SLATEC Routines --- DPIGMR ---


*DECK DPIGMR
      SUBROUTINE DPIGMR (N, R0, SR, SZ, JSCAL, MAXL, MAXLP1, KMP, NRSTS,
     +   JPRE, MATVEC, MSOLVE, NMSL, Z, V, HES, Q, LGMR, RPAR, IPAR, WK,
     +   DL, RHOL, NRMAX, B, BNRM, X, XL, ITOL, TOL, NELT, IA, JA, A,
     +   ISYM, IUNIT, IFLAG, ERR)
C***BEGIN PROLOGUE  DPIGMR
C***SUBSIDIARY
C***PURPOSE  Internal routine for DGMRES.
C***LIBRARY   SLATEC (SLAP)
C***CATEGORY  D2A4, D2B4
C***TYPE      DOUBLE PRECISION (SPIGMR-S, DPIGMR-D)
C***KEYWORDS  GENERALIZED MINIMUM RESIDUAL, ITERATIVE PRECONDITION,
C             NON-SYMMETRIC LINEAR SYSTEM, SLAP, SPARSE
C***AUTHOR  Brown, Peter, (LLNL), pnbrown@llnl.gov
C           Hindmarsh, Alan, (LLNL), alanh@llnl.gov
C           Seager, Mark K., (LLNL), seager@llnl.gov
C             Lawrence Livermore National Laboratory
C             PO Box 808, L-60
C             Livermore, CA 94550 (510) 423-3141
C***DESCRIPTION
C         This routine solves the linear system A * Z = R0 using a
C         scaled preconditioned version of the generalized minimum
C         residual method.  An initial guess of Z = 0 is assumed.
C
C *Usage:
C      INTEGER N, JSCAL, MAXL, MAXLP1, KMP, NRSTS, JPRE, NMSL, LGMR
C      INTEGER IPAR(USER DEFINED), NRMAX, ITOL, NELT, IA(NELT), JA(NELT)
C      INTEGER ISYM, IUNIT, IFLAG
C      DOUBLE PRECISION R0(N), SR(N), SZ(N), Z(N), V(N,MAXLP1),
C     $                 HES(MAXLP1,MAXL), Q(2*MAXL), RPAR(USER DEFINED),
C     $                 WK(N), DL(N), RHOL, B(N), BNRM, X(N), XL(N),
C     $                 TOL, A(NELT), ERR
C      EXTERNAL MATVEC, MSOLVE
C
C      CALL DPIGMR(N, R0, SR, SZ, JSCAL, MAXL, MAXLP1, KMP,
C     $     NRSTS, JPRE, MATVEC, MSOLVE, NMSL, Z, V, HES, Q, LGMR,
C     $     RPAR, IPAR, WK, DL, RHOL, NRMAX, B, BNRM, X, XL,
C     $     ITOL, TOL, NELT, IA, JA, A, ISYM, IUNIT, IFLAG, ERR)
C
C *Arguments:
C N      :IN       Integer
C         The order of the matrix A, and the lengths
C         of the vectors SR, SZ, R0 and Z.
C R0     :IN       Double Precision R0(N)
C         R0 = the right hand side of the system A*Z = R0.
C         R0 is also used as workspace when computing
C         the final approximation.
C         (R0 is the same as V(*,MAXL+1) in the call to DPIGMR.)
C SR     :IN       Double Precision SR(N)
C         SR is a vector of length N containing the non-zero
C         elements of the diagonal scaling matrix for R0.
C SZ     :IN       Double Precision SZ(N)
C         SZ is a vector of length N containing the non-zero
C         elements of the diagonal scaling matrix for Z.
C JSCAL  :IN       Integer
C         A flag indicating whether arrays SR and SZ are used.
C         JSCAL=0 means SR and SZ are not used and the
C                 algorithm will perform as if all
C                 SR(i) = 1 and SZ(i) = 1.
C         JSCAL=1 means only SZ is used, and the algorithm
C                 performs as if all SR(i) = 1.
C         JSCAL=2 means only SR is used, and the algorithm
C                 performs as if all SZ(i) = 1.
C         JSCAL=3 means both SR and SZ are used.
C MAXL   :IN       Integer
C         The maximum allowable order of the matrix H.
C MAXLP1 :IN       Integer
C         MAXPL1 = MAXL + 1, used for dynamic dimensioning of HES.
C KMP    :IN       Integer
C         The number of previous vectors the new vector VNEW
C         must be made orthogonal to.  (KMP .le. MAXL)
C NRSTS  :IN       Integer
C         Counter for the number of restarts on the current
C         call to DGMRES.  If NRSTS .gt. 0, then the residual
C         R0 is already scaled, and so scaling of it is
C         not necessary.
C JPRE   :IN       Integer
C         Preconditioner type flag.
C MATVEC :EXT      External.
C         Name of a routine which performs the matrix vector multiply
C         Y = A*X given A and X.  The name of the MATVEC routine must
C         be declared external in the calling program.  The calling
C         sequence to MATVEC is:
C             CALL MATVEC(N, X, Y, NELT, IA, JA, A, ISYM)
C         where N is the number of unknowns, Y is the product A*X
C         upon return, X is an input vector, and NELT is the number of
C         non-zeros in the SLAP IA, JA, A storage for the matrix A.
C         ISYM is a flag which, if non-zero, denotes that A is
C         symmetric and only the lower or upper triangle is stored.
C MSOLVE :EXT      External.
