G05CAY.
C
C If G05CAY has been implemented simply by selecting suitable
C variant code according to the value of ILIM, then a consistent
C implementation of G05CCF may be obtained by using the variant
C code supplied in comments beginning CAnn where the digits nn
C are the value of ILIM.
C
C If G05CAY has been implemented in machine code, it will still
C be possible on many machines to implement G05CCF in Fortran
C and this will be satisfactory since it is not important for
C G05CCF to be particularly efficient. Essentially the code for
C G05CCF depends only on how the internal variable N is stored in
C the array B in the common block /AG05CA/ and the code given
C below should be applicable provided that N is stored in
C accordance with a particular value of ILIM as defined in the
C text of G05CAY.
C
C ******************************************************************
C
SUBROUTINE G05CFF(IA,NI,XA,NX,IFAIL)
C MARK 14 RE-ISSUE. NAG COPYRIGHT 1989.
C WRITTEN BY N.M.MACLAREN
C UNIVERSITY OF CAMBRIDGE COMPUTER LABORATORY
C THIS SAVES THE GENERATOR STATE IN IA AND XA.
C
C This revised version of G05CFF has been introduced for
C compatibility with the new routine G05FDF, introduced at Mark 14.
C G05CFZ now saves three values in XA(2), XA(3) and XA(4) for use
C by G05FDF, G05DDF and G05DGF respectively, and XA(1) is used for
C check-summing.
C
C Jeremy Du Croz, NAG Ltd, June 1989.
C
SUBROUTINE G05CGF(IA,NI,XA,NX,IFAIL)
C MARK 14 RE-ISSUE. NAG COPYRIGHT 1989.
C WRITTEN BY N.M.MACLAREN
C UNIVERSITY OF CAMBRIDGE COMPUTER LABORATORY
C THIS RESTORES THE GENERATOR STATE FROM IA AND XA.
C
C This revised version of G05CGF has been introduced for
C compatibility with the new routine G05FDF, introduced at Mark 14.
C G05CGZ now restores three values from XA(2), XA(3) and XA(4) for
C use by G05FDF, G05DDF and G05DGF respectively, and XA(1) is used
C for check-summing.
C
C Jeremy Du Croz, NAG Ltd, June 1989.
C
DOUBLE PRECISION FUNCTION G05DAF(A,B)
C MARK 6 RELEASE NAG COPYRIGHT 1976
C MARK 6B REVISED IER-115 (MAR 1978)
C MARK 7 REVISED IER-135 (DEC 1978)
C MARK 7C REVISED IER-188 (MAY 1979)
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C WRITTEN BY N.M.MACLAREN
C UNIVERSITY OF CAMBRIDGE COMPUTER LABORATORY
C THIS RETURNS A REAL NUMBER, UNIFORMLY DISTRIBUTED
C BETWEEN A AND B.
C THE CONTORTED PROGRAMMING IS TO GIVE CORRECT RESULTS AND TO
C AVOID DIAGNOSTICS IN CASES WHERE ROUNDING CAUSES OVERFLOW OF
C THE RANGE (A,B).
DOUBLE PRECISION FUNCTION G05DBF(A)
C MARK 6 RELEASE NAG COPYRIGHT 1976
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C WRITTEN BY N.M.MACLAREN
C UNIVERSITY OF CAMBRIDGE COMPUTER LABORATORY
C THIS RETURNS A REAL NUMBER, NEGATIVE EXPONENTIALLY
C DISTRIBUTED WITH MEAN A.
DOUBLE PRECISION FUNCTION G05DCF(A,B)
C MARK 6 RELEASE NAG COPYRIGHT 1976
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C WRITTEN BY N.M.MACLAREN
C UNIVERSITY OF CAMBRIDGE COMPUTER LABORATORY
C THIS RETURNS A REAL NUMBER, LOGISTICALLY DISTRIBUTED WITH
C MEAN A AND STANDARD DEVIATION B*PI/SQRT(3).
DOUBLE PRECISION FUNCTION G05DDF(A,B)
C MARK 6 RELEASE NAG COPYRIGHT 1976
C MARK 11 REVISED. IER-441 (FEB 1984).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C WRITTEN BY N.M.MACLAREN
C UNIVERSITY OF CAMBRIDGE COMPUTER LABORATORY
C THIS RETURNS A REAL NUMBER, NORMALLY DISTRIBUTED (GAUSSIAN)
C WITH MEAN A AND STANDARD DEVIATION B.
C THE METHOD USED IS A MODIFICATION OF CACM ALGORITHM 488
C (BY R.P.BRENT).
C THE FOLLOWING CAN BE EXTENDED IF A TRUNCATION PROBABILITY
C OF 1.0E-12 IS REGARDED AS UNSATISFACTORY.
C THE CODE ASSUMES THAT THE CONTENTS OF COMMON BLOCK /BG05CA/
C ARE SAVED BETWEEN CALLS OF THIS ROUTINE AND CALLS OF OTHER
C G05 ROUTINES THAT REFERENCE IT. A SAVE STATEMENT
C ENSURES THAT THIS IS SO.
DOUBLE PRECISION FUNCTION G05DEF(A,B)
C MARK 6 RELEASE NAG COPYRIGHT 1976
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C WRITTEN BY N.M.MACLAREN
C UNIVERSITY OF CAMBRIDGE COMPUTER LABORATORY
C THIS RETURNS A REAL NUMBER, LOGNORMALLY DISTRIBUTED WITH
C MEAN EXP(A+B*B/2) AND VARIANCE EXP(2*A+B*B)*(EXP(B*B)-1).
DOUBLE PRECISION FUNCTION G05DFF(A,B)
C MARK 6 RELEASE NAG COPYRIGHT 1976
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C WRITTEN BY N.M.MACLAREN
C UNIVERSITY OF CAMBRIDGE COMPUTER LABORATORY
C THIS RETURNS A REAL NUMBER FROM THE CAUCHY DISTRIBUTION WITH
C MEDIAN A AND SEMI-INTERQUARTILE RANGE B.
DOUBLE PRECISION FUNCTION G05DHF(N,IFAIL)
C MARK 6 RELEASE NAG COPYRIGHT 1976
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 16 REVISED. IER-1128 (JUL 1993).
C WRITTEN BY N.M.MACLAREN
C UNIVERSITY OF CAMBRIDGE COMPUTER LABORATORY
C THIS RETURNS A REAL NUMBER FROM THE CHI-SQUARED DISTRIBUTION
C WITH N DEGREES OF FREEDOM.
DOUBLE PRECISION FUNCTION G05DJF(N,IFAIL)
C MARK 6 RELEASE NAG COPYRIGHT 1976
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 16 REVISED. IER-1129 (JUL 1993).
C WRITTEN BY N.M.MACLAREN
C UNIVERSITY OF CAMBRIDGE COMPUTER LABORATORY
C THIS RETURNS A REAL NUMBER FROM STUDENTS T DISTRIBUTION
C WITH N DEGREES OF FREEDOM.
DOUBLE PRECISION FUNCTION G05DKF(M,N,IFAIL)
C MARK 6 RELEASE NAG COPYRIGHT 1976
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 16 REVISED. IER-1130 (JUL 1993).
C WRITTEN BY N.M.MACLAREN
C UNIVERSITY OF CAMBRIDGE COMPUTER LABORATORY
C THIS RETURNS A REAL NUMBER FROM SNEDECORS F DISTRIBUTION
C WITH M AND N DEGREES OF FREEDOM.
DOUBLE PRECISION FUNCTION G05DPF(A,B,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C G05DPF RETURNS A PSEUDO-RANDOM NUMBER TAKEN FROM A
C TWO-PARAMETER WEIBULL DISTRIBUTION WITH SHAPE
C PARAMETER A AND SCALE PARAMETER B.
C
C WRITTEN BY N.M.MACLAREN
C UNIVERSITY OF CAMBRIDGE COMPUTER LABORATORY
INTEGER FUNCTION G05DRF(ALAMDA,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C
C G05DRF returns a Poisson variate from a distribution with the
C supplied parameter. It returns zero if the supplied parameter
C is negative or greater than MAXINT/2.
C
C
C
C
C
C
C
INTEGER FUNCTION G05DYF(M,N)
C MARK 6 RELEASE NAG COPYRIGHT 1976
C MARK 7 REVISED IER-135 (DEC 1978)
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C WRITTEN BY N.M.MACLAREN
C UNIVERSITY OF CAMBRIDGE COMPUTER LABORATORY
C THIS RETURNS AN INTEGER RESULT, UNIFORMLY DISTRIBUTED
C BETWEEN M AND N INCLUSIVE.
C THE CONTORTED PROGRAMMING IS TO GIVE CORRECT RESULTS AND TO
C AVOID DIAGNOSTICS IN CASES WHERE:
C 1) ROUNDING CAUSES OVERFLOW OF THE RANGE (M,N).
C 2) INTEGER/REAL CONVERSION IS NOT EXACT.
C 3) INT TRUNCATES TOWARDS MINUS INFINITY.
LOGICAL FUNCTION G05DZF(P)
C MARK 6 RELEASE NAG COPYRIGHT 1976
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C WRITTEN BY N.M.MACLAREN
C UNIVERSITY OF CAMBRIDGE COMPUTER LABORATORY
C THIS RETURNS A LOGICAL RESULT, TRUE WITH PROBABILITY P.
SUBROUTINE G05EAF(A,N,C,IC,EPS,R,NR,IFAIL)
C MARK 10 RELEASE. NAG COPYRIGHT 1982.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C
C G05EAF CONVERTS A VECTOR OF MEANS AND A COVARIANCE
C MATRIX INTO A FORM THAT IS EFFICIENT FOR THE GENERATION OF
C MULTIVARIATE NORMAL VECTORS WITH THE SPECIFIED PARAMETERS.
C
C THE REFERENCE VECTOR HAS THE FOLLOWING VALUES AT THE
C FOLLOWING LOCATIONS -
C 1) THE DIMENSION OF THE DISTRIBUTION (N)
C 2) THE MEAN OF THE DISTRIBUTION
C N+2) THE LOWER TRIANGULAR MATRIX L SUCH THAT L.L(T) IS
C THE COVARIANCE MATRIX, STORED BY COLUMNS.
C
C WRITTEN BY N.M.MACLAREN
C UNIVERSITY OF CAMBRIDGE COMPUTER LABORATORY
C
SUBROUTINE G05EBF(M,N,R,NR,IFAIL)
C MARK 6 RELEASE NAG COPYRIGHT 1976
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C WRITTEN BY N.M.MACLAREN
C UNIVERSITY OF CAMBRIDGE COMPUTER LABORATORY
C THIS SETS UP THE REFERENCE VECTOR FOR A UNIFORM DISTRIBUTION.
SUBROUTINE G05ECF(T,R,NR,IFAIL)
C MARK 6 RELEASE NAG COPYRIGHT 1976
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C WRITTEN BY N.M.MACLAREN
C UNIVERSITY OF CAMBRIDGE COMPUTER LABORATORY
C THIS SETS UP THE REFERENCE VECTOR FOR A POISSON DISTRIBUTION.
C ADD AND TIMES CAN BE CHANGED IF A TRUNCATION PROBABILITY OF
C 1.0E-12 IS REGARDED AS UNSATISFACTORY.
SUBROUTINE G05EDF(N,P,R,NR,IFAIL)
C MARK 6 RELEASE NAG COPYRIGHT 1976
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C WRITTEN BY N.M.MACLAREN
C UNIVERSITY OF CAMBRIDGE COMPUTER LABORATORY
C THIS SETS UP THE REFERENCE VECTOR FOR A BINOMIAL
C DISTRIBUTION.
C ADD AND TIMES CAN BE CHANGED IF A TRUNCATION PROBABILITY OF
C 1.0E-12 IS REGARDED AS UNSATISFACTORY.
SUBROUTINE G05EEF(N,P,R,NR,IFAIL)
C MARK 6 RELEASE NAG COPYRIGHT 1976
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C WRITTEN BY N.M.MACLAREN
C UNIVERSITY OF CAMBRIDGE COMPUTER LABORATORY
C THIS SETS UP THE REFERENCE VECTOR FOR A NEGATIVE BINOMIAL
C DISTRIBUTION.
C ADD, TIMES AND TIMES2 CAN BE CHANGED IF A TRUNCATION
C PROBABILITY
C OF 1.0E-12 IS REGARDED AS UNSATISFACTORY.
SUBROUTINE G05EFF(L,M,N,R,NR,IFAIL)
C MARK 6 RELEASE NAG COPYRIGHT 1976
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C WRITTEN BY N.M.MACLAREN
C UNIVERSITY OF CAMBRIDGE COMPUTER LABORATORY
C THIS SETS UP THE REFERENCE VECTOR FOR A HYPERGEOMETRIC
C DISTRIBUTION.
C ADD AND TIMES CAN BE CHANGED IF A TRUNCATION PROBABILITY OF
C 1.0E-12 IS REGARDED AS UNSATISFACTORY.
SUBROUTINE G05EGF(E,A,NA,B,NB,R,NR,VAR,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C MARK 14 REVISED. IER-747 (DEC 1989).
C
C G05EGF SETS UP A REFERENCE VECTOR FOR AN AUTOREGRESSIVE
C MOVING-AVERAGE TIME-SERIES MODEL WITH NORMALLY
C DISTRIBUTED ERRORS, SO THAT G05EWF MAY BE USED TO
C GENERATE SUCCESSIVE TERMS. IT ALSO INITIALISES
C THE SERIES TO A STATIONARY POSITION.
C
C THE METHOD IS FROM A PAPER OF G.TUNNICLIFFE WILSON IN THE
C JOURNAL FOR STATISTICAL COMPUTATION AND SIMULATION (C.1978).
C THE REFERENCE VECTOR CONTAINS THE FOLLOWING VALUES
C 1) NA, THE AUTOREGRESSIVE ORDER.
C 2) NB, THE MOVING AVERAGE ORDER.
C 3) A POINTER TO THE LAST VALUE OF THE PURE AUTOREGRESSIVE
C SERIES.
C 4) E, THE MEAN OF THE ERROR TERM.
C 5) A, THE AUTOREGRESSIVE COEFFICIENTS.
C NA+5) B, THE MOVING-AVERAGE COEFFICIENTS.
C NA+NB+5) THE LAST MAX(NA,NB) VALUES OF THE PURE AUTOREGRESSIVE
C SERIES IN A CIRCULAR BUFFER.
C
C WRITTEN BY N.M.MACLAREN
C UNIVERSITY OF CAMBRIDGE COMPUTER LABORATORY
C
SUBROUTINE G05EHF(INDEX,N,IFAIL)
C MARK 10 RELEASE. NAG COPYRIGHT 1982.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C G05EHF RANDOMLY PERMUTES AN INTEGER VECTOR.
C
C WRITTEN BY N.M.MACLAREN
C UNIVERSITY OF CAMBRIDGE COMPUTER LABORATORY
C
SUBROUTINE G05EJF(IA,N,IZ,M,IFAIL)
C MARK 10 RELEASE. NAG COPYRIGHT 1982.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C G05EJF TAKES A RANDOM SAMPLE OF SIZE M FROM IA (OF
C SIZE N) AND PUTS IT INTO IZ.
C
C WRITTEN BY N.M.MACLAREN
C UNIVERSITY OF CAMBRIDGE COMPUTER LABORATORY
C
DOUBLE PRECISION FUNCTION G05EWF(R,NR,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 14 REVISED. IER-748 (DEC 1989).
C
C G05EWF GENERATES THE NEXT TERM FROM AN AUTOREGRESSIVE
C MOVING-AVERAGE TIME-SERIES USING A REFERENCE VECTOR
C SET UP BY G05EGF.
C WRITTEN BY N.M.MACLAREN
C UNIVERSITY OF CAMBRIDGE COMPUTER LABORATORY
C
SUBROUTINE G05EXF(P,NP,IP,LP,R,NR,IFAIL)
C MARK 6 RELEASE NAG COPYRIGHT 1976
C MARK 7 REVISED IER-134 (DEC 1978)
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C WRITTEN BY N.M.MACLAREN
C UNIVERSITY OF CAMBRIDGE COMPUTER LABORATORY
C THIS SETS UP A REFERENCE VECTOR FROM A PDF OR CDF.
INTEGER FUNCTION G05EYF(R,NR)
C MARK 6 RELEASE NAG COPYRIGHT 1976
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C WRITTEN BY N.M.MACLAREN
C UNIVERSITY OF CAMBRIDGE COMPUTER LABORATORY
C THIS RETURNS AN INTEGER FROM THE REFERENCE VECTOR IN R.
SUBROUTINE G05EZF(Z,N,R,NR,IFAIL)
C MARK 10 RELEASE. NAG COPYRIGHT 1982.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C G05EZF RETURNS A MULTIVARIATE NORMAL VECTOR FROM THE
C PARAMETERS CONVERTED BY ROUTINE G05EAF.
C
C WRITTEN BY N.M.MACLAREN
C UNIVERSITY OF CAMBRIDGE COMPUTER LABORATORY
C
SUBROUTINE G05FAF(A,B,N,X)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C Returns a vector of N pseudo-random numbers uniformly distributed
C between A and B.
C
C Pseudo-random numbers are generated by the auxiliary routine
C G05CAY, 63 at a time, and stored in the array RV in common block
C CG05CA. G05FAF copies numbers from the array RV into X,
C transforming them to the interval (A, B), and calling G05CAY to
C replenish RV when necessary.
C
C A call of G05FAF returns the same sequence of pseudo-random
C numbers as N consecutive calls to G05DAF.
C
C Jeremy Du Croz, NAG Ltd, June 1989.
C
SUBROUTINE G05FBF(A,N,X)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C Returns a vector of N pseudo-random numbers from a (negative)
C exponential distribution with mean A.
C
C Pseudo-random numbers are generated by the auxiliary routine
C G05CAY, 63 at a time, and stored in the array RV in common block
C CG05CA. G05FBF copies numbers from the array RV into X,
C transforming them to the exponential distribution, and calling
C G05CAY to replenish RV when necessary.
C
C A call of G05FBF returns the same sequence of pseudo-random
C numbers as N consecutive calls to G05DBF.
C
C Jeremy Du Croz, NAG Ltd, June 1989.
C
SUBROUTINE G05FDF(A,B,N,X)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C MARK 16A REVISED. IER-1038 (JUN 1993).
C
C Returns a vector of N pseudo-random numbers from a normal
C distribution with mean A and standard deviation B.
C
C Pseudo-random numbers are generated by the auxiliary routine
C G05CAY, 63 at a time, and stored in the array RV in common block
C CG05CA. G05FDF copies numbers from RV to X, using the Box-Muller
C method to transform to the normal distribution two at a time,
C and calling G05CAY to replenish RV when necessary.
C
C A call of G05FDF does *not* return the same sequence of
C pseudo-random numbers as N consecutive calls to G05DDF,
C because G05DDF uses a different transformation (Brent's method).
C G05FDF uses the Box-Muller method because it is more amenable
C to vectorization.
C
C Jeremy Du Croz, NAG Ltd, May 1989.
C
SUBROUTINE G05FEF(A,B,N,X,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C
C G05FEF generates a vector of pseudo-random Beta deviates with
C shape parameters A and B.
C A = alpha
C B = beta
C N = number of observations
C X = the output vector which contains Beta deviates
C
SUBROUTINE G05FFF(A,B,N,X,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C
C G05FFF generates a vector of pseudo-random Gamma deviates with
C shape parameters A and B.
C A = alpha
C B = beta
C N = number of observations
C X = the output vector which contains Gamma deviates
C
SUBROUTINE G05FSF(VK,N,T,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
C
C Subroutine generates random variates in the range
C [-Pi,Pi] from a Von Mises distribution with
C density proportional to exp(VK*COS(VMISES))
C using best and fisher's method
C
C VK=Parameter of distribution (0
G05GAF
SUBROUTINE G05GAF(SIDE,INIT,M,N,A,LDA,WK,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
C
C Based on a LAPACK test routine
C
C
C Purpose
C =======
C
C GO5GAF pre- or post-multiplies an M by N matrix A by a random
C orthogonal matrix U, overwriting A. A may optionally be
C initialized to the identity matrix before multiplying by U.
C U is generated using the method of G.W. Stewart
C ( SIAM J. Numer. Anal. 17, 1980, pp. 403-409 ).
C
C Arguments
C =========
C
C SIDE - CHARACTER*1
C SIDE specifies whether A is multiplied on the left or right
C by U.
C SIDE = 'L' Multiply A on the left (premultiply)
C SIDE = 'R' Multiply A on the right (postmultiply)
C Unchanged on exit.
C
C INIT - CHARACTER*1
C INIT specifies whether or not A should be initialized to
C the identity matrix.
C INIT = 'I' Initialize A to (a section of) the
C identity matrix before applying U.
C INIT = 'N' No initialization. Apply U to the
C input matrix A.
C Unchanged on exit.
C
C M - INTEGER
C Number of rows of A. Unchanged on exit.
C
C N - INTEGER
C Number of columns of A. Unchanged on exit.
C
C A - REAL array of DIMENSION ( LDA, N )
C Input array. Overwritten by U*A ( if SIDE = 'L' )
C or by A*U ( if SIDE = 'R' ) on exit.
C
C LDA - INTEGER
C Leading dimension of A. Must be at least M.
C Unchanged on exit.
C
C WK - REAL array of DIMENSION (2 * MAX( M, N ) )
C Workspace. Of length 2*M if SIDE = 'L' and of length 2*N
C if SIDE = 'R'. Overwritten on exit.
C
C
G05GBF
SUBROUTINE G05GBF(N,D,C,LDC,EPS,WK,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
C
C This algorithm randomly selects a correlation matrix from the
C class of all correlation matrices with specified eigenvalues
C
G05HDF
SUBROUTINE G05HDF(MODE,K,IP,IQ,MEAN,PAR,LPAR,QQ,IK,N,W,REF,LREF,
* IWORK,LIWORK,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
G07AAF
SUBROUTINE G07AAF(N,K,CLEVEL,PL,PU,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C MARK 15B REVISED. IER-956 (NOV 1991).
C
G07ABF
SUBROUTINE G07ABF(N,XMEAN,CLEVEL,TL,TU,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C
G07BBF
SUBROUTINE G07BBF(METHOD,N,X,XC,IC,XMU,XSIG,TOL,MAXIT,SEXMU,
* SEXSIG,CORR,DEV,NOBS,NIT,WK,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C
C PROGRAM COMPUTES MAXIMUM-LIKELIHOOD ESTIMATES FOR PARAMETERS OF
C THE NORMAL DISTRIBUTION FROM GROUPED AND/OR CENSORED DATA USING
C EITHER A NEWTON-RAPHSON OR EXPECTATION-MAXIMIZATION APPROACH.
C
G07BEF
SUBROUTINE G07BEF(CENS,N,X,IC,BETA,GAMMA,TOL,MAXIT,SEBETA,SEGAM,
* CORR,DEV,NIT,WK,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C
C PROGRAM FOR ESTIMATING PARAMETERS OF THE WEIBULL DISTRIBUTION.
C
G07CAF
SUBROUTINE G07CAF(TAIL,EQUAL,NX,NY,XMEAN,YMEAN,XSTD,YSTD,CLEVEL,T,
* DF,PROB,DL,DU,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C
C This routines performs the two sample t-test for both equal
C and unequal variances.
C
G07DAF
SUBROUTINE G07DAF(N,X,Y,XME,XMD,XSD,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C MARK 14A REVISED. IER-692 (DEC 1989).
G07DBF
SUBROUTINE G07DBF(ISIGMA,N,X,IPSI,C,H1,H2,H3,DCHI,THETA,SIGMA,
* MAXIT,TOL,RS,NIT,WRK,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
G07DCF
SUBROUTINE G07DCF(CHI,PSI,ISIGMA,N,X,BETA,THETA,SIGMA,MAXIT,TOL,
* RS,NIT,WRK,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C MARK 13B REVISED. IER-666 (AUG 1988).
G07DDF
SUBROUTINE G07DDF(N,X,ALPHA,TMEAN,WMEAN,TVAR,WVAR,K,SX,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C Given data X(1), X(2), ..., X(N) in non-decreasing order,
C this routine finds the alpha-trimmed mean TMEAN, and the
C estimated variance of TMEAN, TVAR.
C
G07EAF
SUBROUTINE G07EAF(METHOD,N,X,CLEVEL,THETA,THETAL,THETAU,ESTCL,
* WLOWER,WUPPER,WRK,IWRK,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
C
G07EBF
SUBROUTINE G07EBF(METHOD,N,X,M,Y,CLEVEL,THETA,THETAL,THETAU,ESTCL,
* ULOWER,UUPPER,WRK,IWRK,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
C
G08AAF
SUBROUTINE G08AAF(X,Y,N,IS,N1,P,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 9 REVISED. IER-333 (SEP 1981).
C MARK 9A REVISED. IER-352 (NOV 1981)
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 16 REVISED. IER-1123 (JUL 1993).
C G08AAF PERFORMS THE SIGN TEST ON TWO RELATED SAMPLES OF SIZE N
G08ACF
SUBROUTINE G08ACF(X,N,N1,W1,I1,I2,P,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 16 REVISED. IER-1116 (JUL 1993).
C G08ACF PERFORMS THE MEDIAN TEST ON TWO INDEPENDENT
C SAMPLES OF POSSIBLY UNEQUAL SIZE
G08AEF
SUBROUTINE G08AEF(X,IX,K,N,W1,W2,FR,P,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 9 REVISED. IER-335 (SEP 1981).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 16 REVISED. IER 1117 (JUL 1993).
C G08AEF PERFORMS THE FRIEDMAN TWO-WAY ANALYSIS OF VARIANCE
C BY RANKS ON K RELATED SAMPLES OF SIZE N.
G08AFF
SUBROUTINE G08AFF(X,LX,L,K,W1,H,P,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 9 REVISED. IER-336 (SEP 1981).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 16 REVISED. IER-1118 (JUL 1993).
C G08AFF PERFORMS THE KRUSKAL-WALLIS ONE-WAY ANALYSIS OF
C VARIANCE BY RANKS ON K INDEPENDENT SAMPLES OF
C POSSIBLY UNEQUAL SIZE
G08AGF
SUBROUTINE G08AGF(N,X,XME,TAIL,ZEROS,RS,RSNOR,P,NZ1,WRK,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C MARK 15A REVISED. IER-922 (APR 1991).
C
C G08AGF performs the one-sample Wilcoxon signed rank test.
C
G08AHF
SUBROUTINE G08AHF(N1,X,N2,Y,TAIL,U,UNOR,P,TIES,RANKS,WRK,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C G08AHF performs the Mann-Whitney U test on two independent
C samples of possibly unequal size. The tail probaility is
C returned via P and corresponds to the TAIL option chosen.
C P is based on the normal approximation. To calculate the
C exact probability routine G08AHY or G08AHZ must be used.
C
G08AJF
SUBROUTINE G08AJF(N1,N2,TAIL,U,P,WRK,LWRK,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C The tail probaility is returned via P and corresponds to
C the TAIL option chosen. This routine calculates the exact
C probability routine using G08AJZ for the case of no ties.
C
G08AKF
SUBROUTINE G08AKF(N1,N2,TAIL,RANKS,U,P,WRK,LWRK,IWRK,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C THIS ROUTINE CALCULATES THE TAIL PROBABILITY P FOR
C THE WILCOXON-MANN-WHINEY STATISTIC U FOR SAMPLE SIZES
C N1 AND N2 FOR THE CASE OF TIES IN THE POOLED SAMPLE.
C
G08ALF
SUBROUTINE G08ALF(N,K,X,LDX,Q,PROB,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C
C Performs the Cochran Q test.
