DGMRES(3)

SYNOPSIS
SUBROUTINE DGMRES (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)

NAME
dgmres - Preconditioned GMRES iterative sparse Ax=b solver. This routine uses the generalized minimum residual (GMRES) method with preconditioning to solve non-symmetric linear systems of the form: Ax = b.

LIBRARY
SLATEC (SLAP)

CATEGORY
D2A4, D2B4

TYPE
DOUBLE PRECISION (SGMRES-S, DGMRES-D)

KEYWORDS
GENERALIZED MINIMUM RESIDUAL, ITERATIVE PRECONDITION, NON-SYMMETRIC LINEAR SYSTEM, SLAP, SPARSE

AUTHOR
Brown, Peter, (LLNL), pnbrown@llnl.gov Hindmarsh, Alan, (LLNL), alanh@llnl.gov Seager, Mark K., (LLNL), seager@llnl.gov Lawrence Livermore National Laboratory PO Box 808, L-60 Livermore, CA 94550 (510) 423-3141

DESCRIPTION
*Usage: INTEGER N, NELT, IA(NELT), JA(NELT), ISYM, ITOL, ITMAX INTEGER ITER, IERR, IUNIT, LRGW, IGWK(LIGW), LIGW INTEGER IWORK(USER DEFINED) DOUBLE PRECISION B(N), X(N), A(NELT), TOL, ERR, SB(N), SX(N) DOUBLE PRECISION RGWK(LRGW), RWORK(USER DEFINED) EXTERNAL MATVEC, MSOLVE CALL DGMRES(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) *Arguments: N :IN Integer. Order of the Matrix. B :IN Double Precision B(N). Right-hand side vector. X :INOUT Double Precision X(N). On input X is your initial guess for the solution vector. On output X is the final approximate solution. NELT :IN Integer. Number of Non-Zeros stored in A. IA :IN Integer IA(NELT). JA :IN Integer JA(NELT). A :IN Double Precision A(NELT). These arrays contain the matrix data structure for A. It could take any form. See "Description", below, for more details. ISYM :IN Integer. Flag to indicate symmetric storage format. If ISYM=0, all non-zero entries of the matrix are stored. If ISYM=1, the matrix is symmetric, and only the upper or lower triangle of the matrix is stored. MATVEC :EXT External. Name of a routine which performs the matrix vector multiply Y = A*X given A and X. The name of the MATVEC routine must be declared external in the calling program. The calling sequence to MATVEC is: CALL MATVEC(N, X, Y, NELT, IA, JA, A, ISYM) where N is the number of unknowns, Y is the product A*X upon return, X is an input vector, and NELT is the number of non-zeros in the SLAP IA, JA, A storage for the matrix A. ISYM is a flag which, if non-zero, denotes that A is symmetric and only the lower or upper triangle is stored. MSOLVE :EXT External. Name of the routine which solves a linear system Mz = r for z given r with the preconditioning matrix M (M is supplied via RWORK and IWORK arrays. The name of the MSOLVE routine must be declared external in the calling program. The calling sequence to MSOLVE is: CALL MSOLVE(N, R, Z, NELT, IA, JA, A, ISYM, RWORK, IWORK) Where N is the number of unknowns, R is the right-hand side vector and Z is the solution upon return. NELT, IA, JA, A and ISYM are defined as above. RWORK is a double precision array that can be used to pass necessary preconditioning information and/or workspace to MSOLVE. IWORK is an integer work array for the same purpose as RWORK. ITOL :IN Integer. Flag to indicate the type of convergence criterion used. ITOL=0 Means the iteration stops when the test described below on the residual RL is satisfied. This is the "Natural Stopping Criteria" for this routine. Other values of ITOL cause extra, otherwise unnecessary, computation per iteration and are therefore much less efficient. See ISDGMR (the stop test routine) for more information. ITOL=1 Means the iteration stops when the first test described below on the residual RL is satisfied, and there is either right or no preconditioning being used. ITOL=2 Implies that the user is using left preconditioning, and the second stopping criterion below is used. ITOL=3 Means the iteration stops when the third test described below on Minv*Residual is satisfied, and there is either left or no preconditioning being used. ITOL=11 is often useful for checking and comparing different routines. For this case, the user must supply the "exact" solution or a very accurate approximation (one with an error much less than TOL) through a common block, COMMON /DSLBLK/ SOLN( ) If ITOL=11, iteration stops when the 2-norm of the difference between the iterative approximation and the user-supplied solution divided by the 2-norm of the user-supplied solution is less than TOL. Note that this requires the user to set up the "COMMON /DSLBLK/ SOLN(LENGTH)" in the calling routine. The routine with this declaration should be loaded before the stop test so that the correct length is used by the loader. This procedure is not standard Fortran and may not work correctly on your system (although it has worked on every system the authors have tried). If ITOL is not 11 then this common block is indeed standard Fortran. TOL :INOUT Double Precision. Convergence criterion, as described below. If TOL is set to zero on input, then a default value of 500*(the smallest positive magnitude, machine epsilon) is used. ITMAX :DUMMY Integer. Maximum number of iterations in most SLAP