SLATEC Routines --- SGMRES ---
*DECK SGMRES
SUBROUTINE SGMRES (N, B, X, NELT, IA, JA, A, ISYM, MATVEC, MSOLVE,
+ ITOL, TOL, ITMAX, ITER, ERR, IERR, IUNIT, SB, SX, RGWK, LRGW,
+ IGWK, LIGW, RWORK, IWORK)
C***BEGIN PROLOGUE SGMRES
C***PURPOSE Preconditioned GMRES Iterative Sparse Ax=b Solver.
C This routine uses the generalized minimum residual
C (GMRES) method with preconditioning to solve
C non-symmetric linear systems of the form: Ax = b.
C***LIBRARY SLATEC (SLAP)
C***CATEGORY D2A4, D2B4
C***TYPE SINGLE PRECISION (SGMRES-S, DGMRES-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
C *Usage:
C INTEGER N, NELT, IA(NELT), JA(NELT), ISYM, ITOL, ITMAX
C INTEGER ITER, IERR, IUNIT, LRGW, IGWK(LIGW), LIGW
C INTEGER IWORK(USER DEFINED)
C REAL B(N), X(N), A(NELT), TOL, ERR, SB(N), SX(N)
C REAL RGWK(LRGW), RWORK(USER DEFINED)
C EXTERNAL MATVEC, MSOLVE
C
C CALL SGMRES(N, B, X, NELT, IA, JA, A, ISYM, MATVEC, MSOLVE,
C $ ITOL, TOL, ITMAX, ITER, ERR, IERR, IUNIT, SB, SX,
C $ RGWK, LRGW, IGWK, LIGW, RWORK, IWORK)
C
C *Arguments:
C N :IN Integer.
C Order of the Matrix.
C B :IN Real B(N).
C Right-hand side vector.
C X :INOUT Real X(N).
C On input X is your initial guess for the solution vector.
C On output X is the final approximate solution.
C NELT :IN Integer.
C Number of Non-Zeros stored in A.
C IA :IN Integer IA(NELT).
C JA :IN Integer JA(NELT).
C A :IN Real A(NELT).
C These arrays contain the matrix data structure for A.
C It could take any form. See "Description", below,
C for more details.
C ISYM :IN Integer.
C Flag to indicate symmetric storage format.
C If ISYM=0, all non-zero entries of the matrix are stored.
C If ISYM=1, the matrix is symmetric, and only the upper
C or lower triangle of the matrix is stored.
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 RWORK and IWORK 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, RWORK, IWORK)
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 above. RWORK is a real array that can
C be used to pass necessary preconditioning information and/or
C workspace to MSOLVE. IWORK is an integer work array for
C the same purpose as RWORK.
C ITOL :IN Integer.
C Flag to indicate the type of convergence criterion used.
C ITOL=0 Means the iteration stops when the test described
C below on the residual RL is satisfied. This is
C the "Natural Stopping Criteria" for this routine.
C Other values of ITOL cause extra, otherwise
C unnecessary, computation per iteration and are
C therefore much less efficient. See ISSGMR (the
C stop test routine) for more information.
C ITOL=1 Means the iteration stops when the first test
C described below on the residual RL is satisfied,
C and there is either right or no preconditioning
C being used.
C ITOL=2 Implies that the user is using left
C preconditioning, and the second stopping criterion
C below is used.
C ITOL=3 Means the iteration stops when the third test
C described below on Minv*Residual is satisfied, and
C there is either left or no preconditioning being
C used.
C ITOL=11 is often useful for checking and comparing
C different routines. For this case, the user must
C supply the "exact" solution or a very accurate
C approximation (one with an error much less than
C TOL) through a common block,
C COMMON /SSLBLK/ SOLN( )
C If ITOL=11, iteration stops when the 2-norm of the
C difference between the iterative approximation and
C the user-supplied solution divided by the 2-norm
C of the user-supplied solution is less than TOL.
C Note that this requires the user to set up the
C "COMMON /SSLBLK/ SOLN(LENGTH)" in the calling
C routine. The routine with this declaration should
C be loaded before the stop test so that the correct
C length is used by the loader. This procedure is
C not standard Fortran and may not work correctly on
C your system (although it has worked on every
C system the authors have tried). If ITOL is not 11
C then this common block is indeed standard Fortran.