C
G08BAF
SUBROUTINE G08BAF(X,N,N1,R,ITEST,W,V,PW,PV,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 10B REVISED. IER-404 (JAN 1983).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C G08DAF PERFORMS MOODS AND DAVIDS TESTS FOR DISPERSION
C DIFFERENCES BETWEEN TWO INDEPENDENT SAMPLES OF
C POSSIBLY UNEQUAL SIZE.
C
C REFERENCE - B.E.COOPER - STATISTICS FOR EXPERIMENTALISTS
C AUTHOR - J. LLOYD-JONES (U.M.R.C.C.)
C
C PARAMETERS -
C X - DATA FOR TWO SAMPLES
C N - TOTAL SIZE OF BOTH SAMPLES
C N1 - SIZE OF FIRST SAMPLE
C R - VECTOR OF RANKS
C ITEST - TEST REQUIRED - SET TO 1 FOR DAVIDS TEST
C 2 FOR MOODS TEST
C 0 FOR BOTH
C W - MOODS MEASURE OF DISPERSION DIFFERENCES
C V - DAVIDS MEASURE OF DISPERSION DIFFERENCES
C PW - SIGNIFICANCE OF W
C PV - SIGNIFICANCE OF V
C
C CHECK PARAMETERS
G08CBF
SUBROUTINE G08CBF(N,X,DIST,PAR,ESTIMA,NTYPE,D,Z,P,SX,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
G08CCF
SUBROUTINE G08CCF(N,X,CDF,NTYPE,D,Z,P,SX,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C G08CCF performs the Kolmogorov-Smirnov one sample test. The
C test statistic D is computed and the tail probability is
C returned via P. This routine requires the user to provide a
C distribuiton function through CDF.
C
G08CDF
SUBROUTINE G08CDF(N1,X,N2,Y,NTYPE,D,Z,P,SX,SY,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
G08CGF
SUBROUTINE G08CGF(K2,IFREQ,CINT,DIST,PAR,IPARAM,PROB,CHISQ,P,NDF,
* EVAL,CHISQI,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
G08DAF
SUBROUTINE G08DAF(X,IX,K,N,RNK,W,P,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 16 REVISED. IER 1119 (JUL 1993).
C
C G08DAF CALCULATES KENDALLS COEFFICIENT OF CONCORDANCE
C ON K INDEPENDENT RANKINGS OF N OBJECTS OR INDIVIDUALS.
C
C REFERENCE - S SIEGEL - NONPARAMETRIC STATISTICS FOR
C THE BEHAVIORAL SCIENCES ( P 229 )
C AUTHOR - J. LLOYD-JONES (U.M.R.C.C.)
C
C PARAMETERS -
C X - MATRIX OF RANKS
C IX - FIRST DIMENSION OF X
C K - NUMBER OF COMPARISONS
C N - NUMBER OF OBJECTS BEING COMPARED
C RNK - RANKED DATA
C W - COEFFICIENT OF CONCORDANCE
C P - SIGNIFICANCE LEVEL OUTPUT BY G01ECF
C IFAIL - ERROR PARAMETER
C
C LOCAL VARIABLES -
C RMEAN - OVERALL AVERAGE RANK
C IDF - DEGREES OF FREEDOM(=N-1) INPUT TO G01ECF
C CHI - CHI SQUARE VALUE INPUT TO G01ECF
C
C CHECK PARAMETERS
G08EAF
SUBROUTINE G08EAF(CL,N,X,M,MAXR,NRUNS,NCOUNT,EX,C,LDC,CHI,DF,PROB,
* WRK,LWRK,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
G08EBF
SUBROUTINE G08EBF(CL,N,X,MSIZE,LAG,NCOUNT,LDC,EX,CHI,DF,P,WRK,
* IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
G08ECF
SUBROUTINE G08ECF(CL,N,X,MSIZE,NCOUNT,LDC,EX,CHI,DF,P,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
G08EDF
SUBROUTINE G08EDF(CL,N,X,M,K,RL,RU,TIL,NGAPS,NCOUNT,EX,CHI,DF,
* PROB,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
G08RAF
SUBROUTINE G08RAF(NS,NV,NSUM,Y,IP,X,NX,IDIST,NMAX,TOL,PARVAR,
* NPVAR,IRANK,ZIN,ETA,VAPVEC,PAREST,WORK,LWORK,
* IWA,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
G08RBF
SUBROUTINE G08RBF(NS,NV,NSUM,Y,IP,X,NX,ICEN,GAMMA,NMAX,TOL,PARVAR,
* NPVAR,IRANK,ZIN,ETA,VAPVEC,PAREST,WORK,LWORK,
* IWA,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
G10ABF
SUBROUTINE G10ABF(MODE,WEIGHT,N,X,Y,WT,RHO,YHAT,C,LDC,RSS,DF,RES,
* H,WK,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
C
C Fits a cubic smoothing spline
C Option MODE allows different stages to be performed
C Note: WK is used to store results between calls.
C
G10ACF
SUBROUTINE G10ACF(METHOD,WEIGHT,N,X,Y,WT,YHAT,C,LDC,RSS,RDF,RES,H,
* CRIT,RHO,U,TOL,MAXCAL,WK,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
C
C H IS USED TO STORE WWT UNTIL G10ABW
C
C
G10BAF
SUBROUTINE G10BAF(N,X,WINDOW,SLO,SHI,NS,SMOOTH,T,USEFFT,FFT,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
C
C Algorithm AS 176 APPL. STATIST. (1982) VOL. 31, NO.1
C
C Find density estimate by kernel method using Gaussian
C kernel. The interval on which the estimate is evaluated
C has end points SLO and SHI. If USEFT is not zero
C then it is assumed that the routine has been
C called before with the same data and end points
C and that the array FT has not been altered.
C
G10CAF
SUBROUTINE G10CAF(ITYPE,N,Y,SMOOTH,ROUGH,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
C
C EDA SMOOTHERS
C
C ITYPE specifies the smoother to be used
C ITYPE=0 specifies 4253H, twice
C ITYPE=1 specifies 3RSSH, twice
C
G10ZAF
SUBROUTINE G10ZAF(WEIGHT,N,X,Y,WT,NORD,XORD,YORD,WWT,RSS,IWRK,
* IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
C
C Subroutine G10ZAF is an auxiliary to spline fitting routines,
C given an unordered data set, weighted or non-weighted, it
C will order the observations and reweight them if necessary.
C
C
G11AAF
SUBROUTINE G11AAF(NROW,NCOL,NOBST,LDT,EXPT,CHIST,PROB,CHI,G,DF,
* IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
G11SAF
SUBROUTINE G11SAF(N2,N,GPROB,S,X,NROWXR,RL,A,C,IPRINT,CGETOL,
* MAXIT,CHISQR,ISHOW,NITER,ALPHA,GAMMA,VAR,IAA,G,
* EXPP,IA,OBS,P,Y,XL,OB,LL,CHI,IDF,SIGLEV,W,LW,
* IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C MARK 13A REVISED. IER-634 (APR 1988).
C
C MAXIMUM LIKELIHOOD ESTIMATION OF ITEM PARAMETERS VIA THE
C E-M ALGORITHM FOR THE FOLLOWING MODELS
C
C (1) LOGIT
C (2) PROBIT
C
G11SBF
SUBROUTINE G11SBF(N2,N,S,X,NRX,RL,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
G12AAF
SUBROUTINE G12AAF(N,T,IC,FREQ,IFREQ,ND,TP,P,PSIG,IWK,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C
G13AAF
SUBROUTINE G13AAF(X,NX,ND,NDS,NS,XD,NXD,IFAIL)
C MARK 9 RELEASE. NAG COPYRIGHT 1981.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C G13AAF CARRIES OUT NON-SEASONAL AND SEASONAL DIFFERENCING
C ON AN INPUT TIME SERIES.
C
C CONTRIBUTORS - G. TUNNICLIFFE WILSON, C. DALY (LANCASTER
C UNIV.)
C VALIDATOR - T. LAMBERT (NAG CENTRAL OFFICE)
C
C USES NAG LIBRARY ROUTINE P01AAF
C
C
C TEST THAT NONE OF THE ORDERS IS NEGATIVE
C
G13ABF
SUBROUTINE G13ABF(X,NX,NK,XM,XV,R,STAT,IFAIL)
C MARK 9 RELEASE. NAG COPYRIGHT 1981.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 12A REVISED. IER-512 (AUG 1986).
C
C G13ABF COMPUTES THE SAMPLE AUTOCORRELATION FUNCTION
C OF AN INPUT TIME SERIES. IT ALSO COMPUTES THE SAMPLE MEAN,
C THE SAMPLE VARIANCE, AND A STATISTIC WHICH MAY BE USED TO
C TEST THE HYPOTHESIS THAT THE TRUE AUTOCORRELATION
C FUNCTION IS ZERO.
C
C CONTRIBUTORS - G. TUNNICLIFFE WILSON, C. DALY (LANCASTER
C UNIV.)
C VALIDATOR - T. LAMBERT (NAG CENTRAL OFFICE)
C
C USES NAG LIBRARY ROUTINE P01AAF
C
G13ACF
SUBROUTINE G13ACF(R,NK,NL,P,V,AR,NVL,IFAIL)
C MARK 9 RELEASE. NAG COPYRIGHT 1981.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C G13ACF CALCULATES PARTIAL AUTOCORRELATION COEFFICIENTS
C GIVEN A SET OF AUTOCORRELATION COEFFICIENTS. IT ALSO
C CALCULATES THE PREDICTION ERROR VARIANCE RATIOS FOR
C INCREASING ORDER OF FINITE LAG AUTOREGRESSIVE PREDICTOR,
C AND THE AUTOREGRESSIVE PARAMETERS ASSOCIATED WITH THE
C PREDICTOR OF MAXIMUM ORDER.
C
C CONTRIBUTORS - G. TUNNICLIFFE WILSON, C. DALY (LANCASTER
C UNIV.)
C VALIDATOR - T. LAMBERT (NAG CENTRAL OFFICE)
C
C USES NAG LIBRARY ROUTINE P01AAF
C
G13ADF
SUBROUTINE G13ADF(MR,R,NK,XV,NPAR,WA,NWA,PAR,RV,ISF,IFAIL)
C MARK 9 RELEASE. NAG COPYRIGHT 1981.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C G13ADF CALCULATES PRELIMINARY ESTIMATES OF THE
C PARAMETERS OF AN AUTOREGRESSIVE MOVING-AVERAGE
C (ARMA) MODEL FROM AN AUTOCORRELATION FUNCTION.
C
C CONTRIBUTORS - G. TUNNICLIFFE WILSON, M. HURLEY (LANCASTER U.)
C VALIDATOR - T. LAMBERT (NAG CENTRAL OFFICE)
C
C ALGORITHM DUE TO G.E.P. BOX AND G.M. JENKINS,
C TIME SERIES ANALYSIS FORECASTING AND CONTROL
C (HOLDEN DAY) PART 5 PROGRAM 2
C
C USES NAG LIBRARY ROUTINES G13ADZ, P01AAF
C
G13AEF
SUBROUTINE G13AEF(MR,PAR,NPAR,C,KFC,X,NX,ICOUNT,EX,EXR,AL,IEX,S,G,
* IGH,SD,H,IH,ST,IST,NST,PIV,KPIV,NIT,ITC,ZSP,
* KZSP,ISF,WA,IWA,HC,IFAIL)
C MARK 9 RELEASE. NAG COPYRIGHT 1981.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C MARK 16 REVISED. IER-1041 (JUN 1993).
C
C G13AEF FITS A SEASONAL ARIMA MODEL TO AN OBSERVED TIME SERIES
C USING A NON-LINEAR LEAST SQUARES PROCEDURE INCORPORATING
C BACKFORECASTING. PARAMETER ESTIMATES ARE OBTAINED, TOGETHER
C WITH APPROPRIATE STANDARD ERRORS. THE RESIDUAL SERIES IS
C RETURNED, AND INFORMATION FOR USE IN FORECASTING THE SERIES IS
C PRODUCED FOR USE IN THE ROUTINES G13AGF AND G13AHF.
C
C THE ESTIMATION PROCEDURE IS ITERATIVE, STARTING WITH INITIAL
C PARAMETER VALUES SUCH AS MAY BE OBTAINED USING THE ROUTINE
C G13ADF. IT CONTINUES UNTIL A SPECIFIED CONVERGENCE CITERION
C IS SATISFIED, OR UNTIL A SPECIFIED NUMBER OF ITERATIONS HAVE
C BEEN CARRIED OUT. THE PROGRESS OF THE PROCEDURE CAN BE
C MONITORED BY MEANS OF A USER-SUPPLIED ROUTINE.
C
C CONTRIBUTORS - G. TUNNICLIFFE WILSON, C. DALY (LANCASTER
C UNIV.)
C VALIDATOR - T. LAMBERT (NAG CENTRAL OFFICE)
C
C PIV
G13AFF
SUBROUTINE G13AFF(MR,PAR,NPAR,C,KFC,X,NX,S,NDF,SD,NPPC,CM,ICM,ST,
* NST,KPIV,NIT,ITC,ISF,RES,IRES,NRES,IFAIL)
C MARK 9 RELEASE. NAG COPYRIGHT 1981.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C G13AFF IS AN EASY TO USE VERSION OF G13AEF.
C IT FITS A SEASONAL ARIMA MODEL TO AN OBSERVED TIME SERIES,
C USING
C A NON-LINEAR LEAST SQUARES TECHNIQUE INCORPORATING
C BACKFORECASTING
C
C USES NAG LIBRARY ROUTINES G13AEF AND P01AAF
C
G13AGF
SUBROUTINE G13AGF(ST,NST,MR,PAR,NPAR,C,ANX,NUV,ANEXR,WA,NWA,IFAIL)
C MARK 9 RELEASE. NAG COPYRIGHT 1981.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C G13AGF ACCEPTS A SERIES OF NEW OBSERVATIONS OF A TIME SERIES,
C THE MODEL OF WHICH IS ALREADY FULLY SPECIFIED, AND UPDATES THE
C STATE SET INFORMATION FOR USE IN CONSTRUCTING FURTHER
C FORECASTS
C
C USES NAG LIBRARY ROUTINE P01AAF
C
C CONTRIBUTORS - G. TUNNICLIFFE WILSON, C. DALY (LANCASTER
C UNIV.)
C VALIDATOR - T. LAMBERT (NAG CENTRAL OFFICE)
C
C
C ABSTRACT INFORMATION FROM THE ORDERS VECTOR
C
G13AHF
SUBROUTINE G13AHF(ST,NST,MR,PAR,NPAR,C,RMS,NFV,FVA,FSD,WA,NWA,
* IFAIL)
C MARK 9 RELEASE. NAG COPYRIGHT 1981.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C G13AHF PRODUCES FORECASTS OF A TIME SERIES, GIVEN A TIME
C SERIES
C MODEL WHICH HAS ALREADY BEEN FITTED TO THE TIME SERIES USING
C ROUTINE G13AEF OR G13AFF.
C
C USES NAG LIBRARY ROUTINE P01AAF
C
C CONTRIBUTORS - G. TUNNICLIFFE WILSON, C. DALY (LANCASTER
C UNIVERSIT
C VALIDATOR - T. LAMBERT (NAG CENTRAL OFFICE)
C
C
C ABSTRACT INFORMATION FROM ORDERS VECTOR
C
G13AJF
SUBROUTINE G13AJF(MR,PAR,NPAR,C,KFC,X,NX,RMS,ST,IST,NST,NFV,FVA,
* FSD,IFV,ISF,W,IW,IFAIL)
C MARK 10 RELEASE. NAG COPYRIGHT 1982.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C G13AJF APPLIES A SEASONAL ARIMA MODEL TO AN OBSERVED
C TIME SERIES AND DERIVES THE STATE SET FOR USE IN
C FORECASTING, AND ANY SPECIFIED NUMBER OF FORECASTS
C WITH STANDARD ERRORS.
C
C CONTRIBUTORS - G. TUNNICLIFFE WILSON, C. DALY (LANCASTER
C UNIV. )
C VALIDATOR - T. LAMBERT (NAG CENTRAL OFFICE)
C
C USES NAG LIBRARY ROUTINES G13AHZ, G13AJY, G13AJZ, P01AAF
C
G13ASF
SUBROUTINE G13ASF(N,V,MR,M,PAR,NPAR,ISHOW,C,ACFVAR,IM,SUM2,IDF,
* SIGLEV,INTGR,LMAX,WORK,LWORK,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C MARK 15 REVISED. IER-942 (APR 1991).
C MARK 16 REVISED. IER-1121 (JUL 1993).
G13AUF
SUBROUTINE G13AUF(N,Z,M,K,RS,Y,MEAN,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
G13BAF
SUBROUTINE G13BAF(Y,NY,MR,NMR,PAR,NPAR,CY,WA,NWA,B,NB,IFAIL)
C MARK 10 RELEASE. NAG COPYRIGHT 1982.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C G13BAF FILTERS A TIME SERIES BY AN AUTOREGRESSIVE
C INTEGRATED MOVING AVERAGE MODEL.
C
C CONTRIBUTORS - G.TUNNICLIFFE WILSON,M. HURLEY (LANC. UNIV.)
C VALIDATOR - T. LAMBERT ( NAG CENTRAL OFFICE )
C
C USES NAG LIBRARY ROUTINES G13BAY, G13BAZ, P01AAF
C
G13BBF
SUBROUTINE G13BBF(Y,NY,MR,NMR,PAR,NPAR,CY,WA,IWA,B,NB,IFAIL)
C MARK 11 RELEASE. NAG COPYRIGHT 1983.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C ----------------------------------------------------
C
C NAG LIBRARY ROUTINE G13BBF FILTERS A TIME SERIES
C BY AN TF MODEL
C
C ORIGIN OF SOFTWARE - LANCASTER UNIVERSITY
C DATE OF INCEPTION - FEBRUARY 1981
C DATE OF COMPLETION - FEBRUARY 1981
C
C ALGORITHM DUE TO
C G.E.P. BOX AND G.M. JENKINS
C TIME SERIES ANALYSIS FORECASTING AND CONTROL
C HOLDEN-DAY
C BACKFORECASTING FEATURE DUE TO G. TUNICLIFFE-WILSON
C
C M.A.H.(PROGRAMMER)
C
C -----------------------------------------------------
C
C G13BBF CHECKS THE USER SUPPLIED PARAMETERS AND MAKES A
C FUDGED CALL TO AUXILIARY G13BAZ WHICH CARRIES OUT THE
C CALCULATIONS.
C
G13BCF
SUBROUTINE G13BCF(X,Y,NXY,NL,S,R0,R,STAT,IFAIL)
C MARK 10 RELEASE. NAG COPYRIGHT 1982.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C G13BCF CALCULATES CROSS CORRELATIONS BETWEEN TWO TIME SERIES
C
C CONTRIBUTORS - G. TUNNICLIFFE WILSON,M. HURLEY (LANC. UNIV.)
C VALIDATOR - T. LAMBERT ( NAG CENTRAL OFFICE )
C
C USES NAG LIBRARY ROUTINES G13BCZ, P01AAF
C
G13BDF
SUBROUTINE G13BDF(R0,R,NL,NNA,S,NWDS,WA,IWA,WDS,ISF,IFAIL)
C MARK 11 RELEASE. NAG COPYRIGHT 1983.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C -------------------------------------------------------
C
C NAG LIBRARY ROUTINE G13BDF CALCULATES PRELIMINARY
C ESTIMATES FO THE PARAMETERS OF A TRANSFER FUNCTION MODEL.
C
C ORIGIN OF SOFTWARE - LANCASTER UNIVERSITY
C DATE OF INCEPTION - MARCH 1981
C DATE OF COMPLETION - APRIL 1981
C
C ALGORITHM DUE TO
C G.E.P. BOX AND G.M. JENKINS
C TIME SERIES ANALYSIS FORECASTING AND CONTROL
C HOLDEN-DAY
C
C M.A.H. (PROGRAMMER)
C
C --------------------------------------------------------
C
C G13BDF CHECKS THE USER SUPPLIED PARAMETERS AND CALLS THE
C AUXILIARY G13BDZ TO CARRY OUT THE CALCULATIONS
C
G13BEF
SUBROUTINE G13BEF(MR,NSER,MT,PARA,NPARA,KFC,NXXY,XXY,IXXY,KEF,NIT,
* KZSP,ZSP,ITC,SD,CM,ICM,S,D,NDF,KZEF,RES,STTF,
* ISTTF,NSTTF,WA,IWA,MWA,IMWA,KPRIV,IFAIL)
C MARK 11 RELEASE. NAG COPYRIGHT 1983.
C MARK 11A REVISED. IER-452 (JUN 1984).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 12 REVISED. IER-521 (AUG 1986).
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C
C SUBROUTINE G13BEF ACCEPTS AN OUTPUT (Y) SERIES AND
C ANY NUMBER OF INPUT (X) SERIES, THE ORDERS OF ALL
C THESE SERIES BEING SPECIFIED. THE Y SERIES IS CHARACTERISED
C BY AN ARIMA MODEL SPECIFICATION AND A SET OF INITIAL PARAMETER
C ESTIMATES WHICH MAY BE ZERO.
C THE ROUTINE CARRIES OUT AN ITERATIVE OPTIMISATION
C PROCEDURE TO OBTAIN A SET OF FINAL ESTIMATES WHICH
C SATISFY A SPECIFIED CONVERGENCE CRITERION OR WHICH
C ARE THE RESULTS OF A SPECIFIED NUMBER OF ITERATIONS.
C
G13BGF
SUBROUTINE G13BGF(STTF,NSTTF,MR,NSER,MT,PARA,NPARA,NNV,XXYN,IXXYN,
* KZEF,RES,WA,IWA,IFAIL)
C MARK 11 RELEASE. NAG COPYRIGHT 1983.
C MARK 11A REVISED. IER-453 (JUN 1984).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C G13BFF UPDATES A STATE SET WHEN NEW VALUES OF X AND Y
C ARE ADDED TO A FULLY SPECIFIED TRANSFER FUNCTION TIME
C SERIES MODEL
C
C
C BREAK DOWN MR(ORDERS ARRAY FOR Y) INTO COMPONENT PARTS
C
G13BHF
SUBROUTINE G13BHF(STTF,NSTTF,MR,NSER,MT,PARA,NPARA,NFV,XXYN,IXXYN,
* MRX,PARX,IPARX,RMSXY,KZEF,FVA,FSD,WA,IWA,IFAIL)
C MARK 11 RELEASE. NAG COPYRIGHT 1983.
C MARK 11A REVISED. IER-454 (JUN 1984).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C SUBROUTINE G13BHF DERIVES FORECAST VALUES AND THEIR
C STANDARD DEVIATIONS IN A TRANSFER FUNCTION CONTEXT
C
C
C MXA,MXB,MXC AND MXD ARE USED IN DERIVATION OF IWAA
C
G13BJF
SUBROUTINE G13BJF(MR,NSER,MT,PARA,NPARA,KFC,NEV,NFV,XXY,IXXY,KZEF,
* RMSXY,MRX,PARX,IPARX,FVA,FSD,STTF,ISTTF,NSTTF,
* WA,IWA,MWA,IMWA,IFAIL)
C MARK 11 RELEASE. NAG COPYRIGHT 1983.
C MARK 11A REVISED. IER-455 (JUN 1984).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C
C G13BJF PRODUCES FORECASTS WHEN NO STATE SET
C IS AVAILABLE. IT REQUIRES A FULLY SPECIFIED
C MODEL AND A SET OF OBSERVATIONS OF INPUT AND
C OUTPUT SERIES.
C
G13CAF
SUBROUTINE G13CAF(NX,MTX,PX,IW,MW,IC,NC,C,KC,L,LG,NXG,XG,NG,STATS,
* IFAIL)
C MARK 10 RELEASE. NAG COPYRIGHT 1982.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C G13CAF CALCULATES THE SMOOTHED SAMPLE SPECTRUM OF A
C UNIVARIATE TIME SERIES USING ONE OF FOUR LAG WINDOWS
C - RECTANGULAR, BARTLETT, TUKEY OR PARZEN WINDOW.
C
C CONTRIBUTORS - G. TUNNICLIFFE WILSON, M. HURLEY (LANC. UNIV.)
C VALIDATOR - T. LAMBERT ( NAG CENTRAL OFFICE )
C
C USES NAG LIBRARY ROUTINES G13CAZ, P01AAF
C
C
G13CBF
SUBROUTINE G13CBF(NX,MTX,PX,MW,PW,L,KC,LG,XG,NG,STATS,IFAIL)
C MARK 10 RELEASE. NAG COPYRIGHT 1982.
C MARK 10B REVISED. IER-405 (JAN 1983).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C G13CBF CALCULATES THE SMOOTHED SAMPLE SPECTRUM OF A
C UNIVARIATE TIME SERIES USING SPECTRAL SMOOTHING BY
C THE TRAPEZIUM FREQUENCY (DANIELL) WINDOW.
C
C CONTRIBUTORS - G. TUNNICLIFFE WILSON,M. HURLEY (LANC. UNIV.)
C VALIDATOR - T. LAMBERT ( NAG CENTRAL OFFICE )
C
C USES NAG LIBRARY ROUTINES G13CAZ, P01AAF
C
G13CCF
SUBROUTINE G13CCF(NXY,MTXY,PXY,IW,MW,IS,IC,NC,CXY,CYX,KC,L,NXYG,
* XG,YG,NG,IFAIL)
C MARK 10 RELEASE. NAG COPYRIGHT 1982.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C G13CCF CALCULATES THE SMOOTHED SAMPLE CROSS SPECTRUM
C OF A BIVARIATE TIME SERIES USING ONE OF FOUR LAG
C WINDOWS - RECTANGULAR, BARTLETT, TUKEY, OR PARZEN
C WINDOW.
C
C CONTRIBUTORS - G. TUNNICLIFFE WILSON,M. HURLEY (LANC. UNIV.)
C VALIDATOR - T. LAMBERT ( NAG CENTRAL OFFICE )
C
C USES NAG LIBRARY ROUTINES G13CCZ, P01AAF
C
G13CDF
SUBROUTINE G13CDF(NXY,MTXY,PXY,MW,IS,PW,L,KC,XG,YG,NG,IFAIL)
C MARK 10 RELEASE. NAG COPYRIGHT 1982.
C MARK 10B REVISED. IER-406 (JAN 1983).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C G13CDF CALCULATES THE SMOOTHED SAMPLE CROSS SPECTRUM
C OF A BIVARIATE TIME SERIES USING SPECTRAL SMOOTHING
C BY THE TRAPEZIUM FREQUENCY (DANIELL) WINDOW.
C
C CONTRIBUTORS - G. TUNNICLIFFE WILSON,M. HURLEY (LANC. UNIV.)
C VALIDATOR - T. LAMBERT ( NAG CENTRAL OFFICE )
C
C USES NAG LIBRARY ROUTINES G13CCZ, P01AAF
C
G13CEF
SUBROUTINE G13CEF(XG,YG,XYRG,XYIG,NG,STATS,CA,CALW,CAUP,T,SC,SCLW,
* SCUP,IFAIL)
C MARK 10 RELEASE. NAG COPYRIGHT 1982.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C FOR A BIVARIATE TIME SERIES, G13CEF CALCULATES THE
C CROSS AMPLITUDE SPECTRUM AND SQUARED COHERENCY, TOGETHER
C WITH LOWER AND UPPER BOUNDS FROM THE UNIVARIATE AND
C BIVARIATE (CROSS) SPECTRA
C
C CONTRIBUTORS - G. TUNNICLIFFE WILSON,M. HURLEY (LANC. UNIV.)
