SUBROUTINE DSDCG (N, B, X, NELT, IA, JA, A, ISYM, ITOL, TOL, + ITMAX, ITER, ERR, IERR, IUNIT, RWORK, LENW, IWORK, LENIW) C***BEGIN PROLOGUE DSDCG C***PURPOSE Diagonally Scaled Conjugate Gradient Sparse Ax=b Solver. C Routine to solve a symmetric positive definite linear C system Ax = b using the Preconditioned Conjugate C Gradient method. The preconditioner is diagonal scaling. C***LIBRARY SLATEC (SLAP) C***CATEGORY D2B4 C***TYPE DOUBLE PRECISION (SSDCG-S, DSDCG-D) C***KEYWORDS ITERATIVE PRECONDITION, SLAP, SPARSE, C SYMMETRIC LINEAR SYSTEM 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(5*N) C C CALL DSDCG(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 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 is the inverse of the C diagonal of A. 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 => Preconditioning matrix, M, is not positive C definite. (r,z) < 0. C IERR = 6 => Matrix A is not positive definite. (p,Ap) < 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 >= 5*N. C IWORK :WORK Integer IWORK(LENIW). C Used to hold pointers into the double precision workspace, C RWORK. Upon return the following locations of IWORK hold C information 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 conjugate gradient C method on the symmetric positive definite linear system C Ax=b. The preconditioner is M = DIAG(A), the diagonal of C the matrix A. This is the simplest of preconditioners and C vectorizes very well. This routine is simply a driver for C the DCG routine. It calls the DSDS routine to set up the C preconditioning and then calls DCG with the appropriate C MATVEC and MSOLVE routines. 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 DCG, DSICCG C***REFERENCES 1. Louis Hageman and David Young, Applied Iterative C Methods, Academic Press, New York, 1981. C 2. Concus, Golub and O'Leary, A Generalized Conjugate C Gradient Method for the Numerical Solution of C Elliptic Partial Differential Equations, in Sparse C Matrix Computations, Bunch and Rose, Eds., Academic C Press, New York, 1979. C***ROUTINES CALLED DCG, 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***END PROLOGUE DSDCG