*DECK SSDGMR SUBROUTINE SSDGMR (N, B, X, NELT, IA, JA, A, ISYM, NSAVE, ITOL, + TOL, ITMAX, ITER, ERR, IERR, IUNIT, RWORK, LENW, IWORK, LENIW) C***BEGIN PROLOGUE SSDGMR C***PURPOSE Diagonally Scaled GMRES Iterative Sparse Ax=b Solver. C This routine uses the generalized minimum residual C (GMRES) method with diagonal scaling to solve possibly C non-symmetric linear systems of the form: Ax = b. C***LIBRARY SLATEC (SLAP) C***CATEGORY D2A4, D2B4 C***TYPE SINGLE PRECISION (SSDGMR-S, DSDGMR-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, NSAVE, ITOL C INTEGER ITMAX, ITER, IERR, IUNIT, LENW, IWORK(LENIW), LENIW C REAL B(N), X(N), A(NELT), TOL, ERR, RWORK(LENW) C C CALL SSDGMR(N, B, X, NELT, IA, JA, A, ISYM, NSAVE, C $ ITOL, TOL, ITMAX, ITER, ERR, IERR, IUNIT, C $ RWORK, LENW, IWORK, LENIW) 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 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 should hold the matrix A in either the SLAP C Triad format or the SLAP Column format. See "Description", C below. If the SLAP Triad format is chosen it is changed C internally to the SLAP Column format. 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 NSAVE :IN Integer. C Number of direction vectors to save and orthogonalize against. C Must be greater than 1. 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 begin 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 :IN Integer. C Maximum number of iterations. This routine uses the default C of NRMAX = ITMAX/NSAVE to determine when each restart C should occur. See the description of NRMAX and MAXL in C SGMRES for a full and frightfully interesting discussion of C this topic. C ITER :OUT Integer. C Number of iterations required to reach convergence, or C ITMAX+1 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 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 SPIGMR 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 => Inconsistent ITOL and JPRE values. 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 RWORK :WORK Real RWORK(LENW). C Real array of size LENW. C LENW :IN Integer. C Length of the real workspace, RWORK. C LENW >= 1 + N*(NSAVE+7) + NSAVE*(NSAVE+3). C For the recommended values of NSAVE (10), RWORK has size at C least 131 + 17*N. C IWORK :INOUT Integer IWORK(USER DEFINED >= 30). C Used to hold pointers into the RWORK array. C Upon return the following locations of IWORK hold information C which may be of use to the user: C IWORK(9) Amount of Integer workspace actually used. C IWORK(10) Amount of Real workspace actually used. C LENIW :IN Integer. C Length of the integer workspace IWORK. LENIW >= 30. C C *Description: C SSDGMR 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 the diagonal of A. It uses C preconditioned Krylov subpace methods based on the C generalized minimum residual method (GMRES). This routine C is a driver routine which assumes a SLAP matrix data C structure and sets up the necessary information to do C diagonal preconditioning and calls the main GMRES routine C SGMRES for the solution of the linear system. SGMRES 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 SSDGMR 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 GMRES: C SGMRES Contains the matrix structure independent driver C routine for GMRES. 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 RLCALC Computes the scaled residual RL. C XLCALC Computes the solution XL. C ISSGMR User-replaceable stopping routine. C C The Sparse Linear Algebra Package (SLAP) utilizes two matrix C data structures: 1) the SLAP Triad format or 2) the SLAP C Column format. The user can hand this routine either of the C of these data structures and SLAP will figure out which on C is being used and act accordingly. 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 *Side Effects: C The SLAP Triad format (IA, JA, A) is modified internally to be C the SLAP Column format. See above. 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***ROUTINES CALLED SCHKW, SGMRES, SS2Y, SSDI, SSDS, SSMV C***REVISION HISTORY (YYMMDD) C 880615 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 910411 Prologue converted to Version 4.0 format. (BAB) C 920407 COMMON BLOCK renamed SSLBLK. (WRB) C 920511 Added complete declaration section. (WRB) C 920929 Corrected format of references. (FNF) C***END PROLOGUE SSDGMR C The following is for optimized compilation on LLNL/LTSS Crays. CLLL. OPTIMIZE C .. Parameters .. INTEGER LOCRB, LOCIB PARAMETER (LOCRB=1, LOCIB=11) C .. Scalar Arguments .. REAL ERR, TOL INTEGER IERR, ISYM, ITER, ITMAX, ITOL, IUNIT, LENIW, LENW, N, + NELT, NSAVE C .. Array Arguments .. REAL A(NELT), B(N), RWORK(LENW), X(N) INTEGER IA(NELT), IWORK(LENIW), JA(NELT) C .. Local Scalars .. INTEGER LOCDIN, LOCIGW, LOCIW, LOCRGW, LOCW, MYITOL C .. External Subroutines .. EXTERNAL SCHKW, SGMRES, SS2Y, SSDI, SSDS, SSMV C***FIRST EXECUTABLE STATEMENT SSDGMR C IERR = 0 ERR = 0 IF( NSAVE.LE.1 ) THEN IERR = 3 RETURN ENDIF C C Change the SLAP input matrix IA, JA, A to SLAP-Column format. CALL SS2Y( N, NELT, IA, JA, A, ISYM ) C C Set up the workspace. We assume MAXL=KMP=NSAVE. LOCIGW = LOCIB LOCIW = LOCIGW + 20 C LOCDIN = LOCRB LOCRGW = LOCDIN + N LOCW = LOCRGW + 1+N*(NSAVE+6)+NSAVE*(NSAVE+3) C IWORK(4) = LOCDIN IWORK(9) = LOCIW IWORK(10) = LOCW C C Check the workspace allocations. CALL SCHKW( 'SSDGMR', LOCIW, LENIW, LOCW, LENW, IERR, ITER, ERR ) IF( IERR.NE.0 ) RETURN C C Compute the inverse of the diagonal of the matrix. CALL SSDS(N, NELT, IA, JA, A, ISYM, RWORK(LOCDIN)) C C Perform the Diagonally Scaled Generalized Minimum C Residual iteration algorithm. The following SGMRES C defaults are used MAXL = KMP = NSAVE, JSCAL = 0, C JPRE = -1, NRMAX = ITMAX/NSAVE IWORK(LOCIGW ) = NSAVE IWORK(LOCIGW+1) = NSAVE IWORK(LOCIGW+2) = 0 IWORK(LOCIGW+3) = -1 IWORK(LOCIGW+4) = ITMAX/NSAVE MYITOL = 0 C CALL SGMRES( N, B, X, NELT, IA, JA, A, ISYM, SSMV, SSDI, $ MYITOL, TOL, ITMAX, ITER, ERR, IERR, IUNIT, RWORK, RWORK, $ RWORK(LOCRGW), LENW-LOCRGW, IWORK(LOCIGW), 20, $ RWORK, IWORK ) C IF( ITER.GT.ITMAX ) IERR = 2 RETURN C------------- LAST LINE OF SSDGMR FOLLOWS ---------------------------- END