C VALIDATOR - T. LAMBERT ( NAG CENTRAL OFFICE )
C
C USES NAG LIBRARY ROUTINES G13CEZ, P01AAF
C
G13CFF
SUBROUTINE G13CFF(XG,YG,XYRG,XYIG,NG,STATS,GN,GNLW,GNUP,PH,PHLW,
* PHUP,IFAIL)
C MARK 10 RELEASE. NAG COPYRIGHT 1982.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C FOR A BIVARIATE TIME SERIES, G13CFF CALCULATES THE
C GAIN AND THE PHASE TOGETHER WITH LOWER AND UPPER
C BOUNDS FROM THE UNIVARIATE AND BIVARIATE SPECTRA.
C
C CONTRIBUTORS - G. TUNNICLIFFE WILSON,M. HURLEY (LANC. UNIV.)
C
C USES NAG LIBRARY ROUTINES G13CFZ, P01AAF
C
G13CGF
SUBROUTINE G13CGF(XG,YG,XYRG,XYIG,NG,STATS,L,N,ER,ERLW,ERUP,RF,
* RFSE,IFAIL)
C MARK 10 RELEASE. NAG COPYRIGHT 1982.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C FOR A BIVARIATE TIME SERIES, G13CGF CALCULATES THE
C NOISE SPECTRUM TOGETHER WITH MULTIPLYING FACTORS
C FOR THE BOUNDS AND THE IMPULSE RESPONSE FUNCTION
C AND ITS STANDARD ERROR, FROM THE UNIVARIATE
C AND BIVARIATE SPECTRA.
C
C CONTRIBUTORS - G. TUNNICLIFFE WILSON,M. HURLEY (LANC. UNIV.)
C VALIDATOR - T. LAMBERT ( NAG CENTRAL OFFICE )
C
C USES NAG LIBRARY ROUTINES G13CGZ, P01AAF
C
G13DBF
SUBROUTINE G13DBF(C0,C,NSM,NS,NL,NK,P,V0,V,D,DB,W,WB,NVP,WA,IWA,
* IFAIL)
C MARK 11 RELEASE. NAG COPYRIGHT 1983.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C -----------------------------------------------------
C
C NAG LIBRARY ROUTINE G13DBF CALCULATES THE MULTIVARIATE PARTIAL
C AUTOCORRELATION FUNCTION OF A MULTIVARIATE TIME SERIES.
C
C ORIGIN OF SOFTWARE - LANCASTER UNIVERSITY
C DATE OF INCEPTION - OCTOBER 1981
C DATE OF COMPLETION - JANUARY 1982
C
C M.A.H. (PROGRAMMER)
C
C ----------------------------------------------------
C
C G13DBF CHECKS THE USER SUPPLIED PARAMETERS AND CALLS THE
C AUXILIARY G13DBZ TO CARRY OUT THE CALCULATIONS
C
G13DCF
SUBROUTINE G13DCF(K,N,P,Q,MEAN,X,N4,QQ,IK,W,PARHLD,CONDS,IPRINT,
* CGETOL,MAXCAL,ISHOW,NITER,LOGL,V,G,DISP,IDISP,
* W2,LW,IW,LIW,IFAIL)
C MARK 14 RE-ISSUE. NAG COPYRIGHT 1989.
C MARK 16 REVISED. IER-1111 (JUL 1993).
G13DJF
SUBROUTINE G13DJF(K,N,Z,IK,TR,ID,DELTA,IP,IQ,MEAN,PAR,LPAR,QQ,V,
* LMAX,PREDZ,SEFZ,REF,LREF,WORK,LWORK,IWORK,
* LIWORK,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C MARK 15B REVISED. IER-957 (NOV 1991).
G13DKF
SUBROUTINE G13DKF(K,LMAX,M,MLAST,Z,IK,REF,LREF,V,PREDZ,SEFZ,WORK,
* IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
G13DLF
SUBROUTINE G13DLF(K,N,Z,IK,TR,ID,DELTA,W,ND,WORK,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
G13DMF
SUBROUTINE G13DMF(MATRIX,K,N,M,W,IK,WMEAN,R0,R,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C MARK 16A REVISED. IER-1044 (JUN 1993).
C
C This subroutine calculates the sample cross-covariance
C (MATRIX = 'V') or cross-correlation (MATRIX = 'R')
C matrices of a multivariate time series.
C
G13DNF
SUBROUTINE G13DNF(K,N,M,IK,R0,R,MAXLAG,PARLAG,X,PVALUE,WORK,LWORK,
* IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C
C This subroutine calculates the sample partial lag correlation
C matrices of a multivariate time series using the recursive
C algorithm of Heyse and Wei.
C
G13DPF
SUBROUTINE G13DPF(K,N,Z,IK,M,MAXLAG,PARLAG,SE,QQ,X,PVALUE,LOGLHD,
* WORK,LWORK,IWORK,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
G13DSF
SUBROUTINE G13DSF(K,N,V,IK,P,Q,M,PAR,PARHLD,QQ,ISHOW,R0,C,ACFVAR,
* IM,CHI,IDF,SIGLEV,IW,LIW,WORK,LWORK,IFAIL)
C MARK 15 RE-ISSUE. NAG COPYRIGHT 1991.
G13DXF
SUBROUTINE G13DXF(K,IP,PAR,RR,RI,RMOD,WORK,IWORK,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
H02BBF
SUBROUTINE H02BBF(ITMAX,MSGLVL,N,M,A,LDA,BL,BU,INTVAR,CVEC,MAXNOD,
* INTFST,MAXDPT,TOLIV,TOLFES,BIGBND,X,OBJMIP,
* IWORK,LIWORK,RWORK,LRWORK,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C MARK 14B REVISED. IER-843 (MAR 1990).
C MARK 15 REVISED. IER-927 (APR 1991).
C MARK 16 RE-ISSUE. NAG COPYRIGHT 1992.
H02BFF
SUBROUTINE H02BFF(INFILE,MAXN,MAXM,OPTIM,XBLDEF,XBUDEF,MAXDPT,
* MSGLVL,N,M,X,CRNAME,IWORK,LIWORK,RWORK,LRWORK,
* IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
H02BUF
SUBROUTINE H02BUF(INFILE,MAXN,MAXM,OPTIM,XBLDEF,XBUDEF,NMOBJ,
* NMRHS,NMRNG,NMBND,MPSLST,N,M,A,BL,BU,CVEC,X,
* INTVAR,CRNAME,NMPROB,IWORK,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
H02BVF
SUBROUTINE H02BVF(N,M,A,LDA,BL,BU,X,CLAMDA,ISTATE,CRNAME,IFAIL)
C MARK 16 RELEASE. NAG COPYRIGHT 1993.
H02BZF
SUBROUTINE H02BZF(N,M,BL,BU,CLAMDA,ISTATE,IWORK,LIWORK,RWORK,
* LRWORK,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C MARK 16 RE-ISSUE. NAG COPYRIGHT 1992.
H03ABF
SUBROUTINE H03ABF(KOST,MMM,MA,MB,M,K15,MAXIT,K7,K9,NUMIT,K6,K8,
* K11,K12,Z,IFAIL)
C NAG COPYRIGHT 1975
C MARK 4.5 REVISED
C MARK 6 REVISED IER-91
C MARK 7C REVISED IER-189 (MAY 1979)
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C H03ABF SOLVES THE CLASSICAL TRANSPORTATION PROBLEM IN THAT IT
C MINIMIZES THE COST OF SENDING GOODS FROM SOURCES TO
C DESTINATIONS
C SUBJECT TO CONSTRAINTS ON THE AVAILABILITIES AND
C REQUIREMENTS.
M01CAF
SUBROUTINE M01CAF(RV,M1,M2,ORDER,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C M01CAF SORTS A VECTOR OF REAL NUMBERS INTO ASCENDING
C OR DESCENDING ORDER.
C
C M01CAF IS BASED ON SINGLETON'S IMPLEMENTATION OF THE
C 'MEDIAN-OF-THREE' QUICKSORT ALGORITHM, BUT WITH TWO
C ADDITIONAL MODIFICATIONS. FIRST, SMALL SUBFILES ARE
C SORTED BY AN INSERTION SORT ON A SEPARATE FINAL PASS.
C SECOND, IF A SUBFILE IS PARTITIONED INTO TWO VERY
C UNBALANCED SUBFILES, THE LARGER OF THEM IS FLAGGED FOR
C SPECIAL TREATMENT: BEFORE IT IS PARTITIONED, ITS END-
C POINTS ARE SWAPPED WITH TWO RANDOM POINTS WITHIN IT;
C THIS MAKES THE WORST CASE BEHAVIOUR EXTREMELY UNLIKELY.
C
C THE MAXIMUM LENGTH OF A SMALL SUBFILE IS DEFINED BY THE
C VARIABLE MINQIK, SET TO 15.
C
C THE ROUTINE ASSUMES THAT THE NUMBER OF ELEMENTS TO BE
C SORTED DOES NOT EXCEED MINQIK*2**MAXSTK.
C
C WRITTEN BY N.M.MACLAREN, UNIVERSITY OF CAMBRIDGE.
C REVISED BY NAG CENTRAL OFFICE.
C
M01CBF
SUBROUTINE M01CBF(IV,M1,M2,ORDER,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C M01CBF SORTS A VECTOR OF INTEGER NUMBERS INTO ASCENDING
C OR DESCENDING ORDER.
C
C M01CBF IS BASED ON SINGLETON'S IMPLEMENTATION OF THE
C 'MEDIAN-OF-THREE' QUICKSORT ALGORITHM, BUT WITH TWO
C ADDITIONAL MODIFICATIONS. FIRST, SMALL SUBFILES ARE
C SORTED BY AN INSERTION SORT ON A SEPARATE FINAL PASS.
C SECOND, IF A SUBFILE IS PARTITIONED INTO TWO VERY
C UNBALANCED SUBFILES, THE LARGER OF THEM IS FLAGGED FOR
C SPECIAL TREATMENT: BEFORE IT IS PARTITIONED, ITS END-
C POINTS ARE SWAPPED WITH TWO RANDOM POINTS WITHIN IT;
C THIS MAKES THE WORST CASE BEHAVIOUR EXTREMELY UNLIKELY.
C
C THE MAXIMUM LENGTH OF A SMALL SUBFILE IS DEFINED BY THE
C VARIABLE MINQIK, SET TO 15.
C
C THE ROUTINE ASSUMES THAT THE NUMBER OF ELEMENTS TO BE
C SORTED DOES NOT EXCEED MINQIK*2**MAXSTK.
C
C WRITTEN BY N.M.MACLAREN, UNIVERSITY OF CAMBRIDGE.
C REVISED BY NAG CENTRAL OFFICE.
C
M01CCF
SUBROUTINE M01CCF(CH,M1,M2,L1,L2,ORDER,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C M01CCF RE-ARRANGES A VECTOR OF CHARACTER DATA SO THAT A
C SPECIFIED SUBSTRING IS IN ASCII OR REVERSE ASCII ORDER.
C
C M01CCF IS BASED ON SINGLETON'S IMPLEMENTATION OF THE
C 'MEDIAN-OF-THREE' QUICKSORT ALGORITHM, BUT WITH TWO
C ADDITIONAL MODIFICATIONS. FIRST, SMALL SUBFILES ARE
C SORTED BY AN INSERTION SORT ON A SEPARATE FINAL PASS.
C SECOND, IF A LARGER OF THEM IS FLAGGED FOR SPECIAL
C TREATMENT: BEFORE IT IS PARTITIONED, ITS END-POINTS
C ARE SWAPPED WITH TWO RANDOM POINTS WITHIN IT; THIS
C MAKES THE WORST CASE BEHAVIOUR EXTREMELY UNLIKELY.
C
C ONLY THE SUBSTRING (L1:L2) OF EACH ELEMENT OF THE ARRAY
C CH IS USED TO DETERMINE THE SORTED ORDER, BUT THE ENTIRE
C ELEMENTS ARE RE-ARRANGED INTO SORTED ORDER.
C
C THE MAXIMUM LENGTH OF A SMALL SUBFILE IS DEFINED BY THE
C VARIABLE MINQIK, SET TO 15.
C
C THE ROUTINE ASSUMES THAT THE NUMBER OF ELEMENTS TO BE
C SORTED DOES NOT EXCEED MINQIK*2**MAXSTK.
C
C THE MAXIMUM PERMITTED LENGTH OF EACH ELEMENT OF THE ARRAY CH
C IS DEFINED BY THE PARAMETER MAXLCH. THIS RESTRICTION IS
C IMPOSED BY THE NEED TO SPECIFY A LENGTH FOR THE INTERNAL
C CHARACTER VARIABLE A.
C
C WRITTEN BY NAG CENTRAL OFFICE.
C
M01DAF
SUBROUTINE M01DAF(RV,M1,M2,ORDER,IRANK,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C M01DAF RANKS A VECTOR OF REAL NUMBERS IN ASCENDING
C OR DESCENDING ORDER.
C
C M01DAF USES A VARIANT OF LIST-MERGING, AS DESCRIBED
C BY KNUTH. THE ROUTINE TAKES ADVANTAGE OF NATURAL
C ORDERING IN THE DATA, AND USES A SIMPLE LIST INSERTION
C IN A PREPARATORY PASS TO GENERATE ORDERED LISTS OF
C LENGTH AT LEAST 10. THE RANKING IS STABLE: EQUAL ELEMENTS
C PRESERVE THEIR ORDERING IN THE INPUT DATA.
C
C THE MINIMUM LENGTH OF THE LISTS AT THE END OF THE
C PREPARATORY PASS IS DEFINED BY THE VARIABLE MAXINS.
C
C WRITTEN BY N.M.MACLAREN, UNIVERSITY OF CAMBRIDGE.
C REVISED BY NAG CENTRAL OFFICE.
C
M01DBF
SUBROUTINE M01DBF(IV,M1,M2,ORDER,IRANK,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C M01DBF RANKS A VECTOR OF INTEGER NUMBERS IN ASCENDING
C OR DESCENDING ORDER.
C
C M01DBF USES A VARIANT OF LIST-MERGING, AS DESCRIBED
C BY KNUTH. THE ROUTINE TAKES ADVANTAGE OF NATURAL
C ORDERING IN THE DATA, AND USES A SIMPLE LIST INSERTION
C IN A PREPARATORY PASS TO GENERATE ORDERED LISTS OF
C LENGTH AT LEAST 10. THE RANKING IS STABLE: EQUAL ELEMENTS
C PRESERVE THEIR ORDERING IN THE INPUT DATA.
C
C THE MINIMUM LENGTH OF THE LISTS AT THE END OF THE
C PREPARATORY PASS IS DEFINED BY THE VARIABLE MAXINS.
C
C WRITTEN BY N.M.MACLAREN, UNIVERSITY OF CAMBRIDGE.
C REVISED BY NAG CENTRAL OFFICE.
C
M01DCF
SUBROUTINE M01DCF(CH,M1,M2,L1,L2,ORDER,IRANK,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C M01DCF RANKS A VECTOR OF CHARACTER DATA IN ASCII OR
C REVERSE ASCII ORDER OF A SPECIFIED SUBSTRING.
C
C M01DCF USES A VARIANT OF LIST-MERGING, AS DESCRIBED
C BY KNUTH. THE ROUTINE TAKES ADVANTAGE OF NATURAL
C ORDERING IN THE DATA, AND USES A SIMPLE LIST INSERTION
C IN A PREPARATORY PASS TO GENERATE ORDERED LISTS OF
C LENGTH AT LEAST 10. THE RANKING IS STABLE: EQUAL ELEMENTS
C PRESERVE THEIR ORDERING IN THE INPUT DATA.
C
C ONLY THE SUBSTRING (L1:L2) OF EACH ELEMENT OF THE ARRAY
C CH IS USED TO DETERMINE THE RANK ORDER.
C
C THE MINIMUM LENGTH OF THE LISTS AT THE END OF THE
C PREPARATORY PASS IS DEFINED BY THE VARIABLE MAXINS.
C
C THE MAXIMUM PERMITTED LENGTH OF EACH ELEMENT OF THE ARRAY CH
C IS DEFINED BY THE PARAMETER MAXLCH. THIS RESTRICTION IS
C IMPOSED BY THE NEED TO SPECIFY A LENGTH FOR THE INTERNAL
C CHARACTER VARIABLES A, B AND C.
C
C WRITTEN BY NAG CENTRAL OFFICE.
C
M01DEF
SUBROUTINE M01DEF(RM,LDM,M1,M2,N1,N2,ORDER,IRANK,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C M01DEF RANKS THE ROWS OF A MATRIX OF REAL NUMBERS
C IN ASCENDING OR DESCENDING ORDER.
C
C M01DEF RANKS ROWS M1 TO M2 OF A MATRIX, USING THE DATA
C IN COLUMNS N1 TO N2 OF THOSE ROWS. THE ORDERING IS
C DETERMINED BY FIRST RANKING THE DATA IN COLUMN N1,
C THEN RANKING ANY TIED ROWS ACCORDING TO THE DATA IN
C COLUMN N1+1, AND SO ON UP TO COLUMN N2.
C
C M01DEF USES A VARIANT OF LIST-MERGING, AS DESCRIBED BY
C KNUTH. THE ROUTINE TAKES ADVANTAGE OF NATURAL ORDERING
C IN THE DATA, AND USES A SIMPLE LIST INSERTION IN A
C PREPARATORY PASS TO GENERATE ORDERED LISTS OF LENGTH AT
C LEAST 10. THE RANKING IS STABLE: EQUAL ROWS PRESERVE
C THEIR ORDERING IN THE INPUT DATA.
C
C THE MINIMUM LENGTH OF THE LISTS AT THE END OF THE
C PREPARATORY PASS IS DEFINED BY THE VARIABLE MAXINS.
C
C WRITTEN BY N.M.MACLAREN, UNIVERSITY OF CAMBRIDGE.
C REVISED BY NAG CENTRAL OFFICE.
C
M01DFF
SUBROUTINE M01DFF(IV,LDM,M1,M2,N1,N2,ORDER,IRANK,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C M01DFF RANKS THE ROWS OF A MATRIX OF INTEGER NUMBERS
C IN ASCENDING OR DESCENDING ORDER.
C
C M01DFF RANKS ROWS M1 TO M2 OF A MATRIX, USING THE DATA
C IN COLUMNS N1 TO N2 OF THOSE ROWS. THE ORDERING IS
C DETERMINED BY FIRST RANKING THE DATA IN COLUMN N1,
C THEN RANKING ANY TIED ROWS ACCORDING TO THE DATA IN
C COLUMN N1+1, AND SO ON UP TO COLUMN N2.
C
C M01DFF USES A VARIANT OF LIST-MERGING, AS DESCRIBED BY
C KNUTH. THE ROUTINE TAKES ADVANTAGE OF ANY NATURAL ORDERING
C IN THE DATA, AND USES A SIMPLE LIST INSERTION IN A
C PREPARATORY PASS TO GENERATE ORDERED LISTS OF LENGTH AT
C LEAST 10. THE RANKING IS STABLE: EQUAL ROWS PRESERVE
C THEIR ORDERING IN THE INPUT DATA.
C
C THE MINIMUM LENGTH OF THE LISTS AT THE END OF THE
C PREPARATORY PASS IS DEFINED BY THE VARIABLE MAXINS.
C
C WRITTEN BY N.M.MACLAREN, UNIVERSITY OF CAMBRIDGE.
C REVISED BY NAG CENTRAL OFFICE.
C
M01DJF
SUBROUTINE M01DJF(RM,LDM,M1,M2,N1,N2,ORDER,IRANK,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C M01DJF RANKS THE COLUMNS OF A MATRIX OF REAL NUMBERS IN
C ASCENDING OR DESCENDING ORDER.
C
C M01DJF RANKS COLUMNS N1 TO N2 OF A MATRIX, USING THE DATA
C IN ROWS M1 TO M2 OF THOSE COLUMNS. THE ORDERING IS
C DETERMINED BY FIRST RANKING THE DATA IN ROW M1, THEN
C RANKING ANY TIED COLUMNS ACCORDING TO THE DATA IN ROW
C M1+1, AND SO ON UP TO ROW M2.
C
C M01DJF USES A VARIANT OF LIST-MERGING, AS DECRIBED BY
C KNUTH. THE ROUTINE TAKES ADVANTAGE OF ANY NATURAL ORDERING
C IN THE DATA, AND USES A SIMPLE LIST INSERTION IN A
C PREPARATORY PASS TO GENERATE ORDERED LISTS OF LENGTH AT
C LEAST 10. THE RANKING IS STABLE: EQUAL COLUMNS PRESERVE
C THEIR ORDERING IN THE INPUT DATA.
C
C THE MINIMUM LENGTH OF THE LISTS AT THE END OF THE
C PREPARATORY PASS IS DEFINED BY THE VARIABLE MAXINS.
C
C WRITTEN BY N.M.MACLAREN, UNIVERSITY OF CAMBRIDGE.
C REVISED BY NAG CENTRAL OFFICE.
M01DKF
SUBROUTINE M01DKF(IM,LDM,M1,M2,N1,N2,ORDER,IRANK,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C M01DKF RANKS THE COLUMNS OF A MATRIX OF INTEGER NUMBERS IN
C ASCENDING OR DESCENDING ORDER.
C
C M01DKF RANKS COLUMNS N1 TO N2 OF A MATRIX, USING THE DATA
C IN ROWS M1 TO M2 OF THOSE COLUMNS. THE ORDERING IS
C DETERMINED BY FIRST RANKING THE DATA IN ROW M1, THEN
C RANKING ANY TIED COLUMNS ACCORDING TO THE DATA IN ROW
C M1+1, AND SO ON UP TO ROW M2.
C
C M01DKF USES A VARIANT OF LIST-MERGING, AS DECRIBED BY
C KNUTH. THE ROUTINE TAKES ADVANTAGE OF ANY NATURAL ORDERING
C IN THE DATA, AND USES A SIMPLE LIST INSERTION IN A
C PREPARATORY PASS TO GENERATE ORDERED LISTS OF LENGTH AT
C LEAST 10. THE RANKING IS STABLE: EQUAL COLUMNS PRESERVE
C THEIR ORDERING IN THE INPUT DATA.
C
C THE MINIMUM LENGTH OF THE LISTS AT THE END OF THE
C PREPARATORY PASS IS DEFINED BY THE VARIABLE MAXINS.
C
C WRITTEN BY N.M.MACLAREN, UNIVERSITY OF CAMBRIDGE.
C REVISED BY NAG CENTRAL OFFICE.
M01DZF
SUBROUTINE M01DZF(COMPAR,M1,M2,IRANK,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C M01DZF RANKS ARBITRARY DATA ACCORDING TO A USER-SUPPLIED
C COMPARISON ROUTINE.
C
C M01DZF IS A GENERAL-PURPOSE ROUTINE FOR RANKING ARBITRARY
C DATA. M01DZF DOES NOT ACCESS THE DATA DIRECTLY; INSTEAD IT
C CALLS A USER-SUPPLIED ROUTINE COMPAR TO DETERMINE THE RELATIVE
C ORDERING OF ANY TWO DATA ITEMS. THE DATA ITEMS ARE IDENTIFIED
C SIMPLY BY AN INTEGER IN THE RANGE M1 TO M2.
C
C M01DZF USES A VARIANT OF LIST-MERGING, AS DESCRIBED BY KNUTH.
C THE ROUTINE TAKES ADVANTAGE OF ANY NATURAL ORDERING IN THE
C DATA, AND USES A SIMPLE LIST INSERTION IN A PREPARATORY PASS
C TO GENERATE ORDERED LISTS OF LENGTH AT LEAST 10.
C
C THE MINIMUM LENGTH OF THE LISTS AT THE END OF THE
C PREPARATORY PASS IS DEFINED BY THE VARIABLE MAXINS.
C
C WRITTEN BY NAG CENTRAL OFFICE.
C
M01EAF
SUBROUTINE M01EAF(RV,M1,M2,IRANK,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C M01EAF RE-ARRANGES A VECTOR OF REAL NUMBERS INTO
C THE ORDER SPECIFIED BY A VECTOR OF RANKS.
C
C M01EAF IS DESIGNED TO BE USED TYPICALLY IN CONJUNCTION
C WITH THE M01D- RANKING ROUTINES. AFTER ONE OF THE M01D-
C ROUTINES HAS BEEN CALLED TO DETERMINE A VECTOR OF RANKS,
C M01EAF CAN BE CALLED TO RE-ARRANGE A VECTOR OF REAL
C NUMBERS INTO THE RANK ORDER. IF THE VECTOR OF RANKS HAS
C BEEN GENERATED IN SOME OTHER WAY, THEN M01ZBF SHOULD BE
C CALLED TO CHECK ITS VALIDITY BEFORE M01EAF IS CALLED.
C
C WRITTEN BY N.M.MACLAREN, UNIVERSITY OF CAMBRIDGE.
C REVISED BY NAG CENTRAL OFFICE.
C
M01EBF
SUBROUTINE M01EBF(IV,M1,M2,IRANK,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C M01EBF RE-ARRANGES A VECTOR OF INTEGER NUMBERS INTO
C THE ORDER SPECIFIED BY A VECTOR OF RANKS.
C
C M01EBF IS DESIGNED TO BE USED TYPICALLY IN CONJUNCTION
C WITH THE M01D- RANKING ROUTINES. AFTER ONE OF THE M01D-
C ROUTINES HAS BEEN CALLED TO DETERMINE A VECTOR OF RANKS,
C M01EBF CAN BE CALLED TO RE-ARRANGE A VECTOR OF INTEGER
C NUMBERS INTO THE RANK ORDER. IF THE VECTOR OF RANKS HAS
C BEEN GENERATED IN SOME OTHER WAY, THEN M01ZBF SHOULD BE
C CALLED TO CHECK ITS VALIDITY BEFORE M01EBF IS CALLED.
C
C WRITTEN BY N.M.MACLAREN, UNIVERSITY OF CAMBRIDGE.
C REVISED BY NAG CENTRAL OFFICE
C
M01ECF
SUBROUTINE M01ECF(CH,M1,M2,IRANK,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C M01ECF RE-ARRANGES A VECTOR OF CHARACTER DATA INTO
C THE ORDER SPECIFIED BY A VECTOR OF RANKS.
C
C M01ECF IS DESIGNED TO BE USED TYPICALLY IN CONJUNCTION
C WITH THE M01D- RANKING ROUTINES. AFTER ONE OF THE M01D-
C ROUTINES HAS BEEN CALLED TO DETERMINE A VECTOR OF RANKS,
C M01ECF CAN BE CALLED TO RE-ARRANGE A VECTOR OF CHARACTER
C DATA INTO THE RANK ORDER. IF THE VECTOR OF RANKS HAS
C BEEN GENERATED IN SOME OTHER WAY, THEN M01ZBF SHOULD BE
C CALLED TO CHECK ITS VALIDITY BEFORE M01ECF IS CALLED.
C
C THE MAXIMUM PERMITTED LENGTH OF EACH ELEMENT OF THE ARRAY CH
C IS DEFINED BY THE PARAMETER MAXLCH. THIS RESTRICTION IS
C IMPOSED BY THE NEED TO SPECIFY A LENGTH FOR THE INTERNAL
C CHARACTER VARIABLES A AND B.
C
C WRITTEN BY NAG CENTRAL OFFICE
C
M01ZAF
SUBROUTINE M01ZAF(IPERM,M1,M2,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C M01ZAF INVERTS A PERMUTATION, AND HENCE CONVERTS A RANK VECTOR TO
C AN INDEX VECTOR, OR VICE VERSA.
C
C WRITTEN BY N.M.MACLAREN, UNIVERSITY OF CAMBRIDGE.
C REVISED BY NAG CENTRAL OFFICE.
