*DECK DSDCGS SUBROUTINE DSDCGS (N, B, X, NELT, IA, JA, A, ISYM, ITOL, TOL, + ITMAX, ITER, ERR, IERR, IUNIT, RWORK, LENW, IWORK, LENIW) C***BEGIN PROLOGUE DSDCGS C***PURPOSE Diagonally Scaled CGS Sparse Ax=b Solver. C Routine to solve a linear system Ax = b using the C BiConjugate Gradient Squared method with diagonal scaling. C***LIBRARY SLATEC (SLAP) C***CATEGORY D2A4, D2B4 C***TYPE DOUBLE PRECISION (SSDCGS-S, DSDCGS-D) C***KEYWORDS ITERATIVE PRECONDITION, NON-SYMMETRIC LINEAR SYSTEM, SLAP, C SPARSE C***AUTHOR Greenbaum, Anne, (Courant Institute) C Seager, Mark K., (LLNL) C Lawrence Livermore National Laboratory C PO BOX 808, L-60 C Livermore, CA 94550 (510) 423-3141 C seager@llnl.gov C***DESCRIPTION C C *Usage: C INTEGER N, NELT, IA(NELT), JA(NELT), ISYM, ITOL, ITMAX C INTEGER ITER, IERR, IUNIT, LENW, IWORK(10), LENIW C DOUBLE PRECISION B(N), X(N), A(NELT), TOL, ERR, RWORK(8*N) C C CALL DSDCGS(N, B, X, NELT, IA, JA, A, ISYM, ITOL, TOL, C $ ITMAX, ITER, ERR, IERR, IUNIT, RWORK, LENW, IWORK, LENIW ) C C *Arguments: C N :IN Integer C Order of the Matrix. C B :IN Double Precision B(N). C Right-hand side vector. C X :INOUT Double Precision 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 :INOUT Integer IA(NELT). C JA :INOUT Integer JA(NELT). C A :INOUT Double Precision 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 ITOL :IN Integer. C Flag to indicate type of convergence criterion. C If ITOL=1, iteration stops when the 2-norm of the residual C divided by the 2-norm of the right-hand side is less than TOL. C This routine must calculate the residual from R = A*X - B. C This is unnatural and hence expensive for this type of iter- C ative method. ITOL=2 is *STRONGLY* recommended. C If ITOL=2, iteration stops when the 2-norm of M-inv times the C residual divided by the 2-norm of M-inv times the right hand C side is less than TOL, where M-inv time a vector is the pre- C conditioning step. This is the *NATURAL* stopping for this C iterative method and is *STRONGLY* recommended. C ITOL=11 is often useful for checking and comparing different C routines. For this case, the user must supply the "exact" C solution or a very accurate approximation (one with an error C much less than TOL) through a common block, C COMMON /DSLBLK/ SOLN( ) C If ITOL=11, iteration stops when the 2-norm of the difference C between the iterative approximation and the user-supplied C solution divided by the 2-norm of the user-supplied solution C is less than TOL. Note that this requires the user to set up C the "COMMON /DSLBLK/ SOLN(LENGTH)" in the calling routine. C The routine with this declaration should be loaded before the C stop test so that the correct length is used by the loader. C This procedure is not standard Fortran and may not work C correctly on your system (although it has worked on every C system the authors have tried). If ITOL is not 11 then this C common block is indeed standard Fortran. C TOL :INOUT Double Precision. C Convergence criterion, as described above. (Reset if IERR=4.) C ITMAX :IN Integer. C Maximum number of iterations. 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 Double Precision. C Error estimate of error in final approximate solution, as C defined by ITOL. C IERR :OUT Integer. C Return error flag. C IERR = 0 => All went well. C IERR = 1 => Insufficient space allocated for WORK or IWORK. C IERR = 2 => Method failed to converge in ITMAX steps. C IERR = 3 => Error in user input. C Check input values of N, ITOL. C IERR = 4 => User error tolerance set too tight. C Reset to 500*D1MACH(3). Iteration proceeded. C IERR = 5 => Breakdown of the method detected. C (r0,r) approximately 0. C IERR = 6 => Stagnation of the method detected. C (r0,v) approximately 0. 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 Double Precision RWORK(LENW). C Double Precision array used for workspace. C LENW :IN Integer. C Length of the double precision workspace, RWORK. LENW >= 8*N. C IWORK :WORK Integer IWORK(LENIW). 