C TOL :INOUT Real.
C Convergence criterion, as described below. If TOL is set
C to zero on input, then a default value of 500*(the smallest
C positive magnitude, machine epsilon) is used.
C ITMAX :DUMMY Integer.
C Maximum number of iterations in most SLAP routines. In
C this routine this does not make sense. The maximum number
C of iterations here is given by ITMAX = MAXL*(NRMAX+1).
C See IGWK for definitions of MAXL and NRMAX.
C ITER :OUT Integer.
C Number of iterations required to reach convergence, or
C ITMAX if convergence criterion could not be achieved in
C ITMAX iterations.
C ERR :OUT Real.
C Error estimate of error in final approximate solution, as
C defined by ITOL. Letting norm() denote the Euclidean
C norm, ERR is defined as follows..
C
C If ITOL=0, then ERR = norm(SB*(B-A*X(L)))/norm(SB*B),
C for right or no preconditioning, and
C ERR = norm(SB*(M-inverse)*(B-A*X(L)))/
C norm(SB*(M-inverse)*B),
C for left preconditioning.
C If ITOL=1, then ERR = norm(SB*(B-A*X(L)))/norm(SB*B),
C since right or no preconditioning
C being used.
C If ITOL=2, then ERR = norm(SB*(M-inverse)*(B-A*X(L)))/
C norm(SB*(M-inverse)*B),
C since left preconditioning is being
C used.
C If ITOL=3, then ERR = Max |(Minv*(B-A*X(L)))(i)/x(i)|
C i=1,n
C If ITOL=11, then ERR = norm(SB*(X(L)-SOLN))/norm(SB*SOLN).
C IERR :OUT Integer.
C Return error flag.
C IERR = 0 => All went well.
C IERR = 1 => Insufficient storage allocated for
C RGWK or IGWK.
C IERR = 2 => Routine SGMRES failed to reduce the norm
C of the current residual on its last call,
C and so the iteration has stalled. In
C this case, X equals the last computed
C approximation. The user must either
C increase MAXL, or choose a different
C initial guess.
C IERR =-1 => Insufficient length for RGWK array.
C IGWK(6) contains the required minimum
C length of the RGWK array.
C IERR =-2 => Illegal value of ITOL, or ITOL and JPRE
C values are inconsistent.
C For IERR <= 2, RGWK(1) = RHOL, which is the norm on the
C left-hand-side of the relevant stopping test defined
C below associated with the residual for the current
C approximation X(L).
C IUNIT :IN Integer.
C Unit number on which to write the error at each iteration,
C if this is desired for monitoring convergence. If unit
C number is 0, no writing will occur.
C SB :IN Real SB(N).
C Array of length N containing scale factors for the right
C hand side vector B. If JSCAL.eq.0 (see below), SB need
C not be supplied.
C SX :IN Real SX(N).
C Array of length N containing scale factors for the solution
C vector X. If JSCAL.eq.0 (see below), SX need not be
C supplied. SB and SX can be the same array in the calling
C program if desired.
C RGWK :INOUT Real RGWK(LRGW).
C Real array used for workspace by SGMRES.
C On return, RGWK(1) = RHOL. See IERR for definition of RHOL.
C LRGW :IN Integer.
C Length of the real workspace, RGWK.
C LRGW >= 1 + N*(MAXL+6) + MAXL*(MAXL+3).
C See below for definition of MAXL.
C For the default values, RGWK has size at least 131 + 16*N.
C IGWK :INOUT Integer IGWK(LIGW).
C The following IGWK parameters should be set by the user
C before calling this routine.
C IGWK(1) = MAXL. Maximum dimension of Krylov subspace in
C which X - X0 is to be found (where, X0 is the initial
C guess). The default value of MAXL is 10.
C IGWK(2) = KMP. Maximum number of previous Krylov basis
C vectors to which each new basis vector is made orthogonal.
C The default value of KMP is MAXL.
C IGWK(3) = JSCAL. Flag indicating whether the scaling
C arrays SB and SX are to be used.
C JSCAL = 0 => SB and SX are not used and the algorithm
C will perform as if all SB(I) = 1 and SX(I) = 1.