C
M01ZBF
SUBROUTINE M01ZBF(IPERM,M1,M2,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C M01ZBF CHECKS THE VALIDITY OF A PERMUTATION.
C
C M01ZBF CHECKS THAT ELEMENTS M1 TO M2 OF IPERM CONTAIN A VALID
C PERMUTATION OF THE INTEGERS M1 TO M2. THE CONTENTS OF IPERM ARE
C UNCHANGED ON EXIT.
C
C WRITTEN BY N.N.MACLAREN, UNIVERSITY OF CAMBRIDGE.
C REVISED BY NAG CENTRAL OFFICE.
C
M01ZCF
SUBROUTINE M01ZCF(IPERM,M1,M2,ICYCL,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C M01ZCF DECOMPOSES A PERMUTATION INTO CYCLES, AS AN AID TO
C RE-ORDERING RANKED DATA.
C
C GIVEN A VECTOR IRANK WHICH SPECIFIES THE RANKS OF THE DATA
C (AS GENERATED BY THE M01D- ROUTINES), M01ZCF GENERATES A NEW
C VECTOR ICYCL, IN WHICH THE PERMUTATION IS REPRESENTED IN ITS
C COMPONENT CYCLES, WITH THE FIRST ELEMENT IN EACH CYCLE
C NEGATED. FOR EXAMPLE, THE PERMUTATION
C
C 5 7 4 2 1 6 3
C
C IS COMPOSED OF THE CYCLES
C
C (1 5) (2 7 3 4) (6)
C
C AND THE VECTOR ICYCL GENERATED BY M01ZCF CONTAINS
C
C -1 5 -2 7 3 4 -6
C
C WRITTEN BY NAG CENTRAL OFFICE
C
P01ABF
INTEGER FUNCTION P01ABF(IFAIL,IERROR,SRNAME,NREC,REC)
C MARK 11.5(F77) RELEASE. NAG COPYRIGHT 1986.
C MARK 13 REVISED. IER-621 (APR 1988).
C MARK 13B REVISED. IER-668 (AUG 1988).
C
C P01ABF is the error-handling routine for the NAG Library.
C
C P01ABF either returns the value of IERROR through the routine
C name (soft failure), or terminates execution of the program
C (hard failure). Diagnostic messages may be output.
C
C If IERROR = 0 (successful exit from the calling routine),
C the value 0 is returned through the routine name, and no
C message is output
C
C If IERROR is non-zero (abnormal exit from the calling routine),
C the action taken depends on the value of IFAIL.
C
C IFAIL = 1: soft failure, silent exit (i.e. no messages are
C output)
C IFAIL = -1: soft failure, noisy exit (i.e. messages are output)
C IFAIL =-13: soft failure, noisy exit but standard messages from
C P01ABF are suppressed
C IFAIL = 0: hard failure, noisy exit
C
C For compatibility with certain routines included before Mark 12
C P01ABF also allows an alternative specification of IFAIL in which
C it is regarded as a decimal integer with least significant digits
C cba. Then
C
C a = 0: hard failure a = 1: soft failure
C b = 0: silent exit b = 1: noisy exit
C
C except that hard failure now always implies a noisy exit.
C
C S.Hammarling, M.P.Hooper and J.J.du Croz, NAG Central Office.
C
S01BAF
DOUBLE PRECISION FUNCTION S01BAF(X,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C Computes log(1+x) retaining full precision when x is small.
C Following a suggestion of N.M.Temme, uses the Chebyshev expansion
C log((1+p*p+2*p*z)/(1+p*p-2*p*z)) =
C 4 * SUM (k=0 to infinity) p**(2*k+1) * T(2*k+1)(z) / (2*k+1)
C (Lyusternik et al, Handbook for Computing Elementary Functions,
C Pergamon Press, 1965).
C Equating (1+x) with (1+p*p+2*p*z)/(1+p*p-2*p*z), we get
C z = x*(1+p*p)/(2*p*(x+2)).
C Choosing p as (sqrt(sqrt(2))-1)/(sqrt(sqrt(2))+1) =
C 0.0864..., the expansion will converge with a very small number
C of terms, and be valid in the range [1/sqrt(2)-1,sqrt(2)-1].
C Outside this range, the standard Fortran log function is good
C enough.
C
C **************************************************************
C
C TO EXTRACT THE CORRECT CODE FOR A PARTICULAR MACHINE-RANGE,
C ACTIVATE THE STATEMENTS CONTAINED IN COMMENTS BEGINNING CDD ,
C WHERE DD IS THE APPROXIMATE NUMBER OF SIGNIFICANT DECIMAL
C DIGITS REPRESENTED BY THE MACHINE
C DELETE THE ILLEGAL DUMMY STATEMENT OF THE FORM
C * EXPANSION (NNNN) *
C
C **************************************************************
C
S01EAF
COMPLEX*16 FUNCTION S01EAF(Z,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C Returns exp(Z) for complex Z.
S07AAF
DOUBLE PRECISION FUNCTION S07AAF(X,IFAIL)
C MARK 5A REVISED - NAG COPYRIGHT 1976
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C TAN(X)
C
S09AAF
DOUBLE PRECISION FUNCTION S09AAF(X,IFAIL)
C MARK 5A REVISED - NAG COPYRIGHT 1976
C MARK 6A REVISED IER-94
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C ARCSIN(X)
C
S09ABF
DOUBLE PRECISION FUNCTION S09ABF(X,IFAIL)
C MARK 5A REVISED - NAG COPYRIGHT 1976
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C ARCCOS(X)
C
S10AAF
DOUBLE PRECISION FUNCTION S10AAF(X,IFAIL)
C MARK 5A REVISED - NAG COPYRIGHT 1976
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C TANH(X)
C
S10ABF
DOUBLE PRECISION FUNCTION S10ABF(X,IFAIL)
C MARK 5A REVISED - NAG COPYRIGHT 1976
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 14 REVISED. IER-749 (DEC 1989).
C SINH(X)
C **************************************************************
C
C TO EXTRACT THE CORRECT CODE FOR A PARTICULAR MACHINE-RANGE,
C ACTIVATE THE STATEMENTS CONTAINED IN COMMENTS BEGINNING CDD ,
C WHERE DD IS THE APPROXIMATE NUMBER OF SIGNIFICANT DECIMAL
C DIGITS REPRESENTED BY THE MACHINE
C DELETE THE ILLEGAL DUMMY STATEMENTS OF THE FORM
C * EXPANSION (NNNN) *
C
C ALSO INSERT APPROPRIATE DATA STATEMENTS TO DEFINE CONSTANTS
C WHICH DEPEND ON THE RANGE OF NUMBERS REPRESENTED BY THE
C MACHINE, RATHER THAN THE PRECISION (SUITABLE STATEMENTS FOR
C SOME MACHINES ARE CONTAINED IN COMMENTS BEGINNING CRD WHERE
C D IS A DIGIT WHICH SIMPLY DISTINGUISHES A GROUP OF MACHINES).
C DELETE THE ILLEGAL DUMMY DATA STATEMENTS WITH VALUES WRITTEN
C *VALUE*
C
C **************************************************************
C
S10ACF
DOUBLE PRECISION FUNCTION S10ACF(X,IFAIL)
C MARK 5A REVISED - NAG COPYRIGHT 1976
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 14 REVISED. IER-750 (DEC 1989).
C COSH(X)
C **************************************************************
C
C TO EXTRACT THE CORRECT CODE FOR A PARTICULAR MACHINE-RANGE,
C ACTIVATE THE STATEMENTS CONTAINED IN COMMENTS BEGINNING CDD ,
C WHERE DD IS THE APPROXIMATE NUMBER OF SIGNIFICANT DECIMAL
C DIGITS REPRESENTED BY THE MACHINE
C DELETE THE ILLEGAL DUMMY STATEMENTS OF THE FORM
C * EXPANSION (NNNN) *
C
C ALSO INSERT APPROPRIATE DATA STATEMENTS TO DEFINE CONSTANTS
C WHICH DEPEND ON THE RANGE OF NUMBERS REPRESENTED BY THE
C MACHINE, RATHER THAN THE PRECISION (SUITABLE STATEMENTS FOR
C SOME MACHINES ARE CONTAINED IN COMMENTS BEGINNING CRD WHERE
C D IS A DIGIT WHICH SIMPLY DISTINGUISHES A GROUP OF MACHINES).
C DELETE THE ILLEGAL DUMMY DATA STATEMENTS WITH VALUES WRITTEN
C *VALUE*
C
C **************************************************************
C
S11AAF
DOUBLE PRECISION FUNCTION S11AAF(X,IFAIL)
C MARK 5A REVISED - NAG COPYRIGHT 1976
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C ARCTANH(X)
C
S11ABF
DOUBLE PRECISION FUNCTION S11ABF(X,IFAIL)
C MARK 5A REVISED - NAG COPYRIGHT 1976
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C ARCSINH(X)
C
S11ACF
DOUBLE PRECISION FUNCTION S11ACF(X,IFAIL)
C MARK 5A REVISED - NAG COPYRIGHT 1976
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 16 REVISED. IER-1045 (JUN 1993).
C ARCCOSH(X)
C
S13AAF
DOUBLE PRECISION FUNCTION S13AAF(X,IFAIL)
C MARK 5A REVISED - NAG COPYRIGHT 1976
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 14 REVISED. IER-751 (DEC 1989).
C MARK 16 RE-ISSUE. NAG COPYRIGHT 1993.
C EXPONENTIAL INTEGRAL E1(X)
C
C **************************************************************
C
C TO EXTRACT THE CORRECT CODE FOR A PARTICULAR MACHINE-RANGE,
C ACTIVATE THE STATEMENTS CONTAINED IN COMMENTS BEGINNING CDD ,
C WHERE DD IS THE APPROXIMATE NUMBER OF SIGNIFICANT DECIMAL
C DIGITS REPRESENTED BY THE MACHINE
C DELETE THE ILLEGAL DUMMY STATEMENTS OF THE FORM
C * EXPANSION (NNNN) *
C
C ALSO INSERT APPROPRIATE DATA STATEMENTS TO DEFINE CONSTANTS
C WHICH DEPEND ON THE RANGE OF NUMBERS REPRESENTED BY THE
C MACHINE, RATHER THAN THE PRECISION (SUITABLE STATEMENTS FOR
C SOME MACHINES ARE CONTAINED IN COMMENTS BEGINNING CRD WHERE
C D IS A DIGIT WHICH SIMPLY DISTINGUISHES A GROUP OF MACHINES).
C DELETE THE ILLEGAL DUMMY DATA STATEMENTS WITH VALUES WRITTEN
C *VALUE*
C
C **************************************************************
S13ACF
DOUBLE PRECISION FUNCTION S13ACF(X,IFAIL)
C MARK 5A REVISED - NAG COPYRIGHT 1976
C MARK 5C REVISED
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C COSINE INTEGRAL CI(X)
C
S13ADF
DOUBLE PRECISION FUNCTION S13ADF(X,IFAIL)
C MARK 5A REVISED - NAG COPYRIGHT 1976
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 14 REVISED. IER-752 (DEC 1989).
C SINE INTEGRAL SI(X)
C
S14AAF
DOUBLE PRECISION FUNCTION S14AAF(X,IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 7C REVISED IER-184 (MAY 1979)
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C GAMMA FUNCTION
C
C **************************************************************
C
C TO EXTRACT THE CORRECT CODE FOR A PARTICULAR MACHINE-RANGE,
C ACTIVATE THE STATEMENTS CONTAINED IN COMMENTS BEGINNING CDD ,
C WHERE DD IS THE APPROXIMATE NUMBER OF SIGNIFICANT DECIMAL
C DIGITS REPRESENTED BY THE MACHINE
C DELETE THE ILLEGAL DUMMY STATEMENTS OF THE FORM
C * EXPANSION (NNNN) *
C
C ALSO INSERT APPROPRIATE DATA STATEMENTS TO DEFINE CONSTANTS
C WHICH DEPEND ON THE RANGE OF NUMBERS REPRESENTED BY THE
C MACHINE, RATHER THAN THE PRECISION (SUITABLE STATEMENTS FOR
C SOME MACHINES ARE CONTAINED IN COMMENTS BEGINNING CRD WHERE
C D IS A DIGIT WHICH SIMPLY DISTINGUISHES A GROUP OF MACHINES).
C DELETE THE ILLEGAL DUMMY DATA STATEMENTS WITH VALUES WRITTEN
C *VALUE*
C
C **************************************************************
C
S14ABF
DOUBLE PRECISION FUNCTION S14ABF(X,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C LNGAMMA(X) FUNCTION
C ABRAMOWITZ AND STEGUN CH.6
C
C **************************************************************
C
C TO EXTRACT THE CORRECT CODE FOR A PARTICULAR MACHINE-RANGE,
C ACTIVATE THE STATEMENTS CONTAINED IN COMMENTS BEGINNING CDD ,
C WHERE DD IS THE APPROXIMATE NUMBER OF SIGNIFICANT DECIMAL
C DIGITS REPRESENTED BY THE MACHINE
C DELETE THE ILLEGAL DUMMY STATEMENTS OF THE FORM
C * EXPANSION (NNNN) *
C
C ALSO INSERT APPROPRIATE DATA STATEMENTS TO DEFINE CONSTANTS
C WHICH DEPEND ON THE RANGE OF NUMBERS REPRESENTED BY THE
C MACHINE, RATHER THAN THE PRECISION (SUITABLE STATEMENTS FOR
C SOME MACHINES ARE CONTAINED IN COMMENTS BEGINNING CRD WHERE
C D IS A DIGIT WHICH SIMPLY DISTINGUISHES A GROUP OF MACHINES).
C DELETE THE ILLEGAL DUMMY DATA STATEMENTS WITH VALUES WRITTEN
C *VALUE*
C
C **************************************************************
C
C IMPLEMENTATION DEPENDENT CONSTANTS
C
C IF(X.LT.XSMALL)GAMMA(X)=1/X
C I.E. XSMALL*EULGAM.LE.XRELPR
C LNGAM(XVBIG)=GBIG.LE.XOVFLO
C LNR2PI=LN(SQRT(2*PI))
C IF(X.GT.XBIG)LNGAM(X)=(X-0.5)LN(X)-X+LNR2PI
C
S14ACF
DOUBLE PRECISION FUNCTION S14ACF(X,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C Driver routine for S14ACZ.
C Returns psi(X) - log(X).
S14ADF
SUBROUTINE S14ADF(X,N,M,W,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C Driver routine for S14ACZ.
C Returns w(k,x) = ( (-1)**(k+1) * psi(x) SUP (k) ) / k! ,
C where psi(x) SUP (k) is the kth derivative of psi(x),
C for k = N, N+1, ... N + M - 1.
S14BAF
SUBROUTINE S14BAF(A,X,TOL,P,Q,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
S15ABF
DOUBLE PRECISION FUNCTION S15ABF(X,IFAIL)
C MARK 5A REVISED - NAG COPYRIGHT 1976
C MARK 5C REVISED
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 14 REVISED. IER-753 (DEC 1989).
C CUMULATIVE NORMAL DISTRIBUTION P(X)
C ******************************************************************
C TO SELECT THE CORRECT VALUE FOR A PARTICULAR MACHINE-RANGE,
C ACTIVATE THE STATEMENT CONTAINED IN A COMMENT BEGINNING CDD ,
C WHERE DD IS THE APPROXIMATE NUMBER OF SIGNIFICANT DECIMAL
C DIGITS REPRESENTED BY THE MACHINE.
C DELETE THE ILLEGAL DUMMY STATEMENT OF THE FORM
C * EXPANSION (DATA) *
C ******************************************************************
C
S15ACF
DOUBLE PRECISION FUNCTION S15ACF(X,IFAIL)
C MARK 5A REVISED - NAG COPYRIGHT 1976
C MARK 5C REVISED
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 14 REVISED. IER-754 (DEC 1989).
C COMPLEMENT OF CUMULATIVE NORMAL DISTRIBUTION Q(X)
C ******************************************************************
C TO SELECT THE CORRECT VALUE FOR A PARTICULAR MACHINE-RANGE,
C ACTIVATE THE STATEMENT CONTAINED IN A COMMENT BEGINNING CDD ,
C WHERE DD IS THE APPROXIMATE NUMBER OF SIGNIFICANT DECIMAL
C DIGITS REPRESENTED BY THE MACHINE.
C DELETE THE ILLEGAL DUMMY STATEMENT OF THE FORM
C * EXPANSION (DATA) *
C ******************************************************************
C
S15ADF
DOUBLE PRECISION FUNCTION S15ADF(X,IFAIL)
C MARK 5A REVISED - NAG COPYRIGHT 1976
C MARK 5C REVISED
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C COMPLEMENT OF ERROR FUNCTION ERFC(X)
C
S15AEF
DOUBLE PRECISION FUNCTION S15AEF(X,IFAIL)
C MARK 4 RELEASE NAG COPYRIGHT 1974.
C MARK 4.5 REVISED
C MARK 8 REVISED. IER-221 (MAR 1980)
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 14 REVISED. IER-755 (DEC 1989).
C ERF(X)
C
C **************************************************************
C
C TO EXTRACT THE CORRECT CODE FOR A PARTICULAR MACHINE-RANGE,
C ACTIVATE THE STATEMENTS CONTAINED IN COMMENTS BEGINNING CDD ,
C WHERE DD IS THE APPROXIMATE NUMBER OF SIGNIFICANT DECIMAL
C DIGITS REPRESENTED BY THE MACHINE
C DELETE THE ILLEGAL DUMMY STATEMENT OF THE FORM
C * EXPANSION (DATA) *
C
C **************************************************************
S15AFF
DOUBLE PRECISION FUNCTION S15AFF(X,IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 14 REVISED. IER-756 (DEC 1989).
C DAWSONS INTEGRAL
C
C **************************************************************
C
C TO EXTRACT THE CORRECT CODE FOR A PARTICULAR MACHINE-RANGE,
C ACTIVATE THE STATEMENTS CONTAINED IN COMMENTS BEGINNING CDD ,
C WHERE DD IS THE APPROXIMATE NUMBER OF SIGNIFICANT DECIMAL
C DIGITS REPRESENTED BY THE MACHINE
C DELETE THE ILLEGAL DUMMY STATEMENTS OF THE FORM
C * EXPANSION (NNNN) *
C
C ALSO INSERT APPROPRIATE DATA STATEMENTS TO DEFINE CONSTANTS
C WHICH DEPEND ON THE RANGE OF NUMBERS REPRESENTED BY THE
C MACHINE, RATHER THAN THE PRECISION (SUITABLE STATEMENTS FOR
C SOME MACHINES ARE CONTAINED IN COMMENTS BEGINNING CRD WHERE
C D IS A DIGIT WHICH SIMPLY DISTINGUISHES A GROUP OF MACHINES).
C DELETE THE ILLEGAL DUMMY DATA STATEMENTS WITH VALUES WRITTEN
C *VALUE*
C
C **************************************************************
C
S15DDF
COMPLEX*16 FUNCTION S15DDF(Z,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C Evaluates the function w(z) = exp(-z*z)*erfc(-iz), for
C complex z. Derived from ACM Algorithm 363. See W.Gautschi,
C Efficient Computation of the Complex Error Function, SIAM
C J. Numer. Anal. 7, 1970, pp 187-198.
S17ACF
DOUBLE PRECISION FUNCTION S17ACF(X,IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 7C REVISED IER-185 (MAY 1979)
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C BESSEL FUNCTION Y0(X)
C
C **************************************************************
C
C TO EXTRACT THE CORRECT CODE FOR A PARTICULAR MACHINE-RANGE,
C ACTIVATE THE STATEMENTS CONTAINED IN COMMENTS BEGINNING CDD ,
C WHERE DD IS THE APPROXIMATE NUMBER OF SIGNIFICANT DECIMAL
C DIGITS REPRESENTED BY THE MACHINE
C DELETE THE ILLEGAL DUMMY STATEMENTS OF THE FORM
C * EXPANSION (NNNN) *
C
C NOTE - THE VALUE OF THE CONSTANT XBIG (DEFINED IN A DATA
C STATEMENT) SHOULD BE REDUCED IF NECESSARY TO ENSURE THAT SIN
C AND COS WILL RETURN A RESULT WITHOUT AN EXECUTION ERROR FOR
C ALL ARGUMENTS X SUCH THAT ABS(X) .LE. XBIG
C
C **************************************************************
C
S17ADF
DOUBLE PRECISION FUNCTION S17ADF(X,IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 7C REVISED IER-185 (MAY 1979)
C MARK 7D REVISED IER-196 (JUL 1979)
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 14 REVISED. IER-757 (DEC 1989).
C BESSEL FUNCTION Y1(X)
C
C **************************************************************
C
C TO EXTRACT THE CORRECT CODE FOR A PARTICULAR MACHINE-RANGE,
C ACTIVATE THE STATEMENTS CONTAINED IN COMMENTS BEGINNING CDD ,
C WHERE DD IS THE APPROXIMATE NUMBER OF SIGNIFICANT DECIMAL
C DIGITS REPRESENTED BY THE MACHINE
C DELETE THE ILLEGAL DUMMY STATEMENTS OF THE FORM
C * EXPANSION (NNNN) *
C
C ALSO INSERT APPROPRIATE DATA STATEMENTS TO DEFINE CONSTANTS
C WHICH DEPEND ON THE RANGE OF NUMBERS REPRESENTED BY THE
C MACHINE, RATHER THAN THE PRECISION (SUITABLE STATEMENTS FOR
C SOME MACHINES ARE CONTAINED IN COMMENTS BEGINNING CRD WHERE
C D IS A DIGIT WHICH SIMPLY DISTINGUISHES A GROUP OF MACHINES).
C DELETE THE ILLEGAL DUMMY DATA STATEMENTS WITH VALUES WRITTEN
C *VALUE*
C
C NOTE - THE VALUE OF THE CONSTANT XBIG (DEFINED IN A DATA
C STATEMENT) SHOULD BE REDUCED IF NECESSARY TO ENSURE THAT SIN
C AND COS WILL RETURN A RESULT WITHOUT AN EXECUTION ERROR FOR
C ALL ARGUMENTS X SUCH THAT ABS(X) .LE. XBIG
C
C **************************************************************
C
S17AEF
DOUBLE PRECISION FUNCTION S17AEF(X,IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 7C REVISED IER-185 (MAY 1979)
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C BESSEL FUNCTION J0(X)
C
C **************************************************************
C
C TO EXTRACT THE CORRECT CODE FOR A PARTICULAR MACHINE-RANGE,
C ACTIVATE THE STATEMENTS CONTAINED IN COMMENTS BEGINNING CDD ,
C WHERE DD IS THE APPROXIMATE NUMBER OF SIGNIFICANT DECIMAL
C DIGITS REPRESENTED BY THE MACHINE
C DELETE THE ILLEGAL DUMMY STATEMENTS OF THE FORM
C * EXPANSION (NNNN) *
C
C NOTE - THE VALUE OF THE CONSTANT XBIG (DEFINED IN A DATA
C STATEMENT) SHOULD BE REDUCED IF NECESSARY TO ENSURE THAT SIN
C AND COS WILL RETURN A RESULT WITHOUT AN EXECUTION ERROR FOR
C ALL ARGUMENTS X SUCH THAT ABS(X) .LE. XBIG
C
C **************************************************************
C
S17AFF
DOUBLE PRECISION FUNCTION S17AFF(X,IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 7C REVISED IER-185 (MAY 1979)
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 14 REVISED. IER-758 (DEC 1989).
C BESSEL FUNCTION J1(X)
C
C **************************************************************
C
C TO EXTRACT THE CORRECT CODE FOR A PARTICULAR MACHINE-RANGE,
C ACTIVATE THE STATEMENTS CONTAINED IN COMMENTS BEGINNING CDD ,
C WHERE DD IS THE APPROXIMATE NUMBER OF SIGNIFICANT DECIMAL
C DIGITS REPRESENTED BY THE MACHINE
C DELETE THE ILLEGAL DUMMY STATEMENTS OF THE FORM
C * EXPANSION (NNNN) *
C
C NOTE - THE VALUE OF THE CONSTANT XBIG (DEFINED IN A DATA
C STATEMENT) SHOULD BE REDUCED IF NECESSARY TO ENSURE THAT SIN
C AND COS WILL RETURN A RESULT WITHOUT AN EXECUTION ERROR FOR
C ALL ARGUMENTS X SUCH THAT ABS(X) .LE. XBIG
C
C **************************************************************
C
S17AGF
DOUBLE PRECISION FUNCTION S17AGF(X,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 8A REVISED. IER-263 (AUG 1980)
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C
C AIRY FUNCTION AI(X)
C
C **************************************************************
C
C TO EXTRACT THE CORRECT CODE FOR A PARTICULAR MACHINE-RANGE,
C ACTIVATE THE STATEMENTS CONTAINED IN COMMENTS BEGINNING CDD ,
C WHERE DD IS THE APPROXIMATE NUMBER OF SIGNIFICANT DECIMAL
C DIGITS REPRESENTED BY THE MACHINE
C DELETE THE ILLEGAL DUMMY STATEMENTS OF THE FORM
C * EXPANSION (NNNN) *
C
C ALSO INSERT APPROPRIATE DATA STATEMENTS TO DEFINE CONSTANTS
C WHICH DEPEND ON THE RANGE OF NUMBERS REPRESENTED BY THE
C MACHINE, RATHER THAN THE PRECISION (SUITABLE STATEMENTS FOR
C SOME MACHINES ARE CONTAINED IN COMMENTS BEGINNING CRD WHERE
C D IS A DIGIT WHICH SIMPLY DISTINGUISHES A GROUP OF MACHINES).
C DELETE THE ILLEGAL DUMMY DATA STATEMENTS WITH VALUES WRITTEN
C *VALUE*
C
C **************************************************************
C
S17AHF
DOUBLE PRECISION FUNCTION S17AHF(X,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 8A REVISED. IER-263 (AUG 1980)
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C
C AIRY FUNCTION BI(X)
C
C **************************************************************
C
C TO EXTRACT THE CORRECT CODE FOR A PARTICULAR MACHINE-RANGE,
C ACTIVATE THE STATEMENTS CONTAINED IN COMMENTS BEGINNING CDD ,
C WHERE DD IS THE APPROXIMATE NUMBER OF SIGNIFICANT DECIMAL
C DIGITS REPRESENTED BY THE MACHINE
C DELETE THE ILLEGAL DUMMY STATEMENTS OF THE FORM
C * EXPANSION (NNNN) *
C
C ALSO INSERT APPROPRIATE DATA STATEMENTS TO DEFINE CONSTANTS
C WHICH DEPEND ON THE RANGE OF NUMBERS REPRESENTED BY THE
C MACHINE, RATHER THAN THE PRECISION (SUITABLE STATEMENTS FOR
C SOME MACHINES ARE CONTAINED IN COMMENTS BEGINNING CRD WHERE
C D IS A DIGIT WHICH SIMPLY DISTINGUISHES A GROUP OF MACHINES).
C DELETE THE ILLEGAL DUMMY DATA STATEMENTS WITH VALUES WRITTEN
C *VALUE*
C
C **************************************************************
C
S17AJF
DOUBLE PRECISION FUNCTION S17AJF(X,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 8A REVISED. IER-263 (AUG 1980)
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C
C AIRY FUNCTION AIP(X)
C
C **************************************************************
C
C TO EXTRACT THE CORRECT CODE FOR A PARTICULAR MACHINE-RANGE,
C ACTIVATE THE STATEMENTS CONTAINED IN COMMENTS BEGINNING CDD ,
C WHERE DD IS THE APPROXIMATE NUMBER OF SIGNIFICANT DECIMAL
C DIGITS REPRESENTED BY THE MACHINE
C DELETE THE ILLEGAL DUMMY STATEMENTS OF THE FORM
C * EXPANSION (NNNN) *
C
C ALSO INSERT APPROPRIATE DATA STATEMENTS TO DEFINE CONSTANTS
C WHICH DEPEND ON THE RANGE OF NUMBERS REPRESENTED BY THE
C MACHINE, RATHER THAN THE PRECISION (SUITABLE STATEMENTS FOR
C SOME MACHINES ARE CONTAINED IN COMMENTS BEGINNING CRD WHERE
C D IS A DIGIT WHICH SIMPLY DISTINGUISHES A GROUP OF MACHINES).