routines. In this routine this does not make sense. The maximum number of iterations here is given by ITMAX = MAXL*(NRMAX+1). See IGWK for definitions of MAXL and NRMAX. ITER :OUT Integer. Number of iterations required to reach convergence, or ITMAX if convergence criterion could not be achieved in ITMAX iterations. ERR :OUT Double Precision. Error estimate of error in final approximate solution, as defined by ITOL. Letting norm() denote the Euclidean norm, ERR is defined as follows.. If ITOL=0, then ERR = norm(SB*(B-A*X(L)))/norm(SB*B), for right or no preconditioning, and ERR = norm(SB*(M-inverse)*(B-A*X(L)))/ norm(SB*(M-inverse)*B), for left preconditioning. If ITOL=1, then ERR = norm(SB*(B-A*X(L)))/norm(SB*B), since right or no preconditioning being used. If ITOL=2, then ERR = norm(SB*(M-inverse)*(B-A*X(L)))/ norm(SB*(M-inverse)*B), since left preconditioning is being used. If ITOL=3, then ERR = Max |(Minv*(B-A*X(L)))(i)/x(i)| i=1,n If ITOL=11, then ERR = norm(SB*(X(L)-SOLN))/norm(SB*SOLN). IERR :OUT Integer. Return error flag. IERR = 0 => All went well. IERR = 1 => Insufficient storage allocated for RGWK or IGWK. IERR = 2 => Routine DGMRES failed to reduce the norm of the current residual on its last call, and so the iteration has stalled. In this case, X equals the last computed approximation. The user must either increase MAXL, or choose a different initial guess. IERR =-1 => Insufficient length for RGWK array. IGWK(6) contains the required minimum length of the RGWK array. IERR =-2 => Illegal value of ITOL, or ITOL and JPRE values are inconsistent. For IERR <= 2, RGWK(1) = RHOL, which is the norm on the left-hand-side of the relevant stopping test defined below associated with the residual for the current approximation X(L). IUNIT :IN Integer. Unit number on which to write the error at each iteration, if this is desired for monitoring convergence. If unit number is 0, no writing will occur. SB :IN Double Precision SB(N). Array of length N containing scale factors for the right hand side vector B. If JSCAL.eq.0 (see below), SB need not be supplied. SX :IN Double Precision SX(N). Array of length N containing scale factors for the solution vector X. If JSCAL.eq.0 (see below), SX need not be supplied. SB and SX can be the same array in the calling program if desired. RGWK :INOUT Double Precision RGWK(LRGW). Double Precision array used for workspace by DGMRES. On return, RGWK(1) = RHOL. See IERR for definition of RHOL. LRGW :IN Integer. Length of the double precision workspace, RGWK. LRGW >= 1 + N*(MAXL+6) + MAXL*(MAXL+3). See below for definition of MAXL. For the default values, RGWK has size at least 131 + 16*N. IGWK :INOUT Integer IGWK(LIGW). The following IGWK parameters should be set by the user before calling this routine. IGWK(1) = MAXL. Maximum dimension of Krylov subspace in which X - X0 is to be found (where, X0 is the initial guess). The default value of MAXL is 10. IGWK(2) = KMP. Maximum number of previous Krylov basis vectors to which each new basis vector is made orthogonal. The default value of KMP is MAXL. IGWK(3) = JSCAL. Flag indicating whether the scaling arrays SB and SX are to be used. JSCAL = 0 => SB and SX are not used and the algorithm will perform as if all SB(I) = 1 and SX(I) = 1. JSCAL = 1 => Only SX is used, and the algorithm performs as if all SB(I) = 1. JSCAL = 2 => Only SB is used, and the algorithm performs as if all SX(I) = 1. JSCAL = 3 => Both SB and SX are used. IGWK(4) = JPRE. Flag indicating whether preconditioning is being used. JPRE = 0 => There is no preconditioning. JPRE > 0 => There is preconditioning on the right only, and the solver will call routine MSOLVE. JPRE < 0 => There is preconditioning on the left only, and the solver will call routine MSOLVE. IGWK(5) = NRMAX. Maximum number of restarts of the Krylov iteration. The default value of NRMAX = 10. if IWORK(5) = -1, then no restarts are performed (in this case, NRMAX is set to zero internally). The following IWORK parameters are diagnostic information made available to the user after this routine completes. IGWK(6) = MLWK. Required minimum length of RGWK array. IGWK(7) = NMS. The total number of calls to MSOLVE. LIGW :IN Integer. Length of the integer workspace, IGWK. LIGW >= 20. RWORK :WORK Double Precision RWORK(USER DEFINED). Double Precision array that can be used for workspace in MSOLVE. IWORK :WORK Integer IWORK(USER DEFINED). Integer array that can be used for workspace in MSOLVE. *Description: DGMRES solves a linear system A*X = B rewritten in the form: (SB*A*(M-inverse)*(SX-inverse))*(SX*M*X) = SB*B, with right preconditioning, or (SB*(M-inverse)*A*(SX-inverse))*(SX*X) = SB*(M-inverse)*B, with left preconditioning, where A is an N-by-N double precision matrix, X and B are N-vectors, SB and SX are diagonal scaling matrices, and M is a preconditioning matrix. It uses preconditioned Krylov subpace methods based on the generalized minimum residual method (GMRES). This routine optionally performs either