!DECK DSLUGM SUBROUTINE DSLUGM (N, B, X, NELT, IA, JA, A, ISYM, NSAVE, ITOL, & TOL, ITMAX, ITER, ERR, IERR, IUNIT, RWORK, LENW, IWORK, LENIW) !***BEGIN PROLOGUE DSLUGM !***PURPOSE 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) !***END PROLOGUE DSLUGM ! The following is for optimized compilation on LLNL/LTSS Crays. !LLL. OPTIMIZE ! .. Parameters .. INTEGER LOCRB, LOCIB PARAMETER (LOCRB=1, LOCIB=11) ! .. Scalar Arguments .. DOUBLE PRECISION ERR, TOL INTEGER IERR, ISYM, ITER, ITMAX, ITOL, IUNIT, LENIW, LENW, N, & NELT, NSAVE ! .. Array Arguments .. DOUBLE PRECISION A(NELT), B(N), RWORK(LENW), X(N) INTEGER IA(NELT), IWORK(LENIW), JA(NELT) ! .. Local Scalars .. INTEGER ICOL, J, JBGN, JEND, LOCDIN, LOCIGW, LOCIL, LOCIU, LOCIW, & LOCJL, LOCJU, LOCL, LOCNC, LOCNR, LOCRGW, LOCU, LOCW, & MYITOL, NL, NU ! .. External Subroutines .. EXTERNAL DCHKW, DGMRES, DS2Y, DSILUS, DSLUI, DSMV !***FIRST EXECUTABLE STATEMENT DSLUGM ! IERR = 0 ERR = 0 IF( NSAVE.LE.1 ) THEN IERR = 3 RETURN ENDIF ! ! Change the SLAP input matrix IA, JA, A to SLAP-Column format. CALL DS2Y( N, NELT, IA, JA, A, ISYM ) ! ! Count number of Non-Zero elements preconditioner ILU matrix. ! Then set up the work arrays. We assume MAXL=KMP=NSAVE. NL = 0 NU = 0 DO 20 ICOL = 1, N ! Don't count diagonal. JBGN = JA(ICOL)+1 JEND = JA(ICOL+1)-1 IF( JBGN.LE.JEND ) THEN !VD$ NOVECTOR DO 10 J = JBGN, JEND IF( IA(J).GT.ICOL ) THEN NL = NL + 1 IF( ISYM.NE.0 ) NU = NU + 1 ELSE NU = NU + 1 ENDIF 10 CONTINUE ENDIF 20 CONTINUE ! LOCIGW = LOCIB LOCIL = LOCIGW + 20 LOCJL = LOCIL + N+1 LOCIU = LOCJL + NL LOCJU = LOCIU + NU LOCNR = LOCJU + N+1 LOCNC = LOCNR + N LOCIW = LOCNC + N ! LOCL = LOCRB LOCDIN = LOCL + NL LOCU = LOCDIN + N LOCRGW = LOCU + NU LOCW = LOCRGW + 1+N*(NSAVE+6)+NSAVE*(NSAVE+3) ! ! Check the workspace allocations. CALL DCHKW( 'DSLUGM', LOCIW, LENIW, LOCW, LENW, IERR, ITER, ERR ) IF( IERR.NE.0 ) RETURN ! IWORK(1) = LOCIL IWORK(2) = LOCJL IWORK(3) = LOCIU IWORK(4) = LOCJU IWORK(5) = LOCL IWORK(6) = LOCDIN IWORK(7) = LOCU IWORK(9) = LOCIW IWORK(10) = LOCW ! ! Compute the Incomplete LU decomposition. CALL DSILUS( N, NELT, IA, JA, A, ISYM, NL, IWORK(LOCIL), & IWORK(LOCJL), RWORK(LOCL), RWORK(LOCDIN), NU, IWORK(LOCIU), & IWORK(LOCJU), RWORK(LOCU), IWORK(LOCNR), IWORK(LOCNC) ) ! ! Perform the Incomplete LU Preconditioned Generalized Minimum ! Residual iteration algorithm. The following DGMRES ! defaults are used MAXL = KMP = NSAVE, JSCAL = 0, ! 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 ! CALL DGMRES( N, B, X, NELT, IA, JA, A, ISYM, DSMV, DSLUI, & MYITOL, TOL, ITMAX, ITER, ERR, IERR, IUNIT, RWORK, RWORK, & RWORK(LOCRGW), LENW-LOCRGW, IWORK(LOCIGW), 20, & RWORK, IWORK ) ! IF( ITER.GT.ITMAX ) IERR = 2 RETURN !------------- LAST LINE OF DSLUGM FOLLOWS ---------------------------- END