C DELETE THE ILLEGAL DUMMY DATA STATEMENTS WITH VALUES WRITTEN
C *VALUE*
C
C **************************************************************
C
S17AKF
DOUBLE PRECISION FUNCTION S17AKF(X,IFAIL)
C MARK 8 RELEASE. NAG COPYRIGHT 1979.
C MARK 8A REVISED. IER-263 (AUG 1980)
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 13 REVISED. USE OF MARK 12 X02 FUNCTIONS (APR 1988).
C
C AIRY FUNCTION BIP(X)
C
C **************************************************************
C
C TO EXTRACT THE CORRECT CODE FOR A PARTICULAR MACHINE-RANGE,
C ACTIVATE THE STATEMENTS CONTAINED IN COMMENTS BEGINNING CDD ,
C WHERE DD IS THE APPROXIMATE NUMBER OF SIGNIFICANT DECIMAL
C DIGITS REPRESENTED BY THE MACHINE
C DELETE THE ILLEGAL DUMMY STATEMENTS OF THE FORM
C * EXPANSION (NNNN) *
C
C ALSO INSERT APPROPRIATE DATA STATEMENTS TO DEFINE CONSTANTS
C WHICH DEPEND ON THE RANGE OF NUMBERS REPRESENTED BY THE
C MACHINE, RATHER THAN THE PRECISION (SUITABLE STATEMENTS FOR
C SOME MACHINES ARE CONTAINED IN COMMENTS BEGINNING CRD WHERE
C D IS A DIGIT WHICH SIMPLY DISTINGUISHES A GROUP OF MACHINES).
C DELETE THE ILLEGAL DUMMY DATA STATEMENTS WITH VALUES WRITTEN
C *VALUE*
C
C **************************************************************
C
S17DCF
SUBROUTINE S17DCF(FNU,Z,N,SCALE,CY,NZ,CWRK,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C MARK 14 REVISED. IER-759 (DEC 1989).
C
C Original name: CBESY
C
C PURPOSE TO COMPUTE THE Y-BESSEL FUNCTION OF A COMPLEX ARGUMENT
C DESCRIPTION
C ===========
C
C ON SCALE='U', S17DCF COMPUTES AN N MEMBER SEQUENCE OF COMPLEX
C BESSEL FUNCTIONS CY(I)=Y(FNU+I-1,Z) FOR REAL, NONNEGATIVE
C ORDERS FNU+I-1, I=1,...,N AND COMPLEX Z IN THE CUT PLANE
C -PI.LT.ARG(Z).LE.PI. ON SCALE='S', S17DCF RETURNS THE SCALED
C FUNCTIONS
C
C CY(I)=EXP(-ABS(Y))*Y(FNU+I-1,Z) I = 1,...,N , Y=AIMAG(Z)
C
C WHICH REMOVE THE EXPONENTIAL GROWTH IN BOTH THE UPPER AND
C LOWER HALF PLANES FOR Z TO INFINITY. DEFINITIONS AND NOTATION
C ARE FOUND IN THE NBS HANDBOOK OF MATHEMATICAL FUNCTIONS
C (REF. 1).
C
C INPUT
C Z - Z=CMPLX(X,Y), Z.NE.CMPLX(0.,0.),-PI.LT.ARG(Z).LE.PI
C FNU - ORDER OF INITIAL Y FUNCTION, FNU.GE.0.0E0
C SCALE - A PARAMETER TO INDICATE THE SCALING OPTION
C SCALE= 'U' OR 'u' RETURNS
C CY(I)=Y(FNU+I-1,Z), I=1,...,N
C = 'S' OR 's' RETURNS
C CY(I)=Y(FNU+I-1,Z)*EXP(-ABS(Y)), I=1,...,N
C WHERE Y=AIMAG(Z)
C N - NUMBER OF MEMBERS OF THE SEQUENCE, N.GE.1
C CWRK - A COMPLEX WORK VECTOR OF DIMENSION AT LEAST N
C
C OUTPUT
C CY - A COMPLEX VECTOR WHOSE FIRST N COMPONENTS CONTAIN
C VALUES FOR THE SEQUENCE
C CY(I)=Y(FNU+I-1,Z) OR
C CY(I)=Y(FNU+I-1,Z)*EXP(-ABS(Y)) I=1,...,N
C DEPENDING ON PARAMETER SCALE.
C NZ - NZ=0 , A NORMAL RETURN
C NZ.GT.0 , NZ COMPONENTS OF CY SET TO ZERO DUE TO
C UNDERFLOW (GENERALLY ON SCALE='S')
C IFAIL - ERROR FLAG
C IFAIL=0, NORMAL RETURN - COMPUTATION COMPLETED
C IFAIL=1, INPUT ERROR - NO COMPUTATION
C IFAIL=2, OVERFLOW - NO COMPUTATION, CABS(Z) IS
C TOO SMALL
C IFAIL=3, OVERFLOW - NO COMPUTATION, FNU+N-1 IS
C TOO LARGE
C IFAIL=4, CABS(Z) OR FNU+N-1 LARGE - COMPUTATION DON
C BUT LOSSES OF SIGNIFICANCE BY ARGUMENT
C REDUCTION PRODUCE LESS THAN HALF OF MACHIN
C ACCURACY
C IFAIL=5, CABS(Z) OR FNU+N-1 TOO LARGE - NO COMPUTA-
C TION BECAUSE OF COMPLETE LOSSES OF SIGNIFI
C CANCE BY ARGUMENT REDUCTION
C IFAIL=6, ERROR - NO COMPUTATION,
C ALGORITHM TERMINATION CONDITION NOT MET
C
C LONG DESCRIPTION
C ================
C
C THE COMPUTATION IS CARRIED OUT BY THE FORMULA
C
C Y(FNU,Z)=0.5*(H(1,FNU,Z)-H(2,FNU,Z))/I
C
C WHERE I**2 = -1 AND THE HANKEL BESSEL FUNCTIONS H(1,FNU,Z)
C AND H(2,FNU,Z) ARE CALCULATED IN S17DLF.
C
C FOR NEGATIVE ORDERS,THE FORMULA
C
C Y(-FNU,Z) = Y(FNU,Z)*COS(PI*FNU) + J(FNU,Z)*SIN(PI*FNU)
C
C CAN BE USED. HOWEVER,FOR LARGE ORDERS CLOSE TO HALF ODD
C INTEGERS THE FUNCTION CHANGES RADICALLY. WHEN FNU IS A LARGE
C POSITIVE HALF ODD INTEGER,THE MAGNITUDE OF Y(-FNU,Z)=J(FNU,Z)*
C SIN(PI*FNU) IS A LARGE NEGATIVE POWER OF TEN. BUT WHEN FNU IS
C NOT A HALF ODD INTEGER, Y(FNU,Z) DOMINATES IN MAGNITUDE WITH A
C LARGE POSITIVE POWER OF TEN AND THE MOST THAT THE SECOND TERM
C CAN BE REDUCED IS BY UNIT ROUNDOFF FROM THE COEFFICIENT. THUS,
C WIDE CHANGES CAN OCCUR WITHIN UNIT ROUNDOFF OF A LARGE HALF
C ODD INTEGER. HERE, LARGE MEANS FNU.GT.CABS(Z).
C
C IN MOST COMPLEX VARIABLE COMPUTATION, ONE MUST EVALUATE ELE-
C MENTARY FUNCTIONS. WHEN THE MAGNITUDE OF Z OR FNU+N-1 IS
C LARGE, LOSSES OF SIGNIFICANCE BY ARGUMENT REDUCTION OCCUR.
C CONSEQUENTLY, IF EITHER ONE EXCEEDS U1=SQRT(0.5/UR), THEN
C LOSSES EXCEEDING HALF PRECISION ARE LIKELY AND AN ERROR FLAG
C IFAIL=4 IS TRIGGERED WHERE UR=X02AJF()=UNIT ROUNDOFF. ALSO
C IF EITHER IS LARGER THAN U2=0.5/UR, THEN ALL SIGNIFICANCE IS
C LOST AND IFAIL=5. IN ORDER TO USE THE INT FUNCTION, ARGUMENTS
C MUST BE FURTHER RESTRICTED NOT TO EXCEED THE LARGEST MACHINE
C INTEGER, U3=X02BBF(). THUS, THE MAGNITUDE OF Z AND FNU+N-1 IS
C RESTRICTED BY MIN(U2,U3). ON 32 BIT MACHINES, U1,U2, AND U3
C ARE APPROXIMATELY 2.0E+3, 4.2E+6, 2.1E+9 IN SINGLE PRECISION
C ARITHMETIC AND 1.3E+8, 1.8E+16, 2.1E+9 IN DOUBLE PRECISION
C ARITHMETIC RESPECTIVELY. THIS MAKES U2 AND U3 LIMITING IN
C THEIR RESPECTIVE ARITHMETICS. THIS MEANS THAT ONE CAN EXPECT
C TO RETAIN, IN THE WORST CASES ON 32 BIT MACHINES, NO DIGITS
C IN SINGLE AND ONLY 7 DIGITS IN DOUBLE PRECISION ARITHMETIC.
C SIMILAR CONSIDERATIONS HOLD FOR OTHER MACHINES.
C
C THE APPROXIMATE RELATIVE ERROR IN THE MAGNITUDE OF A COMPLEX
C BESSEL FUNCTION CAN BE EXPRESSED BY P*10**S WHERE P=MAX(UNIT
C ROUNDOFF,1.0E-18) IS THE NOMINAL PRECISION AND 10**S REPRE-
C SENTS THE INCREASE IN ERROR DUE TO ARGUMENT REDUCTION IN THE
C ELEMENTARY FUNCTIONS. HERE, S=MAX(1,ABS(LOG10(CABS(Z))),
C ABS(LOG10(FNU))) APPROXIMATELY (I.E. S=MAX(1,ABS(EXPONENT OF
C CABS(Z),ABS(EXPONENT OF FNU)) ). HOWEVER, THE PHASE ANGLE MAY
C HAVE ONLY ABSOLUTE ACCURACY. THIS IS MOST LIKELY TO OCCUR WHEN
C ONE COMPONENT (IN ABSOLUTE VALUE) IS LARGER THAN THE OTHER BY
C SEVERAL ORDERS OF MAGNITUDE. IF ONE COMPONENT IS 10**K LARGER
C THAN THE OTHER, THEN ONE CAN EXPECT ONLY MAX(ABS(LOG10(P))-K,
C 0) SIGNIFICANT DIGITS; OR, STATED ANOTHER WAY, WHEN K EXCEEDS
C THE EXPONENT OF P, NO SIGNIFICANT DIGITS REMAIN IN THE SMALLER
C COMPONENT. HOWEVER, THE PHASE ANGLE RETAINS ABSOLUTE ACCURACY
C BECAUSE, IN COMPLEX ARITHMETIC WITH PRECISION P, THE SMALLER
C COMPONENT WILL NOT (AS A RULE) DECREASE BELOW P TIMES THE
C MAGNITUDE OF THE LARGER COMPONENT. IN THESE EXTREME CASES,
C THE PRINCIPAL PHASE ANGLE IS ON THE ORDER OF +P, -P, PI/2-P,
C OR -PI/2+P.
C
C REFERENCES
C ==========
C HANDBOOK OF MATHEMATICAL FUNCTIONS BY M. ABRAMOWITZ
C AND I. A. STEGUN, NBS AMS SERIES 55, U.S. DEPT. OF
C COMMERCE, 1955.
C
C COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT
C BY D. E. AMOS, SAND83-0083, MAY, 1983.
C
C COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT
C AND LARGE ORDER BY D. E. AMOS, SAND83-0643, MAY, 1983
C
C A SUBROUTINE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX
C ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, SAND85-
C 1018, MAY, 1985
C
C A PORTABLE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX
C ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, TRANS.
C MATH. SOFTWARE, 1986
C
C DATE WRITTEN 830501 (YYMMDD)
C REVISION DATE 830501 (YYMMDD)
C AUTHOR AMOS, DONALD E., SANDIA NATIONAL LABORATORIES
C
S17DEF
SUBROUTINE S17DEF(FNU,Z,N,SCALE,CY,NZ,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C MARK 14 REVISED. IER-760 (DEC 1989).
C
C Original name: CBESJ
C
C PURPOSE TO COMPUTE THE J-BESSEL FUNCTION OF A COMPLEX ARGUMENT
C
C DESCRIPTION
C ===========
C
C ON SCALE='U', S17DEF COMPUTES AN N MEMBER SEQUENCE OF COMPLEX
C BESSEL FUNCTIONS CY(I)=J(FNU+I-1,Z) FOR REAL, NONNEGATIVE
C ORDERS FNU+I-1, I=1,...,N AND COMPLEX Z IN THE CUT PLANE
C -PI.LT.ARG(Z).LE.PI. ON SCALE='S', S17DEF RETURNS THE SCALED
C FUNCTIONS
C
C CY(I)=EXP(-ABS(Y))*J(FNU+I-1,Z) I = 1,...,N , Y=AIMAG(Z)
C
C WHICH REMOVE THE EXPONENTIAL GROWTH IN BOTH THE UPPER AND
C LOWER HALF PLANES FOR Z TO INFINITY. DEFINITIONS AND NOTATION
C ARE FOUND IN THE NBS HANDBOOK OF MATHEMATICAL FUNCTIONS
C (REF. 1).
C
C INPUT
C Z - Z=CMPLX(X,Y), -PI.LT.ARG(Z).LE.PI
C FNU - ORDER OF INITIAL J FUNCTION, FNU.GE.0.0E0
C SCALE - A PARAMETER TO INDICATE THE SCALING OPTION
C SCALE= 'U' OR 'u' RETURNS
C CY(I)=J(FNU+I-1,Z), I=1,...,N
C = 'S' OR 's' RETURNS
C CY(I)=J(FNU+I-1,Z)*EXP(-ABS(Y)), I=1,...
C N - NUMBER OF MEMBERS OF THE SEQUENCE, N.GE.1
C
C OUTPUT
C CY - A COMPLEX VECTOR WHOSE FIRST N COMPONENTS CONTAIN
C VALUES FOR THE SEQUENCE
C CY(I)=J(FNU+I-1,Z) OR
C CY(I)=J(FNU+I-1,Z)*EXP(-ABS(Y)) I=1,...,N
C DEPENDING ON SCALE, Y=AIMAG(Z).
C NZ - NUMBER OF COMPONENTS SET TO ZERO DUE TO UNDERFLOW,
C NZ= 0 , NORMAL RETURN
C NZ.GT.0 , LAST NZ COMPONENTS OF CY SET TO ZERO
C DUE TO UNDERFLOW, CY(I)=CMPLX(0.0,0.0),
C I = N-NZ+1,...,N
C IFAIL - ERROR FLAG
C IFAIL=0, NORMAL RETURN - COMPUTATION COMPLETED
C IFAIL=1, INPUT ERROR - NO COMPUTATION
C IFAIL=2, OVERFLOW - NO COMPUTATION, AIMAG(Z)
C TOO LARGE ON SCALE='U'
C IFAIL=3, CABS(Z) OR FNU+N-1 LARGE - COMPUTATION DONE
C BUT LOSSES OF SIGNIFCANCE BY ARGUMENT
C REDUCTION PRODUCE LESS THAN HALF OF MACHINE
C ACCURACY
C IFAIL=4, CABS(Z) OR FNU+N-1 TOO LARGE - NO COMPUTA-
C TION BECAUSE OF COMPLETE LOSSES OF SIGNIFI-
C CANCE BY ARGUMENT REDUCTION
C IFAIL=5, ERROR - NO COMPUTATION,
C ALGORITHM TERMINATION CONDITION NOT MET
C
C LONG DESCRIPTION
C ================
C
C THE COMPUTATION IS CARRIED OUT BY THE FORMULA
C
C J(FNU,Z)=EXP( FNU*PI*I/2)*I(FNU,-I*Z) AIMAG(Z).GE.0.0
C
C J(FNU,Z)=EXP(-FNU*PI*I/2)*I(FNU, I*Z) AIMAG(Z).LT.0.0
C
C WHERE I**2 = -1 AND I(FNU,Z) IS THE I BESSEL FUNCTION.
C
C FOR NEGATIVE ORDERS,THE FORMULA
C
C J(-FNU,Z) = J(FNU,Z)*COS(PI*FNU) - Y(FNU,Z)*SIN(PI*FNU)
C
C CAN BE USED. HOWEVER,FOR LARGE ORDERS CLOSE TO INTEGERS, THE
C THE FUNCTION CHANGES RADICALLY. WHEN FNU IS A LARGE POSITIVE
C INTEGER,THE MAGNITUDE OF J(-FNU,Z)=J(FNU,Z)*COS(PI*FNU) IS A
C LARGE NEGATIVE POWER OF TEN. BUT WHEN FNU IS NOT AN INTEGER,
C Y(FNU,Z) DOMINATES IN MAGNITUDE WITH A LARGE POSITIVE POWER OF
C TEN AND THE MOST THAT THE SECOND TERM CAN BE REDUCED IS BY
C UNIT ROUNDOFF FROM THE COEFFICIENT. THUS, WIDE CHANGES CAN
C OCCUR WITHIN UNIT ROUNDOFF OF A LARGE INTEGER FOR FNU. HERE,
C LARGE MEANS FNU.GT.CABS(Z).
C
C IN MOST COMPLEX VARIABLE COMPUTATION, ONE MUST EVALUATE ELE-
C MENTARY FUNCTIONS. WHEN THE MAGNITUDE OF Z OR FNU+N-1 IS
C LARGE, LOSSES OF SIGNIFICANCE BY ARGUMENT REDUCTION OCCUR.
C CONSEQUENTLY, IF EITHER ONE EXCEEDS U1=SQRT(0.5/UR), THEN
C LOSSES EXCEEDING HALF PRECISION ARE LIKELY AND AN ERROR FLAG
C IFAIL=3 IS TRIGGERED WHERE UR=X02AJF()=UNIT ROUNDOFF. ALSO
C IF EITHER IS LARGER THAN U2=0.5/UR, THEN ALL SIGNIFICANCE IS
C LOST AND IFAIL=4. IN ORDER TO USE THE INT FUNCTION, ARGUMENTS
C MUST BE FURTHER RESTRICTED NOT TO EXCEED THE LARGEST MACHINE
C INTEGER, U3=X02BBF(). THUS, THE MAGNITUDE OF Z AND FNU+N-1 IS
C RESTRICTED BY MIN(U2,U3). ON 32 BIT MACHINES, U1,U2, AND U3
C ARE APPROXIMATELY 2.0E+3, 4.2E+6, 2.1E+9 IN SINGLE PRECISION
C ARITHMETIC AND 1.3E+8, 1.8E+16, 2.1E+9 IN DOUBLE PRECISION
C ARITHMETIC RESPECTIVELY. THIS MAKES U2 AND U3 LIMITING IN
C THEIR RESPECTIVE ARITHMETICS. THIS MEANS THAT ONE CAN EXPECT
C TO RETAIN, IN THE WORST CASES ON 32 BIT MACHINES, NO DIGITS
C IN SINGLE AND ONLY 7 DIGITS IN DOUBLE PRECISION ARITHMETIC.
C SIMILAR CONSIDERATIONS HOLD FOR OTHER MACHINES.
C
C THE APPROXIMATE RELATIVE ERROR IN THE MAGNITUDE OF A COMPLEX
C BESSEL FUNCTION CAN BE EXPRESSED BY P*10**S WHERE P=MAX(UNIT
C ROUNDOFF,1.0E-18) IS THE NOMINAL PRECISION AND 10**S REPRE-
C SENTS THE INCREASE IN ERROR DUE TO ARGUMENT REDUCTION IN THE
C ELEMENTARY FUNCTIONS. HERE, S=MAX(1,ABS(LOG10(CABS(Z))),
C ABS(LOG10(FNU))) APPROXIMATELY (I.E. S=MAX(1,ABS(EXPONENT OF
C CABS(Z),ABS(EXPONENT OF FNU)) ). HOWEVER, THE PHASE ANGLE MAY
C HAVE ONLY ABSOLUTE ACCURACY. THIS IS MOST LIKELY TO OCCUR WHEN
C ONE COMPONENT (IN ABSOLUTE VALUE) IS LARGER THAN THE OTHER BY
C SEVERAL ORDERS OF MAGNITUDE. IF ONE COMPONENT IS 10**K LARGER
C THAN THE OTHER, THEN ONE CAN EXPECT ONLY MAX(ABS(LOG10(P))-K,
C 0) SIGNIFICANT DIGITS; OR, STATED ANOTHER WAY, WHEN K EXCEEDS
C THE EXPONENT OF P, NO SIGNIFICANT DIGITS REMAIN IN THE SMALLER
C COMPONENT. HOWEVER, THE PHASE ANGLE RETAINS ABSOLUTE ACCURACY
C BECAUSE, IN COMPLEX ARITHMETIC WITH PRECISION P, THE SMALLER
C COMPONENT WILL NOT (AS A RULE) DECREASE BELOW P TIMES THE
C MAGNITUDE OF THE LARGER COMPONENT. IN THESE EXTREME CASES,
C THE PRINCIPAL PHASE ANGLE IS ON THE ORDER OF +P, -P, PI/2-P,
C OR -PI/2+P.
C
C REFERENCES
C ==========
C HANDBOOK OF MATHEMATICAL FUNCTIONS BY M. ABRAMOWITZ
C AND I. A. STEGUN, NBS AMS SERIES 55, U.S. DEPT. OF
C COMMERCE, 1955.
C
C COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT
C BY D. E. AMOS, SAND83-0083, MAY, 1983.
C
C COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT
C AND LARGE ORDER BY D. E. AMOS, SAND83-0643, MAY, 1983
C
C A SUBROUTINE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX
C ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, SAND85-
C 1018, MAY, 1985
C
C A PORTABLE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX
C ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, TRANS.
C MATH. SOFTWARE, 1986
C
C DATE WRITTEN 830501 (YYMMDD)
C REVISION DATE 830501 (YYMMDD)
C AUTHOR AMOS, DONALD E., SANDIA NATIONAL LABORATORIES
C
S17DGF
SUBROUTINE S17DGF(DERIV,Z,SCALE,AI,NZ,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C MARK 14 REVISED. IER-770 (DEC 1989).
C
C Original name: CAIRY
C
C PURPOSE TO COMPUTE AIRY FUNCTIONS AI(Z) AND DAI(Z) FOR COMPLEX Z
C
C DESCRIPTION
C ===========
C
C ON SCALE='U', S17DGF COMPUTES THE COMPLEX AIRY FUNCTION AI(Z)
C OR ITS DERIVATIVE DAI(Z)/DZ ON DERIV='F' OR DERIV='D'
C RESPECTIVELY. ON SCALE='S', A SCALING OPTION
C CEXP(ZTA)*AI(Z) OR CEXP(ZTA)*DAI(Z)/DZ IS PROVIDED TO REMOVE
C THE EXPONENTIAL DECAY IN -PI/3.LT.ARG(Z).LT.PI/3 AND THE
C EXPONENTIAL GROWTH IN PI/3.LT.ABS(ARG(Z)).LT.PI WHERE
C ZTA=(2/3)*Z*CSQRT(Z)
C
C WHILE THE AIRY FUNCTIONS AI(Z) AND DAI(Z)/DZ ARE ANALYTIC IN
C THE WHOLE Z PLANE, THE CORRESPONDING SCALED FUNCTIONS DEFINED
C FOR SCALE='S' HAVE A CUT ALONG THE NEGATIVE REAL AXIS.
C DEFINITIONS AND NOTATION ARE FOUND IN THE NBS HANDBOOK OF
C MATHEMATICAL FUNCTIONS (REF. 1).
C
C INPUT
C Z - Z=CMPLX(X,Y)
C DERIV - RETURN FUNCTION (DERIV='F') OR DERIVATIVE
C (DERIV='D')
C SCALE - A PARAMETER TO INDICATE THE SCALING OPTION
C SCALE = 'U' OR 'u' RETURNS
C AI=AI(Z) ON DERIV='F' OR
C AI=DAI(Z)/DZ ON DERIV='D'
C SCALE = 'S' OR 's' RETURNS
C AI=CEXP(ZTA)*AI(Z) ON DERIV='F' OR
C AI=CEXP(ZTA)*DAI(Z)/DZ ON DERIV='D' WHERE
C ZTA=(2/3)*Z*CSQRT(Z)
C
C OUTPUT
C AI - COMPLEX ANSWER DEPENDING ON THE CHOICES FOR DERIV
C AND SCALE
C NZ - UNDERFLOW INDICATOR
C NZ= 0 , NORMAL RETURN
C NZ= 1 , AI=CMPLX(0.0,0.0) DUE TO UNDERFLOW IN
C -PI/3.LT.ARG(Z).LT.PI/3 ON SCALE='U'
C IFAIL - ERROR FLAG
C IFAIL=0, NORMAL RETURN - COMPUTATION COMPLETED
C IFAIL=1, INPUT ERROR - NO COMPUTATION
C IFAIL=2, OVERFLOW - NO COMPUTATION, REAL(ZTA)
C TOO LARGE WITH SCALE = 'U'
C IFAIL=3, CABS(Z) LARGE - COMPUTATION COMPLETED
C LOSSES OF SIGNIFCANCE BY ARGUMENT REDUCTION
C PRODUCE LESS THAN HALF OF MACHINE ACCURACY
C IFAIL=4, CABS(Z) TOO LARGE - NO COMPUTATION
C COMPLETE LOSS OF ACCURACY BY ARGUMENT
C REDUCTION
C IFAIL=5, ERROR - NO COMPUTATION,
C ALGORITHM TERMINATION CONDITION NOT MET
C
C LONG DESCRIPTION
C ================
C
C AI AND DAI ARE COMPUTED FOR CABS(Z).GT.1.0 FROM THE K BESSEL
C FUNCTIONS BY
C
C AI(Z)=C*SQRT(Z)*K(1/3,ZTA) , DAI(Z)=-C*Z*K(2/3,ZTA)
C C=1.0/(PI*SQRT(3.0))
C ZTA=(2/3)*Z**(3/2)
C
C WITH THE POWER SERIES FOR CABS(Z).LE.1.0.
C
C IN MOST COMPLEX VARIABLE COMPUTATION, ONE MUST EVALUATE ELE-
C MENTARY FUNCTIONS. WHEN THE MAGNITUDE OF Z IS LARGE, LOSSES
C OF SIGNIFICANCE BY ARGUMENT REDUCTION OCCUR. CONSEQUENTLY, IF
C THE MAGNITUDE OF ZETA=(2/3)*Z**1.5 EXCEEDS U1=SQRT(0.5/UR),
C THEN LOSSES EXCEEDING HALF PRECISION ARE LIKELY AND AN ERROR
C FLAG IFAIL=3 IS TRIGGERED WHERE UR=X02AJF()=UNIT ROUNDOFF.