the full orthogonalization version of the GMRES algorithm or an incomplete variant of it. Both versions use restarting of the linear iteration by default, although the user can disable this feature. The GMRES algorithm generates a sequence of approximations X(L) to the true solution of the above linear system. The convergence criteria for stopping the iteration is based on the size of the scaled norm of the residual R(L) = B - A*X(L). The actual stopping test is either: norm(SB*(B-A*X(L))) .le. TOL*norm(SB*B), for right preconditioning, or norm(SB*(M-inverse)*(B-A*X(L))) .le. TOL*norm(SB*(M-inverse)*B), for left preconditioning, where norm() denotes the Euclidean norm, and TOL is a positive scalar less than one input by the user. If TOL equals zero when DGMRES is called, then a default value of 500*(the smallest positive magnitude, machine epsilon) is used. If the scaling arrays SB and SX are used, then ideally they should be chosen so that the vectors SX*X(or SX*M*X) and SB*B have all their components approximately equal to one in magnitude. If one wants to use the same scaling in X and B, then SB and SX can be the same array in the calling program. The following is a list of the other routines and their functions used by DGMRES: DPIGMR Contains the main iteration loop for GMRES. DORTH Orthogonalizes a new vector against older basis vectors. DHEQR Computes a QR decomposition of a Hessenberg matrix. DHELS Solves a Hessenberg least-squares system, using QR factors. DRLCAL Computes the scaled residual RL. DXLCAL Computes the solution XL. ISDGMR User-replaceable stopping routine. This routine does not care what matrix data structure is used for A and M. It simply calls the MATVEC and MSOLVE routines, with the arguments as described above. The user could write any type of structure and the appropriate MATVEC and MSOLVE routines. It is assumed that A is stored in the IA, JA, A arrays in some fashion and that M (or INV(M)) is stored in IWORK and RWORK in some fashion. The SLAP routines DSDCG and DSICCG are examples of this procedure. Two examples of matrix data structures are the: 1) SLAP Triad format and 2) SLAP Column format. =================== S L A P Triad format =================== This routine requires that the matrix A be stored in the SLAP Triad format. In this format only the non-zeros are stored. They may appear in *ANY* order. The user supplies three arrays of length NELT, where NELT is the number of non-zeros in the matrix: (IA(NELT), JA(NELT), A(NELT)). For each non-zero the user puts the row and column index of that matrix element in the IA and JA arrays. The value of the non-zero matrix element is placed in the corresponding location of the A array. This is an extremely easy data structure to generate. On the other hand it is not too efficient on vector computers for the iterative solution of linear systems. Hence, SLAP changes this input data structure to the SLAP Column format for the iteration (but does not change it back). Here is an example of the SLAP Triad storage format for a 5x5 Matrix. Recall that the entries may appear in any order. 5x5 Matrix SLAP Triad format for 5x5 matrix on left. 1 2 3 4 5 6 7 8 9 10 11 |11 12 0 0 15| A: 51 12 11 33 15 53 55 22 35 44 21 |21 22 0 0 0| IA: 5 1 1 3 1 5 5 2 3 4 2 | 0 0 33 0 35| JA: 1 2 1 3 5 3 5 2 5 4 1 | 0 0 0 44 0| |51 0 53 0 55| =================== S L A P Column format ================== This routine requires that the matrix A be stored in the SLAP Column format. In this format the non-zeros are stored counting down columns (except for the diagonal entry, which must appear first in each "column") and are stored in the double precision array A. In other words, for each column in the matrix put the diagonal entry in A. Then put in the other non-zero elements going down the column (except the diagonal) in order. The IA array holds the row index for each non-zero. The JA array holds the offsets into the IA, A arrays for the beginning of each column. That is, IA(JA(ICOL)), A(JA(ICOL)) points to the beginning of the ICOL-th column in IA and A. IA(JA(ICOL+1)-1), A(JA(ICOL+1)-1) points to the end of the ICOL-th column. Note that we always have JA(N+1) = NELT+1, where N is the number of columns in the matrix and NELT is the number of non-zeros in the matrix. Here is an example of the SLAP Column storage format for a 5x5 Matrix (in the A and IA arrays '|' denotes the end of a column): 5x5 Matrix SLAP Column format for 5x5 matrix on left. 1 2 3 4 5 6 7 8 9 10 11 |11 12 0 0 15| A: 11 21 51 | 22 12 | 33 53 | 44 | 55 15 35 |21 22 0 0 0| IA: 1 2 5 | 2 1 | 3 5 | 4 | 5 1 3 | 0 0 33 0 35| JA: 1 4 6 8 9 12 | 0 0 0 44 0| |51 0 53 0 55| *Cautions: This routine will attempt to write to the Fortran logical output unit IUNIT, if IUNIT .ne. 0. Thus, the user must make sure that this logical unit is attached to a file or terminal before calling this routine with a non-zero value for IUNIT. This routine does not check for the validity of a non-zero IUNIT unit number.