C         Name of the routine which solves a linear system Mz = r for
C         z given r with the preconditioning matrix M (M is supplied via
C         RPAR and IPAR arrays.  The name of the MSOLVE routine must
C         be declared external in the calling program.  The calling
C         sequence to MSOLVE is:
C             CALL MSOLVE(N, R, Z, NELT, IA, JA, A, ISYM, RPAR, IPAR)
C         Where N is the number of unknowns, R is the right-hand side
C         vector and Z is the solution upon return.  NELT, IA, JA, A and
C         ISYM are defined as below.  RPAR is a double precision array
C         that can be used to pass necessary preconditioning information
C         and/or workspace to MSOLVE.  IPAR is an integer work array
C         for the same purpose as RPAR.
C NMSL   :OUT      Integer
C         The number of calls to MSOLVE.
C Z      :OUT      Double Precision Z(N)
C         The final computed approximation to the solution
C         of the system A*Z = R0.
C V      :OUT      Double Precision V(N,MAXLP1)
C         The N by (LGMR+1) array containing the LGMR
C         orthogonal vectors V(*,1) to V(*,LGMR).
C HES    :OUT      Double Precision HES(MAXLP1,MAXL)
C         The upper triangular factor of the QR decomposition
C         of the (LGMR+1) by LGMR upper Hessenberg matrix whose
C         entries are the scaled inner-products of A*V(*,I)
C         and V(*,K).
C Q      :OUT      Double Precision Q(2*MAXL)
C         A double precision array of length 2*MAXL containing the
C         components of the Givens rotations used in the QR
C         decomposition of HES.  It is loaded in DHEQR and used in
C         DHELS.
C LGMR   :OUT      Integer
C         The number of iterations performed and
C         the current order of the upper Hessenberg
C         matrix HES.
C RPAR   :IN       Double Precision RPAR(USER DEFINED)
C         Double Precision workspace passed directly to the MSOLVE
C         routine.
C IPAR   :IN       Integer IPAR(USER DEFINED)
C         Integer workspace passed directly to the MSOLVE routine.
C WK     :IN       Double Precision WK(N)
C         A double precision work array of length N used by routines
C         MATVEC and MSOLVE.
C DL     :INOUT    Double Precision DL(N)
C         On input, a double precision work array of length N used for
C         calculation of the residual norm RHO when the method is
C         incomplete (KMP.lt.MAXL), and/or when using restarting.
C         On output, the scaled residual vector RL.  It is only loaded
C         when performing restarts of the Krylov iteration.
C RHOL   :OUT      Double Precision
C         A double precision scalar containing the norm of the final
C         residual.
C NRMAX  :IN       Integer
C         The maximum number of restarts of the Krylov iteration.
C         NRMAX .gt. 0 means restarting is active, while
C         NRMAX = 0 means restarting is not being used.
C B      :IN       Double Precision B(N)
C         The right hand side of the linear system A*X = b.
C BNRM   :IN       Double Precision
C         The scaled norm of b.
C X      :IN       Double Precision X(N)
C         The current approximate solution as of the last
C         restart.
C XL     :IN       Double Precision XL(N)
C         An array of length N used to hold the approximate
C         solution X(L) when ITOL=11.
C ITOL   :IN       Integer
C         A flag to indicate the type of convergence criterion
C         used.  See the driver for its description.
C TOL    :IN       Double Precision
C         The tolerance on residuals R0-A*Z in scaled norm.
C NELT   :IN       Integer
C         The length of arrays IA, JA and A.
C IA     :IN       Integer IA(NELT)
C         An integer array of length NELT containing matrix data.
C         It is passed directly to the MATVEC and MSOLVE routines.
C JA     :IN       Integer JA(NELT)
C         An integer array of length NELT containing matrix data.
C         It is passed directly to the MATVEC and MSOLVE routines.
C A      :IN       Double Precision A(NELT)
C         A double precision array of length NELT containing matrix
C         data. It is passed directly to the MATVEC and MSOLVE routines.
C ISYM   :IN       Integer
C         A flag to indicate symmetric matrix storage.
C         If ISYM=0, all non-zero entries of the matrix are
C         stored.  If ISYM=1, the matrix is symmetric and
C         only the upper or lower triangular part is stored.
C IUNIT  :IN       Integer
C         The i/o unit number for writing intermediate residual
C         norm values.
C IFLAG  :OUT      Integer
C         An integer error flag..
C         0 means convergence in LGMR iterations, LGMR.le.MAXL.
C         1 means the convergence test did not pass in MAXL
C           iterations, but the residual norm is .lt. norm(R0),
C           and so Z is computed.
C         2 means the convergence test did not pass in MAXL
C           iterations, residual .ge. norm(R0), and Z = 0.
C ERR    :OUT      Double Precision.
C         Error estimate of error in final approximate solution, as
C         defined by ITOL.
C
C *Cautions:
C     This routine will attempt to write to the Fortran logical output
C     unit IUNIT, if IUNIT .ne. 0.  Thus, the user must make sure that
C     this logical unit is attached to a file or terminal before calling
C     this routine with a non-zero value for IUNIT.  This routine does
C     not check for the validity of a non-zero IUNIT unit number.
C
C***SEE ALSO  DGMRES
C***ROUTINES CALLED  DAXPY, DCOPY, DHELS, DHEQR, DNRM2, DORTH, DRLCAL,
C                    DSCAL, ISDGMR
C***REVISION HISTORY  (YYMMDD)
C   890404  DATE WRITTEN
C   890404  Previous REVISION DATE
C   890915  Made changes requested at July 1989 CML Meeting.  (MKS)
C   890922  Numerous changes to prologue to make closer to SLATEC
C           standard.  (FNF)
C   890929  Numerous changes to reduce SP/DP differences.  (FNF)
C   910411  Prologue converted to Version 4.0 format.  (BAB)
C   910502  Removed MATVEC and MSOLVE from ROUTINES CALLED list.  (FNF)
C   910506  Made subsidiary to DGMRES.  (FNF)
C   920511  Added complete declaration section.  (WRB)
C***END PROLOGUE  DPIGMR