C ALSO, IF THE MAGNITUDE OF ZETA IS LARGER THAN U2=0.5/UR, THEN
C ALL SIGNIFICANCE IS LOST AND IFAIL=4. IN ORDER TO USE THE INT
C FUNCTION, ZETA MUST BE FURTHER RESTRICTED NOT TO EXCEED THE
C LARGEST INTEGER, U3=X02BBF(). THUS, THE MAGNITUDE OF ZETA
C MUST BE RESTRICTED BY MIN(U2,U3). ON 32 BIT MACHINES, U1,U2,
C AND U3 ARE APPROXIMATELY 2.0E+3, 4.2E+6, 2.1E+9 IN SINGLE
C PRECISION ARITHMETIC AND 1.3E+8, 1.8E+16, 2.1E+9 IN DOUBLE
C PRECISION ARITHMETIC RESPECTIVELY. THIS MAKES U2 AND U3 LIMIT-
C ING IN THEIR RESPECTIVE ARITHMETICS. THIS MEANS THAT THE MAG-
C NITUDE OF Z CANNOT EXCEED 3.1E+4 IN SINGLE AND 2.1E+6 IN
C DOUBLE PRECISION ARITHMETIC. THIS ALSO MEANS THAT ONE CAN
C EXPECT TO RETAIN, IN THE WORST CASES ON 32 BIT MACHINES,
C NO DIGITS IN SINGLE PRECISION AND ONLY 7 DIGITS IN DOUBLE
C PRECISION ARITHMETIC. SIMILAR CONSIDERATIONS HOLD FOR OTHER
C MACHINES.
C
C THE APPROXIMATE RELATIVE ERROR IN THE MAGNITUDE OF A COMPLEX
C BESSEL FUNCTION CAN BE EXPRESSED BY P*10**S WHERE P=MAX(UNIT
C ROUNDOFF,1.0E-18) IS THE NOMINAL PRECISION AND 10**S REPRE-
C SENTS THE INCREASE IN ERROR DUE TO ARGUMENT REDUCTION IN THE
C ELEMENTARY FUNCTIONS. HERE, S=MAX(1,ABS(LOG10(CABS(Z))),
C ABS(LOG10(FNU))) APPROXIMATELY (I.E. S=MAX(1,ABS(EXPONENT OF
C CABS(Z),ABS(EXPONENT OF FNU)) ). HOWEVER, THE PHASE ANGLE MAY
C HAVE ONLY ABSOLUTE ACCURACY. THIS IS MOST LIKELY TO OCCUR WHEN
C ONE COMPONENT (IN ABSOLUTE VALUE) IS LARGER THAN THE OTHER BY
C SEVERAL ORDERS OF MAGNITUDE. IF ONE COMPONENT IS 10**K LARGER
C THAN THE OTHER, THEN ONE CAN EXPECT ONLY MAX(ABS(LOG10(P))-K,
C 0) SIGNIFICANT DIGITS; OR, STATED ANOTHER WAY, WHEN K EXCEEDS
C THE EXPONENT OF P, NO SIGNIFICANT DIGITS REMAIN IN THE SMALLER
C COMPONENT. HOWEVER, THE PHASE ANGLE RETAINS ABSOLUTE ACCURACY
C BECAUSE, IN COMPLEX ARITHMETIC WITH PRECISION P, THE SMALLER
C COMPONENT WILL NOT (AS A RULE) DECREASE BELOW P TIMES THE
C MAGNITUDE OF THE LARGER COMPONENT. IN THESE EXTREME CASES,
C THE PRINCIPAL PHASE ANGLE IS ON THE ORDER OF +P, -P, PI/2-P,
C OR -PI/2+P.
C
C REFERENCES
C ==========
C HANDBOOK OF MATHEMATICAL FUNCTIONS BY M. ABRAMOWITZ
C AND I. A. STEGUN, NBS AMS SERIES 55, U.S. DEPT. OF
C COMMERCE, 1955.
C
C COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT
C AND LARGE ORDER BY D. E. AMOS, SAND83-0643, MAY, 1983
C
C A SUBROUTINE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX
C ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, SAND85-
C 1018, MAY, 1985
C
C A PORTABLE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX
C ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, TRANS.
C MATH. SOFTWARE, 1986
C
C DATE WRITTEN 830501 (YYMMDD)
C REVISION DATE 830501 (YYMMDD)
C AUTHOR AMOS, DONALD E., SANDIA NATIONAL LABORATORIES
C
S17DHF
SUBROUTINE S17DHF(DERIV,Z,SCALE,BI,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C MARK 14 REVISED. IER-780 (DEC 1989).
C
C Original name: CBIRY
C
C PURPOSE TO COMPUTE AIRY FUNCTIONS BI(Z) AND DBI(Z) FOR COMPLEX Z
C DESCRIPTION
C ===========
C
C ON SCALE='U', S17DHF COMPUTES THE COMPLEX AIRY FUNCTION BI(Z)
C OR ITS DERIVATIVE DBI(Z)/DZ ON DERIV='F' OR DERIV='D'
C RESPECTIVELY. ON
C SCALE='S', A SCALING OPTION CEXP(-AXZTA)*BI(Z) OR CEXP(-AXZTA)
C *DBI(Z)/DZ IS PROVIDED TO REMOVE THE EXPONENTIAL BEHAVIOR IN
C BOTH THE LEFT AND RIGHT HALF PLANES WHERE
C ZTA=(2/3)*Z*CSQRT(Z)=CMPLX(XZTA,YZTA) AND AXZTA=ABS(XZTA).
C DEFINITIONS AND NOTATION ARE FOUND IN THE NBS HANDBOOK OF
C MATHEMATICAL FUNCTIONS (REF. 1).
C
C INPUT
C Z - Z=CMPLX(X,Y)
C DERIV - RETURN FUNCTION (DERIV='F') OR DERIVATIVE
C (DERIV='D')
C SCALE - A PARAMETER TO INDICATE THE SCALING OPTION
C SCALE = 'U' OR SCALE = 'u' RETURNS
C BI=BI(Z) ON DERIV='F' OR
C BI=DBI(Z)/DZ ON DERIV='D'
C SCALE = 'S' OR SCALE = 's' RETURNS
C BI=CEXP(-AXZTA)*BI(Z) ON DERIV='F' OR
C BI=CEXP(-AXZTA)*DBI(Z)/DZ ON DERIV='D' WHERE
C ZTA=(2/3)*Z*CSQRT(Z)=CMPLX(XZTA,YZTA)
C AND AXZTA=ABS(XZTA)
C OUTPUT
C BI - COMPLEX ANSWER DEPENDING ON THE CHOICES FOR DERIV
C AND SCALE
C IFAIL - ERROR FLAG
C IFAIL=0, NORMAL RETURN - COMPUTATION COMPLETED
C IFAIL=1, INPUT ERROR - NO COMPUTATION
C IFAIL=2, OVERFLOW - NO COMPUTATION, REAL(Z)
C TOO LARGE WITH SCALE = 'U'
C IFAIL=3, CABS(Z) LARGE - COMPUTATION COMPLETED
C LOSSES OF SIGNIFCANCE BY ARGUMENT REDUCTION
C PRODUCE LESS THAN HALF OF MACHINE ACCURACY
C IFAIL=4, CABS(Z) TOO LARGE - NO COMPUTATION
C COMPLETE LOSS OF ACCURACY BY ARGUMENT
C REDUCTION
C IFAIL=5, ERROR - NO COMPUTATION,
C ALGORITHM TERMINATION CONDITION NOT MET
C
C LONG DESCRIPTION
C ================
C
C BI AND DBI ARE COMPUTED FOR CABS(Z).GT.1.0 FROM THE I BESSEL
C FUNCTIONS BY
C
C BI(Z)=C*SQRT(Z)*( I(-1/3,ZTA) + I(1/3,ZTA) )
C DBI(Z)=C * Z * ( I(-2/3,ZTA) + I(2/3,ZTA) )
C C=1.0/SQRT(3.0)
C ZTA=(2/3)*Z**(3/2)
C
C WITH THE POWER SERIES FOR CABS(Z).LE.1.0.
C
C IN MOST COMPLEX VARIABLE COMPUTATION, ONE MUST EVALUATE ELE-
C MENTARY FUNCTIONS. WHEN THE MAGNITUDE OF Z IS LARGE, LOSSES
C OF SIGNIFICANCE BY ARGUMENT REDUCTION OCCUR. CONSEQUENTLY, IF
C THE MAGNITUDE OF ZETA=(2/3)*Z**1.5 EXCEEDS U1=SQRT(0.5/UR),
C THEN LOSSES EXCEEDING HALF PRECISION ARE LIKELY AND AN ERROR
C FLAG IFAIL=3 IS TRIGGERED WHERE UR=X02AJF()=UNIT ROUNDOFF.
C ALSO, IF THE MAGNITUDE OF ZETA IS LARGER THAN U2=0.5/UR, THEN
C ALL SIGNIFICANCE IS LOST AND IFAIL=4. IN ORDER TO USE THE INT
C FUNCTION, ZETA MUST BE FURTHER RESTRICTED NOT TO EXCEED THE
C LARGEST INTEGER, U3=X02BBF(). THUS, THE MAGNITUDE OF ZETA
C MUST BE RESTRICTED BY MIN(U2,U3). ON 32 BIT MACHINES, U1,U2,
C AND U3 ARE APPROXIMATELY 2.0E+3, 4.2E+6, 2.1E+9 IN SINGLE
C PRECISION ARITHMETIC AND 1.3E+8, 1.8E+16, 2.1E+9 IN DOUBLE
C PRECISION ARITHMETIC RESPECTIVELY. THIS MAKES U2 AND U3 LIMIT-
C ING IN THEIR RESPECTIVE ARITHMETICS. THIS MEANS THAT THE MAG-
C NITUDE OF Z CANNOT EXCEED 3.1E+4 IN SINGLE AND 2.1E+6 IN
C DOUBLE PRECISION ARITHMETIC. THIS ALSO MEANS THAT ONE CAN
C EXPECT TO RETAIN, IN THE WORST CASES ON 32 BIT MACHINES,
C NO DIGITS IN SINGLE PRECISION AND ONLY 7 DIGITS IN DOUBLE
C PRECISION ARITHMETIC.
C
C THE APPROXIMATE RELATIVE ERROR IN THE MAGNITUDE OF A COMPLEX
C BESSEL FUNCTION CAN BE EXPRESSED BY P*10**S WHERE P=MAX(UNIT
C ROUNDOFF,1.0E-18) IS THE NOMINAL PRECISION AND 10**S REPRE-
C SENTS THE INCREASE IN ERROR DUE TO ARGUMENT REDUCTION IN THE
C ELEMENTARY FUNCTIONS. HERE, S=MAX(1,ABS(LOG10(CABS(Z))),
C ABS(LOG10(FNU))) APPROXIMATELY (I.E. S=MAX(1,ABS(EXPONENT OF
C CABS(Z),ABS(EXPONENT OF FNU)) ). HOWEVER, THE PHASE ANGLE MAY
C HAVE ONLY ABSOLUTE ACCURACY. THIS IS MOST LIKELY TO OCCUR WHEN
C ONE COMPONENT (IN ABSOLUTE VALUE) IS LARGER THAN THE OTHER BY
C SEVERAL ORDERS OF MAGNITUDE. IF ONE COMPONENT IS 10**K LARGER
C THAN THE OTHER, THEN ONE CAN EXPECT ONLY MAX(ABS(LOG10(P))-K,
C 0) SIGNIFICANT DIGITS; OR, STATED ANOTHER WAY, WHEN K EXCEEDS
C THE EXPONENT OF P, NO SIGNIFICANT DIGITS REMAIN IN THE SMALLER
C COMPONENT. HOWEVER, THE PHASE ANGLE RETAINS ABSOLUTE ACCURACY
C BECAUSE, IN COMPLEX ARITHMETIC WITH PRECISION P, THE SMALLER
C COMPONENT WILL NOT (AS A RULE) DECREASE BELOW P TIMES THE
C MAGNITUDE OF THE LARGER COMPONENT. IN THESE EXTREME CASES,
C THE PRINCIPAL PHASE ANGLE IS ON THE ORDER OF +P, -P, PI/2-P,
C OR -PI/2+P.
C
C REFERENCES
C ==========
C HANDBOOK OF MATHEMATICAL FUNCTIONS BY M. ABRAMOWITZ
C AND I. A. STEGUN, NBS AMS SERIES 55, U.S. DEPT. OF
C COMMERCE, 1955.
C
C COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT
C AND LARGE ORDER BY D. E. AMOS, SAND83-0643, MAY, 1983
C
C A SUBROUTINE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX
C ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, SAND85-
C 1018, MAY, 1985
C
C A PORTABLE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX
C ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, TRANS.
C MATH. SOFTWARE, 1986
C
C DATE WRITTEN 830501 (YYMMDD)
C REVISION DATE 830501 (YYMMDD)
C AUTHOR AMOS, DONALD E., SANDIA NATIONAL LABORATORIES
C
S17DLF
SUBROUTINE S17DLF(M,FNU,Z,N,SCALE,CY,NZ,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C MARK 14 REVISED. IER-781 (DEC 1989).
C
C Original name: CBESH
C
C PURPOSE TO COMPUTE THE H-BESSEL FUNCTIONS OF A COMPLEX ARGUMENT
C
C DESCRIPTION
C ===========
C
C ON SCALE='U', S17DLF COMPUTES AN N MEMBER SEQUENCE OF COMPLEX
C HANKEL (BESSEL) FUNCTIONS CY(J)=H(M,FNU+J-1,Z) FOR KINDS M=1
C OR 2, REAL, NONNEGATIVE ORDERS FNU+J-1, J=1,...,N, AND COMPLEX
C Z.NE.CMPLX(0.0E0,0.0E0) IN THE CUT PLANE -PI.LT.ARG(Z).LE.PI.
C ON SCALE='S', S17DLF COMPUTES THE SCALED HANKEL FUNCTIONS
C
C CY(I)=H(M,FNU+J-1,Z)*EXP(-MM*Z*I) MM=3-2M, I**2=-1.
C
C WHICH REMOVES THE EXPONENTIAL BEHAVIOR IN BOTH THE UPPER
C AND LOWER HALF PLANES. DEFINITIONS AND NOTATION ARE FOUND IN
C THE NBS HANDBOOK OF MATHEMATICAL FUNCTIONS (REF. 1).
C
C INPUT
C Z - Z=CMPLX(X,Y), Z.NE.CMPLX(0.,0.),-PI.LT.ARG(Z).LE.PI
C FNU - ORDER OF INITIAL H FUNCTION, FNU.GE.0.0E0
C SCALE - A PARAMETER TO INDICATE THE SCALING OPTION
C SCALE = 'U' OR SCALE = 'u' RETURNS
C CY(J)=H(M,FNU+J-1,Z), J=1,...,N
C = 'S' OR SCALE = 's' RETURNS
C CY(J)=H(M,FNU+J-1,Z)*EXP(-I*Z*(3-2M))
C J=1,...,N , I**2=-1
C M - KIND OF HANKEL FUNCTION, M=1 OR 2
C N - NUMBER OF MEMBERS OF THE SEQUENCE, N.GE.1
C
C OUTPUT
C CY - A COMPLEX VECTOR WHOSE FIRST N COMPONENTS CONTAIN
C VALUES FOR THE SEQUENCE
C CY(J)=H(M,FNU+J-1,Z) OR
C CY(J)=H(M,FNU+J-1,Z)*EXP(-I*Z*(3-2M)) J=1,...,N
C DEPENDING ON SCALE, I**2=-1.
C NZ - NUMBER OF COMPONENTS SET TO ZERO DUE TO UNDERFLOW,
C NZ= 0 , NORMAL RETURN
C NZ.GT.0 , FIRST NZ COMPONENTS OF CY SET TO ZERO
C DUE TO UNDERFLOW, CY(J)=CMPLX(0.0,0.0)
C J=1,...,NZ WHEN Y.GT.0.0 AND M=1 OR
C Y.LT.0.0 AND M=2. FOR THE COMPLMENTARY
C HALF PLANES, NZ STATES ONLY THE NUMBER
C OF UNDERFLOWS.
C IERR -ERROR FLAG
C IERR=0, NORMAL RETURN - COMPUTATION COMPLETED
C IERR=1, INPUT ERROR - NO COMPUTATION
C IERR=2, OVERFLOW - NO COMPUTATION,
C CABS(Z) TOO SMALL
C IERR=3 OVERFLOW - NO COMPUTATION,
C FNU+N-1 TOO LARGE
C IERR=4, CABS(Z) OR FNU+N-1 LARGE - COMPUTATION DONE
C BUT LOSSES OF SIGNIFCANCE BY ARGUMENT
C REDUCTION PRODUCE LESS THAN HALF OF MACHINE
C ACCURACY
C IERR=5, CABS(Z) OR FNU+N-1 TOO LARGE - NO COMPUTA-
C TION BECAUSE OF COMPLETE LOSSES OF SIGNIFI-
C CANCE BY ARGUMENT REDUCTION
C IERR=6, ERROR - NO COMPUTATION,
C ALGORITHM TERMINATION CONDITION NOT MET
C
C LONG DESCRIPTION
C ================
C
C THE COMPUTATION IS CARRIED OUT BY THE RELATION
C
C H(M,FNU,Z)=(1/MP)*EXP(-MP*FNU)*K(FNU,Z*EXP(-MP))
C MP=MM*HPI*I, MM=3-2*M, HPI=PI/2, I**2=-1
C
C FOR M=1 OR 2 WHERE THE K BESSEL FUNCTION IS COMPUTED FOR THE
C RIGHT HALF PLANE RE(Z).GE.0.0. THE K FUNCTION IS CONTINUED
C TO THE LEFT HALF PLANE BY THE RELATION
C
C K(FNU,Z*EXP(MP)) = EXP(-MP*FNU)*K(FNU,Z)-MP*I(FNU,Z)
C MP=MR*PI*I, MR=+1 OR -1, RE(Z).GT.0, I**2=-1
C
C WHERE I(FNU,Z) IS THE I BESSEL FUNCTION.
C
C EXPONENTIAL DECAY OF H(M,FNU,Z) OCCURS IN THE UPPER HALF Z
C PLANE FOR M=1 AND THE LOWER HALF Z PLANE FOR M=2. EXPONENTIAL
C GROWTH OCCURS IN THE COMPLEMENTARY HALF PLANES. SCALING
C BY EXP(-MM*Z*I) REMOVES THE EXPONENTIAL BEHAVIOR IN THE
C WHOLE Z PLANE FOR Z TO INFINITY.
C
C FOR NEGATIVE ORDERS,THE FORMULAE
C
C H(1,-FNU,Z) = H(1,FNU,Z)*CEXP( PI*FNU*I)
C H(2,-FNU,Z) = H(2,FNU,Z)*CEXP(-PI*FNU*I)
C I**2=-1
C
C CAN BE USED.
C
C IN MOST COMPLEX VARIABLE COMPUTATION, ONE MUST EVALUATE ELE-
C MENTARY FUNCTIONS. WHEN THE MAGNITUDE OF Z OR FNU+N-1 IS
C LARGE, LOSSES OF SIGNIFICANCE BY ARGUMENT REDUCTION OCCUR.
C CONSEQUENTLY, IF EITHER ONE EXCEEDS U1=SQRT(0.5/UR), THEN
C LOSSES EXCEEDING HALF PRECISION ARE LIKELY AND AN ERROR FLAG
C IERR=4 IS TRIGGERED WHERE UR=X02AJF()=UNIT ROUNDOFF. ALSO
C IF EITHER IS LARGER THAN U2=0.5/UR, THEN ALL SIGNIFICANCE IS
C LOST AND IERR=5. IN ORDER TO USE THE INT FUNCTION, ARGUMENTS
C MUST BE FURTHER RESTRICTED NOT TO EXCEED THE LARGEST MACHINE
C INTEGER, U3=X02BBF(). THUS, THE MAGNITUDE OF Z AND FNU+N-1 IS
C RESTRICTED BY MIN(U2,U3). ON 32 BIT MACHINES, U1,U2, AND U3
C ARE APPROXIMATELY 2.0E+3, 4.2E+6, 2.1E+9 IN SINGLE PRECISION
C ARITHMETIC AND 1.3E+8, 1.8E+16, 2.1E+9 IN DOUBLE PRECISION
C ARITHMETIC RESPECTIVELY. THIS MAKES U2 AND U3 LIMITING IN
C THEIR RESPECTIVE ARITHMETICS. THIS MEANS THAT ONE CAN EXPECT
C TO RETAIN, IN THE WORST CASES ON 32 BIT MACHINES, NO DIGITS
C IN SINGLE AND ONLY 7 DIGITS IN DOUBLE PRECISION ARITHMETIC.
C SIMILAR CONSIDERATIONS HOLD FOR OTHER MACHINES.
C
C THE APPROXIMATE RELATIVE ERROR IN THE MAGNITUDE OF A COMPLEX
C BESSEL FUNCTION CAN BE EXPRESSED BY P*10**S WHERE P=MAX(UNIT
C ROUNDOFF,1.0E-18) IS THE NOMINAL PRECISION AND 10**S REPRE-
C SENTS THE INCREASE IN ERROR DUE TO ARGUMENT REDUCTION IN THE
C ELEMENTARY FUNCTIONS. HERE, S=MAX(1,ABS(LOG10(CABS(Z))),
C ABS(LOG10(FNU))) APPROXIMATELY (I.E. S=MAX(1,ABS(EXPONENT OF
C CABS(Z),ABS(EXPONENT OF FNU)) ). HOWEVER, THE PHASE ANGLE MAY
C HAVE ONLY ABSOLUTE ACCURACY. THIS IS MOST LIKELY TO OCCUR WHEN
C ONE COMPONENT (IN ABSOLUTE VALUE) IS LARGER THAN THE OTHER BY
C SEVERAL ORDERS OF MAGNITUDE. IF ONE COMPONENT IS 10**K LARGER
C THAN THE OTHER, THEN ONE CAN EXPECT ONLY MAX(ABS(LOG10(P))-K,
C 0) SIGNIFICANT DIGITS; OR, STATED ANOTHER WAY, WHEN K EXCEEDS
C THE EXPONENT OF P, NO SIGNIFICANT DIGITS REMAIN IN THE SMALLER
C COMPONENT. HOWEVER, THE PHASE ANGLE RETAINS ABSOLUTE ACCURACY
C BECAUSE, IN COMPLEX ARITHMETIC WITH PRECISION P, THE SMALLER
C COMPONENT WILL NOT (AS A RULE) DECREASE BELOW P TIMES THE
C MAGNITUDE OF THE LARGER COMPONENT. IN THESE EXTREME CASES,
C THE PRINCIPAL PHASE ANGLE IS ON THE ORDER OF +P, -P, PI/2-P,
C OR -PI/2+P.
C
C REFERENCES
C ==========
C HANDBOOK OF MATHEMATICAL FUNCTIONS BY M. ABRAMOWITZ
C AND I. A. STEGUN, NBS AMS SERIES 55, U.S. DEPT. OF
C COMMERCE, 1955.
C
C COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT
C BY D. E. AMOS, SAND83-0083, MAY, 1983.
C
C COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT
C AND LARGE ORDER BY D. E. AMOS, SAND83-0643, MAY, 1983
C
C A SUBROUTINE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX
C ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, SAND85-
C 1018, MAY, 1985
C
C A PORTABLE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX
C ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, TRANS.
C MATH. SOFTWARE, 1986
C
C DATE WRITTEN 830501 (YYMMDD)
C REVISION DATE 830501 (YYMMDD)
C AUTHOR AMOS, DONALD E., SANDIA NATIONAL LABORATORIES
C
S18ACF
DOUBLE PRECISION FUNCTION S18ACF(X,IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C BESSEL FUNCTION K0(X)
C
C **************************************************************
C
C TO EXTRACT THE CORRECT CODE FOR A PARTICULAR MACHINE-RANGE,
C ACTIVATE THE STATEMENTS CONTAINED IN COMMENTS BEGINNING CDD ,
C WHERE DD IS THE APPROXIMATE NUMBER OF SIGNIFICANT DECIMAL
C DIGITS REPRESENTED BY THE MACHINE
C DELETE THE ILLEGAL DUMMY STATEMENTS OF THE FORM
C * EXPANSION (NNNN) *
C
C ALSO INSERT APPROPRIATE DATA STATEMENTS TO DEFINE CONSTANTS
C WHICH DEPEND ON THE RANGE OF NUMBERS REPRESENTED BY THE
C MACHINE, RATHER THAN THE PRECISION (SUITABLE STATEMENTS FOR
C SOME MACHINES ARE CONTAINED IN COMMENTS BEGINNING CRD WHERE
C D IS A DIGIT WHICH SIMPLY DISTINGUISHES A GROUP OF MACHINES).
C DELETE THE ILLEGAL DUMMY DATA STATEMENTS WITH VALUES WRITTEN
C *VALUE*
C
C **************************************************************
C
S18ADF
DOUBLE PRECISION FUNCTION S18ADF(X,IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C BESSEL FUNCTION K1(X)
C
C **************************************************************
C
C TO EXTRACT THE CORRECT CODE FOR A PARTICULAR MACHINE-RANGE,
C ACTIVATE THE STATEMENTS CONTAINED IN COMMENTS BEGINNING CDD ,
C WHERE DD IS THE APPROXIMATE NUMBER OF SIGNIFICANT DECIMAL
C DIGITS REPRESENTED BY THE MACHINE
C DELETE THE ILLEGAL DUMMY STATEMENTS OF THE FORM
C * EXPANSION (NNNN) *
C
C ALSO INSERT APPROPRIATE DATA STATEMENTS TO DEFINE CONSTANTS
C WHICH DEPEND ON THE RANGE OF NUMBERS REPRESENTED BY THE
C MACHINE, RATHER THAN THE PRECISION (SUITABLE STATEMENTS FOR
C SOME MACHINES ARE CONTAINED IN COMMENTS BEGINNING CRD WHERE
C D IS A DIGIT WHICH SIMPLY DISTINGUISHES A GROUP OF MACHINES).
C DELETE THE ILLEGAL DUMMY DATA STATEMENTS WITH VALUES WRITTEN
C *VALUE*
C
C **************************************************************
C
S18AEF
DOUBLE PRECISION FUNCTION S18AEF(X,IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C BESSEL FUNCTION I0(X)
C
C **************************************************************
C
C TO EXTRACT THE CORRECT CODE FOR A PARTICULAR MACHINE-RANGE,
C ACTIVATE THE STATEMENTS CONTAINED IN COMMENTS BEGINNING CDD ,
C WHERE DD IS THE APPROXIMATE NUMBER OF SIGNIFICANT DECIMAL
C DIGITS REPRESENTED BY THE MACHINE
C DELETE THE ILLEGAL DUMMY STATEMENTS OF THE FORM
C * EXPANSION (NNNN) *
C
C ALSO INSERT APPROPRIATE DATA STATEMENTS TO DEFINE CONSTANTS
C WHICH DEPEND ON THE RANGE OF NUMBERS REPRESENTED BY THE
C MACHINE, RATHER THAN THE PRECISION (SUITABLE STATEMENTS FOR
C SOME MACHINES ARE CONTAINED IN COMMENTS BEGINNING CRD WHERE
C D IS A DIGIT WHICH SIMPLY DISTINGUISHES A GROUP OF MACHINES).
C DELETE THE ILLEGAL DUMMY DATA STATEMENTS WITH VALUES WRITTEN
C *VALUE*
C
C **************************************************************
C
S18AFF
DOUBLE PRECISION FUNCTION S18AFF(X,IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 8C REVISED. IER-269 (OCT.1980).
