*DECK SSICCG
SUBROUTINE SSICCG (N, B, X, NELT, IA, JA, A, ISYM, ITOL, TOL,
+ ITMAX, ITER, ERR, IERR, IUNIT, RWORK, LENW, IWORK, LENIW)
C***BEGIN PROLOGUE SSICCG
C***PURPOSE Incomplete Cholesky Conjugate Gradient Sparse Ax=b Solver.
C Routine to solve a symmetric positive definite linear
C system Ax = b using the incomplete Cholesky
C Preconditioned Conjugate Gradient method.
C***LIBRARY SLATEC (SLAP)
C***CATEGORY D2B4
C***TYPE SINGLE PRECISION (SSICCG-S, DSICCG-D)
C***KEYWORDS INCOMPLETE CHOLESKY, 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(NL+2*N+1), LENIW
C REAL B(N), X(N), A(NELT), TOL, ERR, RWORK(NL+5*N)
C
C CALL SSICCG(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 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 :INOUT Integer IA(NELT).
C JA :INOUT Integer JA(NELT).
C A :INOUT 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 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 /SSLBLK/ 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 /SSLBLK/ 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 Real.
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 Real.
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*R1MACH(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 IERR = 7 => Incomplete factorization broke down and was
C fudged. Resulting preconditioning may be less
C than the best.
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 used for workspace.
C LENW :IN Integer.
C Length of the real workspace, RWORK. LENW >= NL+5*N.
C NL is the number of non-zeros in the lower triangle of the
C matrix (including the diagonal).
C IWORK :WORK Integer IWORK(LENIW).
C Integer array used for workspace.
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 >= NL+N+11.
C NL is the number of non-zeros in the lower triangle of the
C matrix (including the diagonal).
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 the incomplete Cholesky (IC)
C factorization of the matrix A. See SSICS for details about
C the incomplete factorization algorithm. One should note
C here however, that the IC factorization is a slow process
C and that one should save factorizations for reuse, if
C possible. The MSOLVE operation (handled in SSLLTI) does
C vectorize on machines with hardware gather/scatter and is
C quite fast.
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 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***SEE ALSO SCG, SSLLTI
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 SCG, SCHKW, SS2Y, SSICS, SSLLTI, SSMV, XERMSG
C***REVISION HISTORY (YYMMDD)
C 871119 DATE WRITTEN
C 881213 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 900805 Changed XERRWV calls to calls to XERMSG. (RWC)
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 921019 Corrected NEL to NL. (FNF)
C***END PROLOGUE SSICCG
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
C .. Array Arguments ..
REAL A(NELT), B(N), RWORK(LENW), X(N)
INTEGER IA(NELT), IWORK(LENIW), JA(NELT)
C .. Local Scalars ..
INTEGER LOCDIN, LOCDZ, LOCEL, LOCIEL, LOCIW, LOCJEL, LOCP, LOCR,
+ LOCW, LOCZ, NL
CHARACTER XERN1*8
C .. External Subroutines ..
EXTERNAL SCG, SCHKW, SS2Y, SSICS, SSLLTI, SSMV, XERMSG
C***FIRST EXECUTABLE STATEMENT SSICCG
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 SS2Y( N, NELT, IA, JA, A, ISYM )
C
C Count number of elements in lower triangle of the matrix.
C Then set up the work arrays.
IF( ISYM.EQ.0 ) THEN
NL = (NELT + N)/2
ELSE
NL = NELT
ENDIF
C
LOCJEL = LOCIB
LOCIEL = LOCJEL + NL
LOCIW = LOCIEL + N + 1
C
LOCEL = LOCRB
LOCDIN = LOCEL + NL
LOCR = LOCDIN + N
LOCZ = LOCR + N
LOCP = LOCZ + N
LOCDZ = LOCP + N
LOCW = LOCDZ + N
C
C Check the workspace allocations.
CALL SCHKW( 'SSICCG', LOCIW, LENIW, LOCW, LENW, IERR, ITER, ERR )
IF( IERR.NE.0 ) RETURN
C
IWORK(1) = NL
IWORK(2) = LOCJEL
IWORK(3) = LOCIEL
IWORK(4) = LOCEL
IWORK(5) = LOCDIN
IWORK(9) = LOCIW
IWORK(10) = LOCW
C
C Compute the Incomplete Cholesky decomposition.
C
CALL SSICS(N, NELT, IA, JA, A, ISYM, NL, IWORK(LOCIEL),
$ IWORK(LOCJEL), RWORK(LOCEL), RWORK(LOCDIN),
$ RWORK(LOCR), IERR )
IF( IERR.NE.0 ) THEN
WRITE (XERN1, '(I8)') IERR
CALL XERMSG ('SLATEC', 'SSICCG',
$ 'IC factorization broke down on step ' // XERN1 //
$ '. Diagonal was set to unity and factorization proceeded.',
$ 1, 1)
IERR = 7
ENDIF
C
C Do the Preconditioned Conjugate Gradient.
CALL SCG(N, B, X, NELT, IA, JA, A, ISYM, SSMV, SSLLTI,
$ ITOL, TOL, ITMAX, ITER, ERR, IERR, IUNIT, RWORK(LOCR),
$ RWORK(LOCZ), RWORK(LOCP), RWORK(LOCDZ), RWORK(1),
$ IWORK(1))
RETURN
C------------- LAST LINE OF SSICCG FOLLOWS ----------------------------
END