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 Double Precision workspace actually used. C LENIW :IN Integer. C Length of the integer workspace, IWORK. LENIW >= 10. C C *Description: C This routine performs preconditioned BiConjugate gradient C method on the Non-Symmetric positive definite linear system C Ax=b. The preconditioner is M = DIAG(A), the diagonal of the C matrix A. This is the simplest of preconditioners and C vectorizes very well. 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 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 double precision array A. In other words, for each column C in the matrix put the diagonal entry in A. Then put in the C other non-zero elements going down the column (except the C diagonal) in order. The IA array holds the row index for C each non-zero. The JA array holds the offsets into the IA, C A arrays for the beginning of each column. That is, C IA(JA(ICOL)), A(JA(ICOL)) points to the beginning of the C ICOL-th column in IA and A. IA(JA(ICOL+1)-1), C A(JA(ICOL+1)-1) points to the end of the ICOL-th column. C Note that we always have JA(N+1) = NELT+1, where N is the C number of columns in the matrix and NELT is the number of C 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 C be 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***SEE ALSO DCGS, DLUBCG C***REFERENCES 1. P. Sonneveld, CGS, a fast Lanczos-type solver C for nonsymmetric linear systems, Delft University C of Technology Report 84-16, Department of Mathe- C matics and Informatics, Delft, The Netherlands. C 2. E. F. Kaasschieter, The solution of non-symmetric C linear systems by biconjugate gradients or conjugate C gradients squared, Delft University of Technology C Report 86-21, Department of Mathematics and Informa- C tics, Delft, The Netherlands. C***ROUTINES CALLED DCGS, DCHKW, DS2Y, DSDI, DSDS, DSMV C***REVISION HISTORY (YYMMDD) C 890404 DATE WRITTEN C 890404 Previous REVISION DATE C 890915 Made changes requested at July 1989 CML Meeting. (MKS) C 890921 Removed TeX from comments. (FNF) 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 DSLBLK. (WRB) C 920511 Added complete declaration section. (WRB) C 920929 Corrected format of references. (FNF) C 921113 Corrected C***CATEGORY line. (FNF) C***END PROLOGUE DSDCGS C .. Parameters .. INTEGER LOCRB, LOCIB PARAMETER (LOCRB=1, LOCIB=11) C .. Scalar Arguments .. DOUBLE PRECISION ERR, TOL INTEGER IERR, ISYM, ITER, ITMAX, ITOL, IUNIT, LENIW, LENW, N, NELT C .. Array Arguments .. DOUBLE PRECISION A(N), B(N), RWORK(LENW), X(N) INTEGER IA(NELT), IWORK(LENIW), JA(NELT) C .. Local Scalars .. INTEGER LOCDIN, LOCIW, LOCP, LOCQ, LOCR, LOCR0, LOCU, LOCV1, + LOCV2, LOCW C .. External Subroutines .. EXTERNAL DCGS, DCHKW, DS2Y, DSDI, DSDS, DSMV C***FIRST EXECUTABLE STATEMENT DSDCGS C IERR = 0 IF( N.LT.1 .OR. NELT.LT.1 ) THEN IERR = 3 RETURN ENDIF C C Change the SLAP input matrix IA, JA, A to SLAP-Column format. CALL DS2Y( N, NELT, IA, JA, A, ISYM ) C C Set up the workspace. LOCIW = LOCIB C LOCDIN = LOCRB LOCR = LOCDIN + N LOCR0 = LOCR + N LOCP = LOCR0 + N LOCQ = LOCP + N LOCU = LOCQ + N LOCV1 = LOCU + N LOCV2 = LOCV1 + N LOCW = LOCV2 + N C C Check the workspace allocations. CALL DCHKW( 'DSDCGS', LOCIW, LENIW, LOCW, LENW, IERR, ITER, ERR ) IF( IERR.NE.0 ) RETURN C IWORK(4) = LOCDIN IWORK(9) = LOCIW IWORK(10) = LOCW C C Compute the inverse of the diagonal of the matrix. CALL DSDS(N, NELT, IA, JA, A, ISYM, RWORK(LOCDIN)) C C Perform the Diagonally Scaled C BiConjugate Gradient Squared algorithm. CALL DCGS(N, B, X, NELT, IA, JA, A, ISYM, DSMV, $ DSDI, ITOL, TOL, ITMAX, ITER, ERR, IERR, IUNIT, $ RWORK(LOCR), RWORK(LOCR0), RWORK(LOCP), $ RWORK(LOCQ), RWORK(LOCU), RWORK(LOCV1), $ RWORK(LOCV2), RWORK(1), IWORK(1)) RETURN C------------- LAST LINE OF DSDCGS FOLLOWS ---------------------------- END