DSLUGM(3)

SYNOPSIS
SUBROUTINE DSLUGM (N, B, X, NELT, IA, JA, A, ISYM, NSAVE, ITOL, + TOL, ITMAX, ITER, ERR, IERR, IUNIT, RWORK, LENW, IWORK, LENIW)

NAME
dslugm - Incomplete LU GMRES iterative sparse Ax=b solver. This routine uses the generalized minimum residual (GMRES) method with incomplete LU factorization for preconditioning to solve possibly non-symmetric linear systems of the form: Ax = b.

LIBRARY
SLATEC (SLAP)

CATEGORY
D2A4, D2B4

TYPE
DOUBLE PRECISION (SSLUGM-S, DSLUGM-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, NSAVE, ITOL INTEGER ITMAX, ITER, IERR, IUNIT, LENW, IWORK(LENIW), LENIW DOUBLE PRECISION B(N), X(N), A(NELT), TOL, ERR, RWORK(LENW) CALL DSLUGM(N, B, X, NELT, IA, JA, A, ISYM, NSAVE, $ ITOL, TOL, ITMAX, ITER, ERR, IERR, IUNIT, $ RWORK, LENW, IWORK, LENIW) *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 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 should hold the matrix A in either the SLAP Triad format or the SLAP Column format. See "Description", below. If the SLAP Triad format is chosen it is changed internally to the SLAP Column format. 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. NSAVE :IN Integer. Number of direction vectors to save and orthogonalize against. Must be greater than 1. 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 begin 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 :IN Integer. Maximum number of iterations. This routine uses the default of NRMAX = ITMAX/NSAVE to determine the when each restart should occur. See the description of NRMAX and MAXL in DGMRES for a full and frightfully interesting discussion of this topic. ITER :OUT Integer. Number of iterations required to reach convergence, or ITMAX+1 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 DPIGMR 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 => Inconsistent ITOL and JPRE values. 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. RWORK :WORK Double Precision RWORK(LENW). Double Precision array of size LENW. LENW :IN Integer. Length of the double precision workspace, RWORK. LENW >= 1 + N*(NSAVE+7) + NSAVE*(NSAVE+3)+NL+NU. Here NL is the number of non-zeros in the lower triangle of the matrix (including the diagonal) and NU is the number of non-zeros in the upper triangle of the matrix (including the diagonal). For the recommended values, RWORK has size at least 131 + 17*N + NL + NU. IWORK :INOUT Integer IWORK(LENIW). Used to hold pointers into the RWORK array. Upon return the following locations of IWORK hold information which may be of use to the user: IWORK(9) Amount of Integer workspace actually used. IWORK(10) Amount of Double Precision workspace actually used. LENIW :IN Integer. Length of the integer workspace, IWORK. LENIW >= NL+NU+4*N+32. *Description: DSLUGM 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 the Incomplete LU factorization of A. It uses preconditioned Krylov subpace methods based on the generalized minimum residual method (GMRES). This routine is a driver routine which assumes a SLAP matrix data structure and sets up the necessary information to do diagonal preconditioning and calls the main GMRES routine DGMRES for the solution of the linear system. DGMRES 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 DSLUGM 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 GMRES: DGMRES Contains the matrix structure independent driver routine for GMRES. 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. RLCALC Computes the scaled residual RL. XLCALC Computes the solution XL. ISDGMR User-replaceable stopping routine. The Sparse Linear Algebra Package (SLAP) utilizes two matrix data structures: 1) the SLAP Triad format or 2) the SLAP Column format. The user can hand this routine either of the of these data structures and SLAP will figure out which on is being used and act accordingly. =================== 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| *Side Effects: The SLAP Triad format (IA, JA, A) is modified internally to be the SLAP Column format. See above. *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.

ROUTINES CALLED
DCHKW, DGMRES, DS2Y, DSILUS, DSLUI, DSMV

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)
910411 Prologue converted to Version 4.0 format. (BAB)
920407 COMMON BLOCK renamed DSLBLK. (WRB)
920511 Added complete declaration section. (WRB)
920929 Corrected format of references. (FNF)
921019 Corrected NEL to NL. (FNF)