C JSCAL = 1 => Only SX is used, and the algorithm
C performs as if all SB(I) = 1.
C JSCAL = 2 => Only SB is used, and the algorithm
C performs as if all SX(I) = 1.
C JSCAL = 3 => Both SB and SX are used.
C IGWK(4) = JPRE. Flag indicating whether preconditioning
C is being used.
C JPRE = 0 => There is no preconditioning.
C JPRE > 0 => There is preconditioning on the right
C only, and the solver will call routine MSOLVE.
C JPRE < 0 => There is preconditioning on the left
C only, and the solver will call routine MSOLVE.
C IGWK(5) = NRMAX. Maximum number of restarts of the
C Krylov iteration. The default value of NRMAX = 10.
C if IWORK(5) = -1, then no restarts are performed (in
C this case, NRMAX is set to zero internally).
C The following IWORK parameters are diagnostic information
C made available to the user after this routine completes.
C IGWK(6) = MLWK. Required minimum length of RGWK array.
C IGWK(7) = NMS. The total number of calls to MSOLVE.
C LIGW :IN Integer.
C Length of the integer workspace, IGWK. LIGW >= 20.
C RWORK :WORK Real RWORK(USER DEFINED).
C Real array that can be used for workspace in MSOLVE.
C IWORK :WORK Integer IWORK(USER DEFINED).
C Integer array that can be used for workspace in MSOLVE.
C
C *Description:
C SGMRES solves a linear system A*X = B rewritten in the form:
C
C (SB*A*(M-inverse)*(SX-inverse))*(SX*M*X) = SB*B,
C
C with right preconditioning, or
C
C (SB*(M-inverse)*A*(SX-inverse))*(SX*X) = SB*(M-inverse)*B,
C
C with left preconditioning, where A is an N-by-N real matrix,
C X and B are N-vectors, SB and SX are diagonal scaling
C matrices, and M is a preconditioning matrix. It uses
C preconditioned Krylov subpace methods based on the
C generalized minimum residual method (GMRES). This routine
C optionally performs either the full orthogonalization
C version of the GMRES algorithm or an incomplete variant of
C it. Both versions use restarting of the linear iteration by
C default, although the user can disable this feature.
C
C The GMRES algorithm generates a sequence of approximations
C X(L) to the true solution of the above linear system. The
C convergence criteria for stopping the iteration is based on
C the size of the scaled norm of the residual R(L) = B -
C A*X(L). The actual stopping test is either:
C
C norm(SB*(B-A*X(L))) .le. TOL*norm(SB*B),
C
C for right preconditioning, or
C
C norm(SB*(M-inverse)*(B-A*X(L))) .le.
C TOL*norm(SB*(M-inverse)*B),
C
C for left preconditioning, where norm() denotes the Euclidean
C norm, and TOL is a positive scalar less than one input by
C the user. If TOL equals zero when SGMRES is called, then a
C default value of 500*(the smallest positive magnitude,
C machine epsilon) is used. If the scaling arrays SB and SX
C are used, then ideally they should be chosen so that the
C vectors SX*X(or SX*M*X) and SB*B have all their components
C approximately equal to one in magnitude. If one wants to
C use the same scaling in X and B, then SB and SX can be the
C same array in the calling program.
C
C The following is a list of the other routines and their
C functions used by SGMRES:
C SPIGMR Contains the main iteration loop for GMRES.
C SORTH Orthogonalizes a new vector against older basis vectors.
C SHEQR Computes a QR decomposition of a Hessenberg matrix.
C SHELS Solves a Hessenberg least-squares system, using QR
C factors.
C SRLCAL Computes the scaled residual RL.
C SXLCAL Computes the solution XL.
C ISSGMR User-replaceable stopping routine.
C
C This routine does not care what matrix data structure is
C used for A and M. It simply calls the MATVEC and MSOLVE
C routines, with the arguments as described above. The user
C could write any type of structure and the appropriate MATVEC
C and MSOLVE routines. It is assumed that A is stored in the
C IA, JA, A arrays in some fashion and that M (or INV(M)) is
C stored in IWORK and RWORK in some fashion. The SLAP
C routines SSDCG and SSICCG are examples of this procedure.