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C BESSEL FUNCTION I1(X)
C
C **************************************************************
C
C TO EXTRACT THE CORRECT CODE FOR A PARTICULAR MACHINE-RANGE,
C ACTIVATE THE STATEMENTS CONTAINED IN COMMENTS BEGINNING CDD ,
C WHERE DD IS THE APPROXIMATE NUMBER OF SIGNIFICANT DECIMAL
C DIGITS REPRESENTED BY THE MACHINE
C DELETE THE ILLEGAL DUMMY STATEMENTS OF THE FORM
C * EXPANSION (NNNN) *
C
C ALSO INSERT APPROPRIATE DATA STATEMENTS TO DEFINE CONSTANTS
C WHICH DEPEND ON THE RANGE OF NUMBERS REPRESENTED BY THE
C MACHINE, RATHER THAN THE PRECISION (SUITABLE STATEMENTS FOR
C SOME MACHINES ARE CONTAINED IN COMMENTS BEGINNING CRD WHERE
C D IS A DIGIT WHICH SIMPLY DISTINGUISHES A GROUP OF MACHINES).
C DELETE THE ILLEGAL DUMMY DATA STATEMENTS WITH VALUES WRITTEN
C *VALUE*
C
C **************************************************************
C
S18CCF
DOUBLE PRECISION FUNCTION S18CCF(X,IFAIL)
C MARK 10 RELEASE. NAG COPYRIGHT 1982.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C SCALED BESSEL FUNCTION K0(X)EXP(X)
C
C **************************************************************
C
C TO EXTRACT THE CORRECT CODE FOR A PARTICULAR MACHINE-RANGE,
C ACTIVATE THE STATEMENTS CONTAINED IN COMMENTS BEGINNING CDD ,
C WHERE DD IS THE APPROXIMATE NUMBER OF SIGNIFICANT DECIMAL
C DIGITS REPRESENTED BY THE MACHINE
C DELETE THE ILLEGAL DUMMY STATEMENTS OF THE FORM
C * EXPANSION (NNNN) *
C
C **************************************************************
C
S18CDF
DOUBLE PRECISION FUNCTION S18CDF(X,IFAIL)
C MARK 10 RELEASE. NAG COPYRIGHT 1982.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C SCALED BESSEL FUNCTION K1(X)EXP(X)
C
C **************************************************************
C
C TO EXTRACT THE CORRECT CODE FOR A PARTICULAR MACHINE-RANGE,
C ACTIVATE THE STATEMENTS CONTAINED IN COMMENTS BEGINNING CDD ,
C WHERE DD IS THE APPROXIMATE NUMBER OF SIGNIFICANT DECIMAL
C DIGITS REPRESENTED BY THE MACHINE
C DELETE THE ILLEGAL DUMMY STATEMENTS OF THE FORM
C * EXPANSION (NNNN) *
C
C ALSO INSERT APPROPRIATE DATA STATEMENTS TO DEFINE CONSTANTS
C WHICH DEPEND ON THE RANGE OF NUMBERS REPRESENTED BY THE
C MACHINE, RATHER THAN THE PRECISION (SUITABLE STATEMENTS FOR
C SOME MACHINES ARE CONTAINED IN COMMENTS BEGINNING CRD WHERE
C D IS A DIGIT WHICH SIMPLY DISTINGUISHES A GROUP OF MACHINES).
C DELETE THE ILLEGAL DUMMY DATA STATEMENTS WITH VALUES WRITTEN
C *VALUE*
C
C **************************************************************
C
S18CEF
DOUBLE PRECISION FUNCTION S18CEF(X,IFAIL)
C MARK 10 RELEASE. NAG COPYRIGHT 1982.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C SCALED BESSEL FUNCTION I0(X)*EXP(-ABS(X))
C
C **************************************************************
C
C TO EXTRACT THE CORRECT CODE FOR A PARTICULAR MACHINE-RANGE,
C ACTIVATE THE STATEMENTS CONTAINED IN COMMENTS BEGINNING CDD ,
C WHERE DD IS THE APPROXIMATE NUMBER OF SIGNIFICANT DECIMAL
C DIGITS REPRESENTED BY THE MACHINE
C DELETE THE ILLEGAL DUMMY STATEMENTS OF THE FORM
C * EXPANSION (NNNN) *
C
C **************************************************************
C
S18CFF
DOUBLE PRECISION FUNCTION S18CFF(X,IFAIL)
C MARK 10 RELEASE. NAG COPYRIGHT 1982.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C SCALED BESSEL FUNCTION I1(X)*EXP(-ABS(X))
C
C **************************************************************
C
C TO EXTRACT THE CORRECT CODE FOR A PARTICULAR MACHINE-RANGE,
C ACTIVATE THE STATEMENTS CONTAINED IN COMMENTS BEGINNING CDD ,
C WHERE DD IS THE APPROXIMATE NUMBER OF SIGNIFICANT DECIMAL
C DIGITS REPRESENTED BY THE MACHINE
C DELETE THE ILLEGAL DUMMY STATEMENTS OF THE FORM
C * EXPANSION (NNNN) *
C
C **************************************************************
C
S18DCF
SUBROUTINE S18DCF(FNU,Z,N,SCALE,CY,NZ,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C MARK 14 REVISED. IER-784 (DEC 1989).
C
C Original name: CBESK
C
C PURPOSE TO COMPUTE K-BESSEL FUNCTIONS OF COMPLEX ARGUMENT
C
C DESCRIPTION
C ===========
C
C ON SCALE='U', S18DCF COMPUTES AN N MEMBER SEQUENCE OF COMPLEX
C BESSEL FUNCTIONS CY(J)=K(FNU+J-1,Z) FOR REAL, NONNEGATIVE
C ORDERS FNU+J-1, J=1,...,N AND COMPLEX Z.NE.CMPLX(0.0,0.0)
C IN THE CUT PLANE -PI.LT.ARG(Z).LE.PI. ON SCALE='S', S18DCF
C RETURNS THE SCALED K FUNCTIONS,
C
C CY(J)=EXP(Z)*K(FNU+J-1,Z) , J=1,...,N,
C
C WHICH REMOVE THE EXPONENTIAL BEHAVIOR IN BOTH THE LEFT AND
C RIGHT HALF PLANES FOR Z TO INFINITY. DEFINITIONS AND
C NOTATION ARE FOUND IN THE NBS HANDBOOK OF MATHEMATICAL
C FUNCTIONS (REF. 1).
C
C INPUT
C Z - Z=CMPLX(X,Y),Z.NE.CMPLX(0.,0.),-PI.LT.ARG(Z).LE.PI
C FNU - ORDER OF INITIAL K FUNCTION, FNU.GE.0.0E0
C N - NUMBER OF MEMBERS OF THE SEQUENCE, N.GE.1
C SCALE - A PARAMETER TO INDICATE THE SCALING OPTION
C SCALE = 'U' OR SCALE = 'u' RETURNS
C CY(I)=K(FNU+I-1,Z), I=1,...,N
C SCALE = 'S' OR SCALE = 's' RETURNS
C CY(I)=K(FNU+I-1,Z)*EXP(Z), I=1,...,N
C
C OUTPUT
C CY - A COMPLEX VECTOR WHOSE FIRST N COMPONENTS CONTAIN
C VALUES FOR THE SEQUENCE
C CY(I)=K(FNU+I-1,Z), I=1,...,N OR
C CY(I)=K(FNU+I-1,Z)*EXP(Z), I=1,...,N
C DEPENDING ON SCALE
C NZ - NUMBER OF COMPONENTS SET TO ZERO DUE TO UNDERFLOW.
C NZ= 0 , NORMAL RETURN
C NZ.GT.0 , FIRST NZ COMPONENTS OF CY SET TO ZERO
C DUE TO UNDERFLOW, CY(I)=CMPLX(0.0,0.0),
C I=1,...,N WHEN X.GE.0.0. WHEN X.LT.0.0
C NZ STATES ONLY THE NUMBER OF UNDERFLOWS
C IN THE SEQUENCE.
C IFAIL - ERROR FLAG
C IFAIL=0, NORMAL RETURN - COMPUTATION COMPLETED
C IFAIL=1, INPUT ERROR - NO COMPUTATION
C IFAIL=2, OVERFLOW - NO COMPUTATION, CABS(Z) IS
C TOO SMALL
C IFAIL=3, OVERFLOW - NO COMPUTATION, FNU+N-1 IS
C TOO LARGE
C IFAIL=4, CABS(Z) OR FNU+N-1 LARGE - COMPUTATION DONE
C BUT LOSSES OF SIGNIFCANCE BY ARGUMENT
C REDUCTION PRODUCE LESS THAN HALF OF MACHINE
C ACCURACY
C IFAIL=5, CABS(Z) OR FNU+N-1 TOO LARGE - NO COMPUTA-
C TION BECAUSE OF COMPLETE LOSSES OF SIGNIFI-
C CANCE BY ARGUMENT REDUCTION
C IFAIL=6, ERROR - NO COMPUTATION,
C ALGORITHM TERMINATION CONDITION NOT MET
C
C LONG DESCRIPTION
C ================
C
C EQUATIONS OF THE REFERENCE ARE IMPLEMENTED FOR SMALL ORDERS
C DNU AND DNU+1.0 IN THE RIGHT HALF PLANE X.GE.0.0. FORWARD
C RECURRENCE GENERATES HIGHER ORDERS. K IS CONTINUED TO THE LEFT
C HALF PLANE BY THE RELATION
C
C K(FNU,Z*EXP(MP)) = EXP(-MP*FNU)*K(FNU,Z)-MP*I(FNU,Z)
C MP=MR*PI*I, MR=+1 OR -1, RE(Z).GT.0, I**2=-1
C
C WHERE I(FNU,Z) IS THE I BESSEL FUNCTION.
C
C FOR LARGE ORDERS, FNU.GT.FNUL, THE K FUNCTION IS COMPUTED
C BY MEANS OF ITS UNIFORM ASYMPTOTIC EXPANSIONS.
C
C FOR NEGATIVE ORDERS, THE FORMULA
C
C K(-FNU,Z) = K(FNU,Z)
C
C CAN BE USED.
C
C S18DCF ASSUMES THAT A SIGNIFICANT DIGIT SINH(X) FUNCTION IS
C AVAILABLE.
C
C IN MOST COMPLEX VARIABLE COMPUTATION, ONE MUST EVALUATE ELE-
C MENTARY FUNCTIONS. WHEN THE MAGNITUDE OF Z OR FNU+N-1 IS
C LARGE, LOSSES OF SIGNIFICANCE BY ARGUMENT REDUCTION OCCUR.
C CONSEQUENTLY, IF EITHER ONE EXCEEDS U1=SQRT(0.5/UR), THEN
C LOSSES EXCEEDING HALF PRECISION ARE LIKELY AND AN ERROR FLAG
C IFAIL=4 IS TRIGGERED WHERE UR=X02AJF()=UNIT ROUNDOFF. ALSO
C IF EITHER IS LARGER THAN U2=0.5/UR, THEN ALL SIGNIFICANCE IS
C LOST AND IFAIL=5. IN ORDER TO USE THE INT FUNCTION, ARGUMENTS
C MUST BE FURTHER RESTRICTED NOT TO EXCEED THE LARGEST MACHINE
C INTEGER, U3=X02BBF(). THUS, THE MAGNITUDE OF Z AND FNU+N-1 IS
C RESTRICTED BY MIN(U2,U3). ON 32 BIT MACHINES, U1,U2, AND U3
C ARE APPROXIMATELY 2.0E+3, 4.2E+6, 2.1E+9 IN SINGLE PRECISION
C ARITHMETIC AND 1.3E+8, 1.8E+16, 2.1E+9 IN DOUBLE PRECISION
C ARITHMETIC RESPECTIVELY. THIS MAKES U2 AND U3 LIMITING IN
C THEIR RESPECTIVE ARITHMETICS. THIS MEANS THAT ONE CAN EXPECT
C TO RETAIN, IN THE WORST CASES ON 32 BIT MACHINES, NO DIGITS
C IN SINGLE AND ONLY 7 DIGITS IN DOUBLE PRECISION ARITHMETIC.
C SIMILAR CONSIDERATIONS HOLD FOR OTHER MACHINES.
C
C THE APPROXIMATE RELATIVE ERROR IN THE MAGNITUDE OF A COMPLEX
C BESSEL FUNCTION CAN BE EXPRESSED BY P*10**S WHERE P=MAX(UNIT
C ROUNDOFF,1.0E-18) IS THE NOMINAL PRECISION AND 10**S REPRE-
C SENTS THE INCREASE IN ERROR DUE TO ARGUMENT REDUCTION IN THE
C ELEMENTARY FUNCTIONS. HERE, S=MAX(1,ABS(LOG10(CABS(Z))),
C ABS(LOG10(FNU))) APPROXIMATELY (I.E. S=MAX(1,ABS(EXPONENT OF
C CABS(Z),ABS(EXPONENT OF FNU)) ). HOWEVER, THE PHASE ANGLE MAY
C HAVE ONLY ABSOLUTE ACCURACY. THIS IS MOST LIKELY TO OCCUR WHEN
C ONE COMPONENT (IN ABSOLUTE VALUE) IS LARGER THAN THE OTHER BY
C SEVERAL ORDERS OF MAGNITUDE. IF ONE COMPONENT IS 10**K LARGER
C THAN THE OTHER, THEN ONE CAN EXPECT ONLY MAX(ABS(LOG10(P))-K,
C 0) SIGNIFICANT DIGITS; OR, STATED ANOTHER WAY, WHEN K EXCEEDS
C THE EXPONENT OF P, NO SIGNIFICANT DIGITS REMAIN IN THE SMALLER
C COMPONENT. HOWEVER, THE PHASE ANGLE RETAINS ABSOLUTE ACCURACY
C BECAUSE, IN COMPLEX ARITHMETIC WITH PRECISION P, THE SMALLER
C COMPONENT WILL NOT (AS A RULE) DECREASE BELOW P TIMES THE
C MAGNITUDE OF THE LARGER COMPONENT. IN THESE EXTREME CASES,
C THE PRINCIPAL PHASE ANGLE IS ON THE ORDER OF +P, -P, PI/2-P,
C OR -PI/2+P.
C
C REFERENCES
C ==========
C HANDBOOK OF MATHEMATICAL FUNCTIONS BY M. ABRAMOWITZ
C AND I. A. STEGUN, NBS AMS SERIES 55, U.S. DEPT. OF
C COMMERCE, 1955.
C
C COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT
C BY D. E. AMOS, SAND83-0083, MAY, 1983.
C
C COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT
C AND LARGE ORDER BY D. E. AMOS, SAND83-0643, MAY, 1983.
C
C A SUBROUTINE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX
C ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, SAND85-
C 1018, MAY, 1985
C
C A PORTABLE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX
C ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, TRANS.
C MATH. SOFTWARE, 1986
C
C DATE WRITTEN 830501 (YYMMDD)
C REVISION DATE 830501 (YYMMDD)
C AUTHOR AMOS, DONALD E., SANDIA NATIONAL LABORATORIES
C
S18DEF
SUBROUTINE S18DEF(FNU,Z,N,SCALE,CY,NZ,IFAIL)
C MARK 13 RELEASE. NAG COPYRIGHT 1988.
C MARK 14 REVISED. IER-787 (DEC 1989).
C
C Original name: CBESI
C
C PURPOSE TO COMPUTE I-BESSEL FUNCTIONS OF COMPLEX ARGUMENT
C
C DESCRIPTION
C ===========
C
C ON SCALE='U', S18DEF COMPUTES AN N MEMBER SEQUENCE OF COMPLEX
C BESSEL FUNCTIONS CY(J)=I(FNU+J-1,Z) FOR REAL, NONNEGATIVE
C ORDERS FNU+J-1, J=1,...,N AND COMPLEX Z IN THE CUT PLANE
C -PI.LT.ARG(Z).LE.PI. ON SCALE='S', S18DEF RETURNS THE SCALED
C FUNCTIONS
C
C CY(J)=EXP(-ABS(X))*I(FNU+J-1,Z) J = 1,...,N , X=REAL(Z)
C
C WITH THE EXPONENTIAL GROWTH REMOVED IN BOTH THE LEFT AND
C RIGHT HALF PLANES FOR Z TO INFINITY. DEFINITIONS AND
C NOTATION ARE FOUND IN THE NBS HANDBOOK OF MATHEMATICAL
C FUNCTIONS (REF.1)
C
C INPUT
C Z - Z=CMPLX(X,Y), -PI.LT.ARG(Z).LE.PI
C FNU - ORDER OF INITIAL I FUNCTION, FNU.GE.0.0E0
C SCALE - A PARAMETER TO INDICATE THE SCALING OPTION
C SCALE = 'U' OR SCALE = 'u' RETURNS
C CY(J)=I(FNU+J-1,Z), J=1,...,N
C SCALE = 'S' OR SCALE = 's' RETURNS
C CY(J)=I(FNU+J-1,Z)*EXP(-ABS(X)), J=1,...,N
C N - NUMBER OF MEMBERS OF THE SEQUENCE, N.GE.1
C
C OUTPUT
C CY - A COMPLEX VECTOR WHOSE FIRST N COMPONENTS CONTAIN
C VALUES FOR THE SEQUENCE
C CY(J)=I(FNU+J-1,Z) OR
C CY(J)=I(FNU+J-1,Z)*EXP(-ABS(X)) J=1,...,N
C DEPENDING ON SCALE, X=REAL(Z)
C NZ - NUMBER OF COMPONENTS SET TO ZERO DUE TO UNDERFLOW,
C NZ= 0 , NORMAL RETURN
C NZ.GT.0 , LAST NZ COMPONENTS OF CY SET TO ZERO
C DUE TO UNDERFLOW, CY(J)=CMPLX(0.0,0.0),
C J = N-NZ+1,...,N
C IFAIL - ERROR FLAG
C IFAIL=0, NORMAL RETURN - COMPUTATION COMPLETED
C IFAIL=1, INPUT ERROR - NO COMPUTATION
C IFAIL=2, OVERFLOW - NO COMPUTATION, REAL(Z) TOO
C LARGE ON SCALE='U'
C IFAIL=3, CABS(Z) OR FNU+N-1 LARGE - COMPUTATION DONE
C BUT LOSSES OF SIGNIFCANCE BY ARGUMENT
C REDUCTION PRODUCE LESS THAN HALF OF MACHINE
C ACCURACY
C IFAIL=4, CABS(Z) OR FNU+N-1 TOO LARGE - NO COMPUTA-
C TION BECAUSE OF COMPLETE LOSSES OF SIGNIFI-
C CANCE BY ARGUMENT REDUCTION
C IFAIL=5, ERROR - NO COMPUTATION,
C ALGORITHM TERMINATION CONDITION NOT MET
C
C LONG DESCRIPTION
C ================
C
C THE COMPUTATION IS CARRIED OUT BY THE POWER SERIES FOR
C SMALL CABS(Z), THE ASYMPTOTIC EXPANSION FOR LARGE CABS(Z),
C THE MILLER ALGORITHM NORMALIZED BY THE WRONSKIAN AND A
C NEUMANN SERIES FOR IMTERMEDIATE MAGNITUDES, AND THE
C UNIFORM ASYMPTOTIC EXPANSIONS FOR I(FNU,Z) AND J(FNU,Z)
C FOR LARGE ORDERS. BACKWARD RECURRENCE IS USED TO GENERATE
C SEQUENCES OR REDUCE ORDERS WHEN NECESSARY.
C
C THE CALCULATIONS ABOVE ARE DONE IN THE RIGHT HALF PLANE AND
C CONTINUED INTO THE LEFT HALF PLANE BY THE FORMULA
C
C I(FNU,Z*EXP(M*PI)) = EXP(M*PI*FNU)*I(FNU,Z) REAL(Z).GT.0.0
C M = +I OR -I, I**2=-1
C
C FOR NEGATIVE ORDERS,THE FORMULA
C
C I(-FNU,Z) = I(FNU,Z) + (2/PI)*SIN(PI*FNU)*K(FNU,Z)
C
C CAN BE USED. HOWEVER,FOR LARGE ORDERS CLOSE TO INTEGERS, THE
C THE FUNCTION CHANGES RADICALLY. WHEN FNU IS A LARGE POSITIVE
C INTEGER,THE MAGNITUDE OF I(-FNU,Z)=I(FNU,Z) IS A LARGE
C NEGATIVE POWER OF TEN. BUT WHEN FNU IS NOT AN INTEGER,
C K(FNU,Z) DOMINATES IN MAGNITUDE WITH A LARGE POSITIVE POWER OF
C TEN AND THE MOST THAT THE SECOND TERM CAN BE REDUCED IS BY
C UNIT ROUNDOFF FROM THE COEFFICIENT. THUS, WIDE CHANGES CAN
C OCCUR WITHIN UNIT ROUNDOFF OF A LARGE INTEGER FOR FNU. HERE,
C LARGE MEANS FNU.GT.CABS(Z).
C
C IN MOST COMPLEX VARIABLE COMPUTATION, ONE MUST EVALUATE ELE-
C MENTARY FUNCTIONS. WHEN THE MAGNITUDE OF Z OR FNU+N-1 IS
C LARGE, LOSSES OF SIGNIFICANCE BY ARGUMENT REDUCTION OCCUR.
C CONSEQUENTLY, IF EITHER ONE EXCEEDS U1=SQRT(0.5/UR), THEN
C LOSSES EXCEEDING HALF PRECISION ARE LIKELY AND AN ERROR FLAG
C IFAIL=3 IS TRIGGERED WHERE UR=X02AJF()=UNIT ROUNDOFF. ALSO
C IF EITHER IS LARGER THAN U2=0.5/UR, THEN ALL SIGNIFICANCE IS
C LOST AND IFAIL=4. IN ORDER TO USE THE INT FUNCTION, ARGUMENTS
C MUST BE FURTHER RESTRICTED NOT TO EXCEED THE LARGEST MACHINE
C INTEGER, U3=X02BBF(). THUS, THE MAGNITUDE OF Z AND FNU+N-1 IS
C RESTRICTED BY MIN(U2,U3). ON 32 BIT MACHINES, U1,U2, AND U3
C ARE APPROXIMATELY 2.0E+3, 4.2E+6, 2.1E+9 IN SINGLE PRECISION
C ARITHMETIC AND 1.3E+8, 1.8E+16, 2.1E+9 IN DOUBLE PRECISION
C ARITHMETIC RESPECTIVELY. THIS MAKES U2 AND U3 LIMITING IN
C THEIR RESPECTIVE ARITHMETICS. THIS MEANS THAT ONE CAN EXPECT
C TO RETAIN, IN THE WORST CASES ON 32 BIT MACHINES, NO DIGITS
C IN SINGLE AND ONLY 7 DIGITS IN DOUBLE PRECISION ARITHMETIC.
C SIMILAR CONSIDERATIONS HOLD FOR OTHER MACHINES.
C
C THE APPROXIMATE RELATIVE ERROR IN THE MAGNITUDE OF A COMPLEX
C BESSEL FUNCTION CAN BE EXPRESSED BY P*10**S WHERE P=MAX(UNIT
C ROUNDOFF,1.0E-18) IS THE NOMINAL PRECISION AND 10**S REPRE-
C SENTS THE INCREASE IN ERROR DUE TO ARGUMENT REDUCTION IN THE
C ELEMENTARY FUNCTIONS. HERE, S=MAX(1,ABS(LOG10(CABS(Z))),
C ABS(LOG10(FNU))) APPROXIMATELY (I.E. S=MAX(1,ABS(EXPONENT OF
C CABS(Z),ABS(EXPONENT OF FNU)) ). HOWEVER, THE PHASE ANGLE MAY
C HAVE ONLY ABSOLUTE ACCURACY. THIS IS MOST LIKELY TO OCCUR WHEN
C ONE COMPONENT (IN ABSOLUTE VALUE) IS LARGER THAN THE OTHER BY
C SEVERAL ORDERS OF MAGNITUDE. IF ONE COMPONENT IS 10**K LARGER
C THAN THE OTHER, THEN ONE CAN EXPECT ONLY MAX(ABS(LOG10(P))-K,
C 0) SIGNIFICANT DIGITS; OR, STATED ANOTHER WAY, WHEN K EXCEEDS
C THE EXPONENT OF P, NO SIGNIFICANT DIGITS REMAIN IN THE SMALLER
C COMPONENT. HOWEVER, THE PHASE ANGLE RETAINS ABSOLUTE ACCURACY
C BECAUSE, IN COMPLEX ARITHMETIC WITH PRECISION P, THE SMALLER
C COMPONENT WILL NOT (AS A RULE) DECREASE BELOW P TIMES THE
C MAGNITUDE OF THE LARGER COMPONENT. IN THESE EXTREME CASES,
C THE PRINCIPAL PHASE ANGLE IS ON THE ORDER OF +P, -P, PI/2-P,
C OR -PI/2+P.
C
C REFERENCES
C ==========
C HANDBOOK OF MATHEMATICAL FUNCTIONS BY M. ABRAMOWITZ
C AND I. A. STEGUN, NBS AMS SERIES 55, U.S. DEPT. OF
C COMMERCE, 1955.
C
C COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT
C BY D. E. AMOS, SAND83-0083, MAY, 1983.
C
C COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT
C AND LARGE ORDER BY D. E. AMOS, SAND83-0643, MAY, 1983
C
C A SUBROUTINE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX
C ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, SAND85-
C 1018, MAY, 1985
C
C A PORTABLE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX
C ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, TRANS.
C MATH. SOFTWARE, 1986
C
C DATE WRITTEN 830501 (YYMMDD)
C REVISION DATE 830501 (YYMMDD)
C AUTHOR AMOS, DONALD E., SANDIA NATIONAL LABORATORIES
C
S19AAF
DOUBLE PRECISION FUNCTION S19AAF(X,IFAIL)
C MARK 11 RELEASE. NAG COPYRIGHT 1983.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C KELVIN FUNCTION BER(X)
C
C **************************************************************
C
C TO EXTRACT THE CORRECT CODE FOR A PARTICULAR MACHINE-RANGE,
C ACTIVATE THE STATEMENTS CONTAINED IN COMMENTS BEGINNING CDD ,
C WHERE DD IS THE APPROXIMATE NUMBER OF SIGNIFICANT DECIMAL
C DIGITS REPRESENTED BY THE MACHINE
C DELETE THE ILLEGAL DUMMY STATEMENTS OF THE FORM
C * EXPANSION (NNNN) *
C
C **************************************************************
C
S19ABF
DOUBLE PRECISION FUNCTION S19ABF(X,IFAIL)
C MARK 11 RELEASE. NAG COPYRIGHT 1983.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C KELVIN FUNCTION BEI(X)
C
C **************************************************************
C
C TO EXTRACT THE CORRECT CODE FOR A PARTICULAR MACHINE-RANGE,
C ACTIVATE THE STATEMENTS CONTAINED IN COMMENTS BEGINNING CDD ,
C WHERE DD IS THE APPROXIMATE NUMBER OF SIGNIFICANT DECIMAL
C DIGITS REPRESENTED BY THE MACHINE
C DELETE THE ILLEGAL DUMMY STATEMENTS OF THE FORM
C * EXPANSION (NNNN) *
C
C ALSO INSERT APPROPRIATE DATA STATEMENTS TO DEFINE CONSTANTS
C WHICH DEPEND ON THE RANGE OF NUMBERS REPRESENTED BY THE
C MACHINE, RATHER THAN THE PRECISION (SUITABLE STATEMENTS FOR
C SOME MACHINES ARE CONTAINED IN COMMENTS BEGINNING CRD WHERE
C D IS A DIGIT WHICH SIMPLY DISTINGUISHES A GROUP OF MACHINES).
C DELETE THE ILLEGAL DUMMY DATA STATEMENTS WITH VALUES WRITTEN
C *VALUE*
C
C **************************************************************
C
S19ACF
DOUBLE PRECISION FUNCTION S19ACF(X,IFAIL)
C MARK 11 RELEASE. NAG COPYRIGHT 1983.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 12B REVISED. IER-544 (FEB 1987).