REFERENCES
1. Peter N. Brown and A. C. Hindmarsh, Reduced Storage Matrix Methods in Stiff ODE Systems, Lawrence Liver- more National Laboratory Report UCRL-95088, Rev. 1, Livermore, California, June 1987. 2. Mark K. Seager, A SLAP for the Masses, in G. F. Carey, Ed., Parallel Supercomputing: Methods, Algorithms and Applications, Wiley, 1989, pp.135-155.

ROUTINES CALLED
D1MACH, DCOPY, DNRM2, DPIGMR

REVISION HISTORY (YYMMDD)
890404 DATE WRITTEN
890404 Previous REVISION DATE
890915 Made changes requested at July 1989 CML Meeting. (MKS)
890922 Numerous changes to prologue to make closer to SLATEC standard. (FNF)
890929 Numerous changes to reduce SP/DP differences. (FNF)
891004 Added new reference.
910411 Prologue converted to Version 4.0 format. (BAB)
910506 Corrected errors in C***ROUTINES CALLED list. (FNF)
920407 COMMON BLOCK renamed DSLBLK. (WRB)
920511 Added complete declaration section. (WRB)
920929 Corrected format of references. (FNF)
921019 Changed 500.0 to 500 to reduce SP/DP differences. (FNF)
921026 Added check for valid value of ITOL. (FNF)