C
C Two examples of matrix data structures are the: 1) SLAP
C Triad format and 2) SLAP Column format.
C
C =================== S L A P Triad format ===================
C This routine requires that the matrix A be stored in the
C SLAP Triad format. In this format only the non-zeros are
C stored. They may appear in *ANY* order. The user supplies
C three arrays of length NELT, where NELT is the number of
C non-zeros in the matrix: (IA(NELT), JA(NELT), A(NELT)). For
C each non-zero the user puts the row and column index of that
C matrix element in the IA and JA arrays. The value of the
C non-zero matrix element is placed in the corresponding
C location of the A array. This is an extremely easy data
C structure to generate. On the other hand it is not too
C efficient on vector computers for the iterative solution of
C linear systems. Hence, SLAP changes this input data
C structure to the SLAP Column format for the iteration (but
C does not change it back).
C
C Here is an example of the SLAP Triad storage format for a
C 5x5 Matrix. Recall that the entries may appear in any order.
C
C 5x5 Matrix SLAP Triad format for 5x5 matrix on left.
C 1 2 3 4 5 6 7 8 9 10 11
C |11 12 0 0 15| A: 51 12 11 33 15 53 55 22 35 44 21
C |21 22 0 0 0| IA: 5 1 1 3 1 5 5 2 3 4 2
C | 0 0 33 0 35| JA: 1 2 1 3 5 3 5 2 5 4 1
C | 0 0 0 44 0|
C |51 0 53 0 55|
C
C =================== S L A P Column format ==================
C
C This routine requires that the matrix A be stored in the
C SLAP Column format. In this format the non-zeros are stored
C counting down columns (except for the diagonal entry, which
C must appear first in each "column") and are stored in the
C real array A. In other words, for each column in the matrix
C put the diagonal entry in A. Then put in the other non-zero
C elements going down the column (except the diagonal) in
C order. The IA array holds the row index for each non-zero.
C The JA array holds the offsets into the IA, A arrays for the
C beginning of each column. That is, IA(JA(ICOL)),
C A(JA(ICOL)) points to the beginning of the ICOL-th column in
C IA and A. IA(JA(ICOL+1)-1), A(JA(ICOL+1)-1) points to the
C end of the ICOL-th column. Note that we always have
C JA(N+1) = NELT+1, where N is the number of columns in the
C matrix and NELT is the number of non-zeros in the matrix.
C
C Here is an example of the SLAP Column storage format for a
C 5x5 Matrix (in the A and IA arrays '|' denotes the end of a
C column):
C
C 5x5 Matrix SLAP Column format for 5x5 matrix on left.
C 1 2 3 4 5 6 7 8 9 10 11
C |11 12 0 0 15| A: 11 21 51 | 22 12 | 33 53 | 44 | 55 15 35
C |21 22 0 0 0| IA: 1 2 5 | 2 1 | 3 5 | 4 | 5 1 3
C | 0 0 33 0 35| JA: 1 4 6 8 9 12
C | 0 0 0 44 0|
C |51 0 53 0 55|
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***REFERENCES 1. Peter N. Brown and A. C. Hindmarsh, Reduced Storage
C Matrix Methods in Stiff ODE Systems, Lawrence Liver-
C more National Laboratory Report UCRL-95088, Rev. 1,
C Livermore, California, June 1987.
C 2. Mark K. Seager, A SLAP for the Masses, in
C G. F. Carey, Ed., Parallel Supercomputing: Methods,
C Algorithms and Applications, Wiley, 1989, pp.135-155.
C***ROUTINES CALLED R1MACH, SCOPY, SNRM2, SPIGMR
C***REVISION HISTORY (YYMMDD)
C 871001 DATE WRITTEN
C 881213 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 891004 Added new reference.
C 910411 Prologue converted to Version 4.0 format. (BAB)
C 910506 Corrected errors in C***ROUTINES CALLED list. (FNF)
C 920407 COMMON BLOCK renamed SSLBLK. (WRB)
C 920511 Added complete declaration section. (WRB)
C 920929 Corrected format of references. (FNF)
C 921019 Changed 500.0 to 500 to reduce SP/DP differences. (FNF)
C 921026 Added check for valid value of ITOL. (FNF)
C***END PROLOGUE SGMRES