*DECK DSDGMR
SUBROUTINE DSDGMR (N, B, X, NELT, IA, JA, A, ISYM, NSAVE, ITOL,
+ TOL, ITMAX, ITER, ERR, IERR, IUNIT, RWORK, LENW, IWORK, LENIW)
C***BEGIN PROLOGUE DSDGMR
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 DOUBLE 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 DOUBLE PRECISION B(N), X(N), A(NELT), TOL, ERR, RWORK(LENW)
C
C CALL DSDGMR(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 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 :IN Integer IA(NELT).
C JA :IN Integer JA(NELT).
C A :IN 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 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 ISDGMR (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 /DSLBLK/ 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 /DSLBLK/ 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 Double Precision.
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 DGMRES 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 Double Precision.
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 DPIGMR 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 Double Precision RWORK(LENW).
C Double Precision array of size LENW.
C LENW :IN Integer.
C Length of the double precision 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 Double Precision workspace actually used.
C LENIW :IN Integer.
C Length of the integer workspace IWORK. LENIW >= 30.
C
C *Description:
C DSDGMR 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 double precision
C matrix, 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 DGMRES for the solution of the linear system. DGMRES
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 DSDGMR 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 DGMRES Contains the matrix structure independent driver
C routine for GMRES.
C DPIGMR Contains the main iteration loop for GMRES.
C DORTH Orthogonalizes a new vector against older basis vectors.
C DHEQR Computes a QR decomposition of a Hessenberg matrix.
C DHELS Solves a Hessenberg least-squares system, using QR
C factors.
C RLCALC Computes the scaled residual RL.
C XLCALC Computes the solution XL.
C ISDGMR 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 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 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 DCHKW, DGMRES, 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 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 DSDGMR
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 ..
DOUBLE PRECISION ERR, TOL
INTEGER IERR, ISYM, ITER, ITMAX, ITOL, IUNIT, LENIW, LENW, N,
+ NELT, NSAVE
C .. Array Arguments ..
DOUBLE PRECISION 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 DCHKW, DGMRES, DS2Y, DSDI, DSDS, DSMV
C***FIRST EXECUTABLE STATEMENT DSDGMR
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 DS2Y( 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 DCHKW( 'DSDGMR', LOCIW, LENIW, LOCW, LENW, IERR, ITER, ERR )
IF( IERR.NE.0 ) RETURN
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 Generalized Minimum
C Residual iteration algorithm. The following DGMRES
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 DGMRES( N, B, X, NELT, IA, JA, A, ISYM, DSMV, DSDI,
$ 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 DSDGMR FOLLOWS ----------------------------
END