C KELVIN FUNCTION KER(X)
C
C **************************************************************
C
C TO EXTRACT THE CORRECT CODE FOR A PARTICULAR MACHINE-RANGE,
C ACTIVATE THE STATEMENTS CONTAINED IN COMMENTS BEGINNING CDD ,
C WHERE DD IS THE APPROXIMATE NUMBER OF SIGNIFICANT DECIMAL
C DIGITS REPRESENTED BY THE MACHINE
C DELETE THE ILLEGAL DUMMY STATEMENTS OF THE FORM
C * EXPANSION (NNNN) *
C
C ALSO INSERT APPROPRIATE DATA STATEMENTS TO DEFINE CONSTANTS
C WHICH DEPEND ON THE RANGE OF NUMBERS REPRESENTED BY THE
C MACHINE, RATHER THAN THE PRECISION (SUITABLE STATEMENTS FOR
C SOME MACHINES ARE CONTAINED IN COMMENTS BEGINNING CRD WHERE
C D IS A DIGIT WHICH SIMPLY DISTINGUISHES A GROUP OF MACHINES).
C DELETE THE ILLEGAL DUMMY DATA STATEMENTS WITH VALUES WRITTEN
C *VALUE*
C
C **************************************************************
C
S19ADF
DOUBLE PRECISION FUNCTION S19ADF(X,IFAIL)
C MARK 11 RELEASE. NAG COPYRIGHT 1983.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C KELVIN FUNCTION KEI(X)
C
C **************************************************************
C
C TO EXTRACT THE CORRECT CODE FOR A PARTICULAR MACHINE-RANGE,
C ACTIVATE THE STATEMENTS CONTAINED IN COMMENTS BEGINNING CDD ,
C WHERE DD IS THE APPROXIMATE NUMBER OF SIGNIFICANT DECIMAL
C DIGITS REPRESENTED BY THE MACHINE
C DELETE THE ILLEGAL DUMMY STATEMENTS OF THE FORM
C * EXPANSION (NNNN) *
C
C ALSO INSERT APPROPRIATE DATA STATEMENTS TO DEFINE CONSTANTS
C WHICH DEPEND ON THE RANGE OF NUMBERS REPRESENTED BY THE
C MACHINE, RATHER THAN THE PRECISION (SUITABLE STATEMENTS FOR
C SOME MACHINES ARE CONTAINED IN COMMENTS BEGINNING CRD WHERE
C D IS A DIGIT WHICH SIMPLY DISTINGUISHES A GROUP OF MACHINES).
C DELETE THE ILLEGAL DUMMY DATA STATEMENTS WITH VALUES WRITTEN
C *VALUE*
C
C **************************************************************
C
S20ACF
DOUBLE PRECISION FUNCTION S20ACF(X,IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C FRESNEL INTEGRAL S(X)
C
C **************************************************************
C
C TO EXTRACT THE CORRECT CODE FOR A PARTICULAR MACHINE-RANGE,
C ACTIVATE THE STATEMENTS CONTAINED IN COMMENTS BEGINNING CDD ,
C WHERE DD IS THE APPROXIMATE NUMBER OF SIGNIFICANT DECIMAL
C DIGITS REPRESENTED BY THE MACHINE
C DELETE THE ILLEGAL DUMMY STATEMENTS OF THE FORM
C * EXPANSION (NNNN) *
C
C ALSO INSERT APPROPRIATE DATA STATEMENTS TO DEFINE CONSTANTS
C WHICH DEPEND ON THE RANGE OF NUMBERS REPRESENTED BY THE
C MACHINE, RATHER THAN THE PRECISION (SUITABLE STATEMENTS FOR
C SOME MACHINES ARE CONTAINED IN COMMENTS BEGINNING CRD WHERE
C D IS A DIGIT WHICH SIMPLY DISTINGUISHES A GROUP OF MACHINES).
C DELETE THE ILLEGAL DUMMY DATA STATEMENTS WITH VALUES WRITTEN
C *VALUE*
C
C **************************************************************
C
S20ADF
DOUBLE PRECISION FUNCTION S20ADF(X,IFAIL)
C MARK 7 RELEASE. NAG COPYRIGHT 1978.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C FRESNEL INTEGRAL C(X)
C
C **************************************************************
C
C TO EXTRACT THE CORRECT CODE FOR A PARTICULAR MACHINE-RANGE,
C ACTIVATE THE STATEMENTS CONTAINED IN COMMENTS BEGINNING CDD ,
C WHERE DD IS THE APPROXIMATE NUMBER OF SIGNIFICANT DECIMAL
C DIGITS REPRESENTED BY THE MACHINE
C DELETE THE ILLEGAL DUMMY STATEMENTS OF THE FORM
C * EXPANSION (NNNN) *
C
C ALSO INSERT APPROPRIATE DATA STATEMENTS TO DEFINE CONSTANTS
C WHICH DEPEND ON THE RANGE OF NUMBERS REPRESENTED BY THE
C MACHINE, RATHER THAN THE PRECISION (SUITABLE STATEMENTS FOR
C SOME MACHINES ARE CONTAINED IN COMMENTS BEGINNING CRD WHERE
C D IS A DIGIT WHICH SIMPLY DISTINGUISHES A GROUP OF MACHINES).
C DELETE THE ILLEGAL DUMMY DATA STATEMENTS WITH VALUES WRITTEN
C *VALUE*
C
C **************************************************************
C
S21BAF
DOUBLE PRECISION FUNCTION S21BAF(X,Y,IFAIL)
C MARK 14 RE-ISSUE. NAG COPYRIGHT 1989.
C
C ******************************************************************
C
C Calculates an approximate value for the function
C Rc(x,y) related to the elliptic integrals.
C The function is as defined by Carlson - Special
C Functions of Applied Mathematics - Academic Press(1977)
C
C Rc(x,y)=integral(zero to infinity) of
C 0.5/(sqrt(t+x)*(t+y)) dt
C
C subject to restrictions
C x.ge.0.0 and y.ne.0.0
C ******************************************************************
C Precision-dependent constant
C
C ACC Controls final truncation accuracy. An accuracy
C close to full machine precision can be obtained
C for all legal arguments if ACC is chosen so that
C 16.0*ACC**6/(1.0-2.0*ACC).le.X02AJF/X02BHF
C
C To select the correct value for a particular machine-range,
C activate the statement contained in a comment beginning CDD ,
C where DD is the approximate number of significant decimal
C digits represented by the machine.
C * EXPANSION (DATA) *
C ******************************************************************
C
S21BBF
DOUBLE PRECISION FUNCTION S21BBF(X,Y,Z,IFAIL)
C MARK 14 RE-ISSUE. NAG COPYRIGHT 1989.
C
C ******************************************************************
C
C Calculates an approximate value for the elliptic integral of
C the first kind, Rf(X,Y,Z), as defined by B.C.Carlson.
C - Special Functions of Applied Maths - Academic Press(1977).
C The algorithm is also due to Carlson but has been recoded by
C J.L.Schonfelder to avoid intermediate under and overflows.
C
C Rf(X,Y,Z)=integral(zero to infinity) of
C 0.5/sqrt((t+X)*(t+Y)*(t+Z)) dt
C
C for X,Y,Z all .ge. zero and at most one equal to zero.
C
C ******************************************************************
C Precision-dependent constant
C
C ACC Controls final truncation accuracy. An accuracy
C close to full machine precision can be obtained
C for all legal arguments if ACC is chosen so that
C 0.25*ACC**6/(1.0-ACC).le.X02AJF/X02BHF
C
C To select the correct value for a particular machine-range,
C activate the statement contained in a comment beginning CDD ,
C where DD is the approximate number of significant decimal
C digits represented by the machine.
C * EXPANSION (DATA) *
C ******************************************************************
C
S21BCF
DOUBLE PRECISION FUNCTION S21BCF(X,Y,Z,IFAIL)
C MARK 14 RE-ISSUE. NAG COPYRIGHT 1989.
C
C ******************************************************************
C
C Calculates an approximate value for the elliptic integral of
C the second kind, Rd(X,Y,Z), as defined by B.C.Carlson.
C - Special Functions of Applied Maths - Academic Press(1977).
C The algorithm is also due to Carlson.
C
C Rd(X,Y,Z)=integral(zero to infinity) of
C 1.5/sqrt((t+X)*(t+Y)*(t+Z)**3) dt
C
C for X.ge.0.0 , Y.ge.0.0, Z.gt.0.0 and at most one of X
C and Y equal to 0.0.
C
C ******************************************************************
C Precision-dependent constant
C
C ACC Controls final truncation accuracy. An accuracy
C close to full machine precision can be obtained
C for all legal arguments if ACC is chosen so that
C 3.0*ACC**6/(1.0-ACC)**1.5.le.X02AJF/X02BHF
C
C To select the correct value for a particular machine-range,
C activate the statement contained in a comment beginning CDD ,
C where DD is the approximate number of significant decimal
C digits represented by the machine.
C * EXPANSION (DATA) *
C ******************************************************************
C
S21BDF
DOUBLE PRECISION FUNCTION S21BDF(X,Y,Z,R,IFAIL)
C MARK 14 RE-ISSUE. NAG COPYRIGHT 1989.
C
C ******************************************************************
C
C Calculates an approximate value for the elliptic integral of
C the third kind, Rj(X,Y,Z,R), as defined by B.C.Carlson.
C - Special Functions of Applied Maths - Academic Press(1977).
C The algorithm is also due to Carlson
C
C Rj(X,Y,Z,R)=integral(zero to infinity) of
C 1.5/(sqrt((t+X)*(t+Y)*(t+Z))*(t+R)) dt
C
C for X.ge.0.0 , Y.ge.0.0 , Z.ge.0.0
C X+Y.gt.0.0
C Y+Z.gt.0.0
C Z+X.gt.0.0
C R.ne.0.0
C
C ******************************************************************
C Precision-dependent constant
C
C ACC Controls final truncation accuracy. An accuracy
C close to full machine precision can be obtained
C for all legal arguments if ACC is chosen so that
C 3.0*ACC**6/(1.0-ACC)**1.5.le.X02AJF/X02BHF
C
C To select the correct value for a particular machine-range,
C activate the statement contained in a comment beginning CDD,
C where DD is the approximate number of significant decimal
C digits represented by the machine.
C * EXPANSION (DATA) *
C ******************************************************************
C
S21CAF
SUBROUTINE S21CAF(U,M,SN,CN,DN,IFAIL)
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C
C Returns Jacobian elliptic functions sn, cn, dn.
C Based on routine by R. Bulirsch (Num. Math. 7, pp 76-90, 1965).
C
X01AAF
DOUBLE PRECISION FUNCTION X01AAF(X)
C MARK 8 RE-ISSUE. NAG COPYRIGHT 1980.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C RETURNS THE VALUE OF THE MATHEMATICAL CONSTANT PI.
C
C X IS A DUMMY ARGUMENT
C
C IT MAY BE NECESSARY TO ROUND THE REAL CONSTANT IN THE
C ASSIGNMENT STATEMENT TO A SMALLER NUMBER OF SIGNIFICANT
C DIGITS IN ORDER TO AVOID COMPILATION PROBLEMS. IF SO, THEN
C THE NUMBER OF DIGITS RETAINED SHOULD NOT BE LESS THAN
C . 2 + INT(FLOAT(IT)*ALOG10(IB))
C WHERE IB IS THE BASE FOR THE REPRESENTATION OF FLOATING-
C . POINT NUMBERS
C . AND IT IS THE NUMBER OF IB-ARY DIGITS IN THE MANTISSA OF
C . A FLOATING-POINT NUMBER.
C
X01ABF
DOUBLE PRECISION FUNCTION X01ABF(X)
C MARK 8 RE-ISSUE. NAG COPYRIGHT 1980.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C RETURNS THE VALUE OF THE MATHEMATICAL CONSTANT GAMMA.
C
C X IS A DUMMY ARGUMENT
C
C IT MAY BE NECESSARY TO ROUND THE REAL CONSTANT IN THE
C ASSIGNMENT STATEMENT TO A SMALLER NUMBER OF SIGNIFICANT
C DIGITS IN ORDER TO AVOID COMPILATION PROBLEMS. IF SO, THEN
C THE NUMBER OF DIGITS RETAINED SHOULD NOT BE LESS THAN
C . 2 + INT(FLOAT(IT)*ALOG10(IB))
C WHERE IB IS THE BASE FOR THE REPRESENTATION OF FLOATING-
C . POINT NUMBERS
C . AND IT IS THE NUMBER OF IB-ARY DIGITS IN THE MANTISSA OF
C . A FLOATING-POINT NUMBER.
C
X02AHF
DOUBLE PRECISION FUNCTION X02AHF(X)
C MARK 9 RELEASE. NAG COPYRIGHT 1981.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C * MAXIMUM ARGUMENT FOR SIN AND COS *
C RETURNS THE LARGEST POSITIVE REAL NUMBER MAXSC SUCH THAT
C SIN(MAXSC) AND COS(MAXSC) CAN BE SUCCESSFULLY COMPUTED
C BY THE COMPILER SUPPLIED SIN AND COS ROUTINES.
C
X02AJF
DOUBLE PRECISION FUNCTION X02AJF()
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C RETURNS (1/2)*B**(1-P) IF ROUNDS IS .TRUE.
C RETURNS B**(1-P) OTHERWISE
C
DOUBLE PRECISION X02CON
INTEGER Z(2)
EQUIVALENCE (Z,X02CON)
C DATA X02CON /Z'3CA0000000000001' /
DATA Z /Z'00000001',Z'3CA00000' /
X02AKF
DOUBLE PRECISION FUNCTION X02AKF()
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C RETURNS B**(EMIN-1) (THE SMALLEST POSITIVE MODEL NUMBER)
C
DOUBLE PRECISION X02CON
INTEGER Z(2)
EQUIVALENCE (Z,X02CON)
C DATA X02CON /Z'0010000000000000' /
DATA Z /Z'00000000',Z'00100000' /
X02ALF
DOUBLE PRECISION FUNCTION X02ALF()
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C RETURNS (1 - B**(-P)) * B**EMAX (THE LARGEST POSITIVE MODEL
C NUMBER)
C
DOUBLE PRECISION X02CON
INTEGER Z(2)
EQUIVALENCE (Z,X02CON)
C DATA X02CON /Z'7FEFFFFFFFFFFFFF' /
DATA Z /Z'FFFFFFFF',Z'7FEFFFFF' /
X02AMF
DOUBLE PRECISION FUNCTION X02AMF()
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C RETURNS THE 'SAFE RANGE' PARAMETER
C I.E. THE SMALLEST POSITIVE MODEL NUMBER Z SUCH THAT
C FOR ANY X WHICH SATISFIES X.GE.Z AND X.LE.1/Z
C THE FOLLOWING CAN BE COMPUTED WITHOUT OVERFLOW, UNDERFLOW OR OTHER
C ERROR
C
C -X
C 1.0/X
C SQRT(X)
C LOG(X)
C EXP(LOG(X))
C Y**(LOG(X)/LOG(Y)) FOR ANY Y
C
DOUBLE PRECISION X02CON
INTEGER Z(2)
EQUIVALENCE (Z,X02CON)
C DATA X02CON /Z'0010000000000000' /
DATA Z /Z'00000000',Z'00100000' /
X02ANF
DOUBLE PRECISION FUNCTION X02ANF()
C MARK 15 RELEASE. NAG COPYRIGHT 1991.
C
C Returns the 'safe range' parameter for complex numbers,
C i.e. the smallest positive model number Z such that
C for any X which satisfies X.ge.Z and X.le.1/Z
C the following can be computed without overflow, underflow or other
C error
C
C -W
C 1.0/W
C SQRT(W)
C LOG(W)
C EXP(LOG(W))
C Y**(LOG(W)/LOG(Y)) for any Y
C ABS(W)
C
C where W is any of cmplx(X,0), cmplx(0,X), cmplx(X,X),
C cmplx(1/X,0), cmplx(0,1/X), cmplx(1/X,1/X).
C
DOUBLE PRECISION X02CON
INTEGER Z(2)
EQUIVALENCE (Z,X02CON)
C DATA X02CON /Z'0020000000000000' /
DATA Z /Z'00000000',Z'00200000' /
X02BBF
INTEGER FUNCTION X02BBF(X)
C NAG COPYRIGHT 1975
C MARK 4.5 RELEASE
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C * MAXINT *
C RETURNS THE LARGEST INTEGER REPRESENTABLE ON THE COMPUTER
C THE X PARAMETER IS NOT USED
X02BEF
INTEGER FUNCTION X02BEF(X)
C NAG COPYRIGHT 1975
C MARK 4.5 RELEASE
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C * MAXDEC *
C RETURNS THE MAXIMUM NUMBER OF DECIMAL DIGITS THAT CAN BE
C ACCURATELY REPRESENTED IN THE COMPUTER OVER THE WHOLE RANGE
C OF FLOATING-POINT NUMBERS.
C THE X PARAMETER IS NOT USED.
X02BHF
INTEGER FUNCTION X02BHF()
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C RETURNS THE MODEL PARAMETER, B.
C
X02BJF
INTEGER FUNCTION X02BJF()
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C RETURNS THE MODEL PARAMETER, p.
C
X02BKF
INTEGER FUNCTION X02BKF()
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C RETURNS THE MODEL PARAMETER, EMIN.
C
X02BLF
INTEGER FUNCTION X02BLF()
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C RETURNS THE MODEL PARAMETER, EMAX.
C
X02DAF
LOGICAL FUNCTION X02DAF(X)
C MARK 8 RELEASE. NAG COPYRIGHT 1980.
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C
C RETURNS .FALSE. IF THE SYSTEM SETS UNDERFLOWING QUANTITIES
C TO ZERO, WITHOUT ANY ERROR INDICATION OR UNDESIRABLE WARNING
C OR SYSTEM OVERHEAD.
C RETURNS .TRUE. OTHERWISE, IN WHICH CASE CERTAIN LIBRARY
C ROUTINES WILL TAKE SPECIAL PRECAUTIONS TO AVOID UNDERFLOW
C (USUALLY AT SOME COST IN EFFICIENCY).
C
C X IS A DUMMY ARGUMENT
C
X02DJF
LOGICAL FUNCTION X02DJF()
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C RETURNS THE MODEL PARAMETER, ROUNDS.
C
X03AAF
SUBROUTINE X03AAF(A,ISIZEA,B,ISIZEB,N,ISTEPA,ISTEPB,C1,C2,D1,D2,
* SW,IFAIL)
C MARK 10 RE-ISSUE. NAG COPYRIGHT 1982
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 12 REVISED. IER-524 (AUG 1986).
C DOUBLE PRECISION BASE VERSION
C
C CALCULATES THE VALUE OF A SCALAR PRODUCT USING BASIC OR
C ADDITIONAL PRECISION AND ADDS IT TO A BASIC OR ADDITIONAL
C PRECISION INITIAL VALUE.
C
C FOR THIS DOUBLE PRECISION VERSION, ALL ADDITIONAL (I.E.
C QUADRUPLE) PRECISION COMPUTATION IS PERFORMED BY THE AUXILIARY
C ROUTINE X03AAY. SEE THE COMMENTS AT THE HEAD OF THAT ROUTINE
C CONCERNING IMPLEMENTATION.
C
C
X03ABF
SUBROUTINE X03ABF(A,ISIZEA,B,ISIZEB,N,ISTEPA,ISTEPB,CX,DX,SW,
* IFAIL)
C MARK 10 RE-ISSUE. NAG COPYRIGHT 1982
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C DOUBLE PRECISION COMPLEX BASE VERSION
C
C CALCULATES THE VALUE OF A COMPLEX SCALAR PRODUCT USING BASIC
C OR ADDITIONAL PRECISION AND ADDS IT TO A BASIC PRECISION
C INITIAL VALUE.
C
C FOR THIS DOUBLE PRECISION COMPLEX VERSION, ALL ADDITIONAL
C (I.E. QUADRUPLE) PRECISION COMPUTATION IS PERFORMED BY THE
C AUXILIARY ROUTINE X03ABY. SEE THE COMMENTS AT THE HEAD OF
C THAT ROUTINE CONCERNING IMPLEMENTATION.
C
C
X04AAF
SUBROUTINE X04AAF(I,NERR)
C MARK 7 RELEASE. NAG COPYRIGHT 1978
C MARK 7C REVISED IER-190 (MAY 1979)
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 14 REVISED. IER-829 (DEC 1989).
C IF I = 0, SETS NERR TO CURRENT ERROR MESSAGE UNIT NUMBER
C (STORED IN NERR1).
C IF I = 1, CHANGES CURRENT ERROR MESSAGE UNIT NUMBER TO
C VALUE SPECIFIED BY NERR.
C
X04ABF
SUBROUTINE X04ABF(I,NADV)
C MARK 7 RELEASE. NAG COPYRIGHT 1978
C MARK 7C REVISED IER-190 (MAY 1979)
C MARK 11.5(F77) REVISED. (SEPT 1985.)
C MARK 14 REVISED. IER-830 (DEC 1989).
C IF I = 0, SETS NADV TO CURRENT ADVISORY MESSAGE UNIT NUMBER
C (STORED IN NADV1).
C IF I = 1, CHANGES CURRENT ADVISORY MESSAGE UNIT NUMBER TO
C VALUE SPECIFIED BY NADV.
C
X04BAF
SUBROUTINE X04BAF(NOUT,REC)
C MARK 11.5(F77) RELEASE. NAG COPYRIGHT 1986.
C
C X04BAF writes the contents of REC to the unit defined by NOUT.
C
C Trailing blanks are not output, except that if REC is entirely
C blank, a single blank character is output.
C If NOUT.lt.0, i.e. if NOUT is not a valid Fortran unit identifier,
C then no output occurs.
C
X04BBF
SUBROUTINE X04BBF(NIN,REC,IFAIL)
C MARK 12 RELEASE. NAG COPYRIGHT 1986.
C
C X04BBF reads a record from device NIN into REC.
C IFAIL is set to unity if an End-Of-File is found.
C
X04CAF
SUBROUTINE X04CAF(MATRIX,DIAG,M,N,A,LDA,TITLE,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C Prints a general real matrix.
C Easy to use driver for X04CBF.
X04CBF
SUBROUTINE X04CBF(MATRIX,DIAG,M,N,A,LDA,FORMAT,TITLE,LABROW,RLABS,
* LABCOL,CLABS,NCOLS,INDENT,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C MARK 16A REVISED. IER-1047 (JUN 1993).
C Prints a general real matrix.
X04CCF
SUBROUTINE X04CCF(UPLO,DIAG,N,A,TITLE,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C Prints a real triangular matrix stored in packed vector form.
C Easy to use driver for X04CDF.
X04CDF
SUBROUTINE X04CDF(UPLO,DIAG,N,A,FORMAT,TITLE,LABROW,RLABS,LABCOL,
* CLABS,NCOLS,INDENT,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C MARK 16A REVISED. IER-1048 (JUN 1993).
C Prints a real triangular matrix stored in packed vector form.
X04CEF
SUBROUTINE X04CEF(M,N,KL,KU,A,LDA,TITLE,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C Prints a real banded matrix stored in packed form.
C Easy to use driver for X04CFF.
X04CFF
SUBROUTINE X04CFF(M,N,KL,KU,A,LDA,FORMAT,TITLE,LABROW,RLABS,
* LABCOL,CLABS,NCOLS,INDENT,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C Prints a real banded matrix stored in packed form.
X04DAF
SUBROUTINE X04DAF(MATRIX,DIAG,M,N,A,LDA,TITLE,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C Prints a general complex matrix.
C Easy to use driver for X04DBF.
X04DBF
SUBROUTINE X04DBF(MATRIX,DIAG,M,N,A,LDA,USEFRM,FORMAT,TITLE,
* LABROW,RLABS,LABCOL,CLABS,NCOLS,INDENT,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C MARK 16A REVISED. IER-1049 (JUN 1993).
C Prints a general complex matrix.
X04DCF
SUBROUTINE X04DCF(UPLO,DIAG,N,A,TITLE,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C Prints a complex triangular matrix stored in packed vector form.
C Easy to use driver for X04DDF.
X04DDF
SUBROUTINE X04DDF(UPLO,DIAG,N,A,USEFRM,FORMAT,TITLE,LABROW,RLABS,
* LABCOL,CLABS,NCOLS,INDENT,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C Prints a complex triangular matrix stored in packed vector form.
X04DEF
SUBROUTINE X04DEF(M,N,KL,KU,A,LDA,TITLE,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C Prints a complex banded matrix stored in packed form.
C Easy to use driver for X04DFF.
X04DFF
SUBROUTINE X04DFF(M,N,KL,KU,A,LDA,USEFRM,FORMAT,TITLE,LABROW,
* RLABS,LABCOL,CLABS,NCOLS,INDENT,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C Prints a complex banded matrix stored in packed form.
X04EAF
SUBROUTINE X04EAF(MATRIX,DIAG,M,N,A,LDA,TITLE,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C Prints a general integer matrix.
C Easy to use driver for X04EBF.
X04EBF
SUBROUTINE X04EBF(MATRIX,DIAG,M,N,A,LDA,FORMAT,TITLE,LABROW,RLABS,
* LABCOL,CLABS,NCOLS,INDENT,IFAIL)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C MARK 16A REVISED. IER-1050 (JUN 1993).
C Prints a general integer matrix.
X05AAF
SUBROUTINE X05AAF(ITIME)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C Returns the current date and time in the integer array ITIME.
C On exit, ITIME should contain the following values:
C ITIME(1) : the current year.
C ITIME(2) : the current month, in the range 1 - 12.
C ITIME(3) : the current day, in the range 1 - 31.
C ITIME(4) : the current hour, in the range 0 - 23.
C ITIME(5) : the current minute, in the range 0 - 59.
C ITIME(6) : the current second, in the range 0 - 59.
C ITIME(7) : the current millisecond, in the range 0 - 999.
C
INTEGER TIME,C
C INTEGER*8 TARRAY(9)
INTEGER*4 TARRAY(9)
INTEGER ITIME(7)
EXTERNAL TIME,LTIME
C=TIME()
CALL LTIME(C,TARRAY)
ITIME(1) = TARRAY(6)+1900
ITIME(2) = TARRAY(5)+1
ITIME(3) = TARRAY(4)
ITIME(4) = TARRAY(3)
ITIME(5) = TARRAY(2)
ITIME(6) = TARRAY(1)
ITIME(7) = 0
C
RETURN
END
X05ABF
CHARACTER*30 FUNCTION X05ABF(ITIME)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C Converts from the seven-integer format date/time in ITIME, as
C returned by subroutine X05AAF, into a character string.
C The character string has the format, for example,
C 'Thu 20th Apr 1989 16:37:57.320' .
C
X05ACF
INTEGER FUNCTION X05ACF(CTIME1,CTIME2)
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C Compares the date/time strings CTIME1 and CTIME2.
C Returns -1 if CTIME1 is earlier than CTIME2,
C 0 if CTIME1 is the same as CTIME2,
C and 1 if CTIME1 is later than CTIME2.
C
X05BAF
DOUBLE PRECISION FUNCTION X05BAF()
C MARK 14 RELEASE. NAG COPYRIGHT 1989.
C
C Returns the amount of CPU time used since some previous time,
C in seconds. The previous time is arbitrary, but may be the time
C that the job or the program started, for example. The difference
C between two separate calls of this routine is then (approximately)
C the CPU time used between the calls.
C
C
EXTERNAL ETIME
REAL ETIME
X05BAF = ETIME(0)
RETURN
END