# domn.f

          SUBROUTINE DOMN (N, B, X, NELT, IA, JA, A, ISYM, MATVEC, MSOLVE,
         +   NSAVE, ITOL, TOL, ITMAX, ITER, ERR, IERR, IUNIT, R, Z, P, AP,
         +   EMAP, DZ, CSAV, RWORK, IWORK)
    C***BEGIN PROLOGUE  DOMN
    C***PURPOSE  Preconditioned Orthomin Sparse Iterative Ax=b Solver.
    C            Routine to solve a general linear system  Ax = b  using
    C            the Preconditioned Orthomin method.
    C***LIBRARY   SLATEC (SLAP)
    C***CATEGORY  D2A4, D2B4
    C***TYPE      DOUBLE PRECISION (SOMN-S, DOMN-D)
    C***KEYWORDS  ITERATIVE PRECONDITION, NON-SYMMETRIC LINEAR SYSTEM,
    C             ORTHOMIN, SLAP, 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, NSAVE, ITOL, ITMAX
    C     INTEGER  ITER, IERR, IUNIT, IWORK(USER DEFINED)
    C     DOUBLE PRECISION B(N), X(N), A(NELT), TOL, ERR, R(N), Z(N)
    C     DOUBLE PRECISION P(N,0:NSAVE), AP(N,0:NSAVE), EMAP(N,0:NSAVE)
    C     DOUBLE PRECISION DZ(N), CSAV(NSAVE), RWORK(USER DEFINED)
    C     EXTERNAL MATVEC, MSOLVE
    C
    C     CALL DOMN(N, B, X, NELT, IA, JA, A, ISYM, MATVEC, MSOLVE,
    C    $     NSAVE, ITOL, TOL, ITMAX, ITER, ERR, IERR, IUNIT, R,
    C    $     Z, P, AP, EMAP, DZ, CSAV, RWORK, IWORK)
    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 contain the matrix data structure for A.
    C         It could take any form.  See "Description", below, for more
    C         details.
    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 MATVEC :EXT      External.
    C         Name of a routine which performs the matrix vector multiply
    C         Y = A*X given A and X.  The name of the MATVEC routine must
    C         be declared external in the calling program.  The calling
    C         sequence to MATVEC is:
    C             CALL MATVEC( N, X, Y, NELT, IA, JA, A, ISYM )
    C         Where N is the number of unknowns, Y is the product A*X
    C         upon return X is an input vector, NELT is the number of
    C         non-zeros in the SLAP IA, JA, A storage for the matrix A.
    C         ISYM is a flag which, if non-zero, denotest that A is
    C         symmetric and only the lower or upper triangle is stored.
    C MSOLVE :EXT      External.
    C         Name of a routine which solves a linear system MZ = R for
    C         Z given R with the preconditioning matrix M (M is supplied via
    C         RWORK and IWORK arrays).  The name of the MSOLVE routine must
    C         be declared external in the calling program.  The calling
    C         sequence to MSOLVE is:
    C             CALL MSOLVE(N, R, Z, NELT, IA, JA, A, ISYM, RWORK, IWORK)
    C         Where N is the number of unknowns, R is the right-hand side
    C         vector and Z is the solution upon return.  NELT, IA, JA, A and
    C         ISYM are defined as above.  RWORK is a double precision array
    C         that can be used to pass necessary preconditioning information
    C         and/or workspace to MSOLVE.  IWORK is an integer work array
    C         for the same purpose as RWORK.
    C NSAVE  :IN       Integer.
    C         Number of  direction vectors to save and orthogonalize
    C         against.  NSAVE >= 0.
    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 => Breakdown of method detected.
    C                       (p,Ap) < epsilon**2.
    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 R      :WORK     Double Precision R(N).
    C Z      :WORK     Double Precision Z(N).
    C P      :WORK     Double Precision P(N,0:NSAVE).
    C AP     :WORK     Double Precision AP(N,0:NSAVE).
    C EMAP   :WORK     Double Precision EMAP(N,0:NSAVE).
    C DZ     :WORK     Double Precision DZ(N).
    C CSAV   :WORK     Double Precision CSAV(NSAVE)
    C         Double Precision arrays used for workspace.
    C RWORK  :WORK     Double Precision RWORK(USER DEFINED).
    C         Double Precision array that can be used for workspace in
    C         MSOLVE.
    C IWORK  :WORK     Integer IWORK(USER DEFINED).
    C         Integer array that can be used for workspace in MSOLVE.
    C
    C *Description
    C       This routine does  not care  what matrix data   structure is
    C       used for  A and M.  It simply   calls  the MATVEC and MSOLVE
    C       routines, with  the arguments as  described above.  The user
    C       could write any type of structure and the appropriate MATVEC
    C       and MSOLVE routines.  It is assumed  that A is stored in the
    C       IA, JA, A  arrays in some fashion and  that M (or INV(M)) is
    C       stored  in  IWORK  and  RWORK)  in  some fashion.   The SLAP
    C       routines DSDOMN and DSLUOM are examples of this procedure.
    C
    C       Two  examples  of  matrix  data structures  are the: 1) SLAP
    C       Triad  format and 2) SLAP Column format.
    C
    C       =================== S L A P Triad format ===================
    C       In  this   format only the  non-zeros are  stored.  They may
    C       appear  in *ANY* order.   The user  supplies three arrays of
    C       length NELT, where  NELT  is the number  of non-zeros in the
    C       matrix:  (IA(NELT), JA(NELT),  A(NELT)).  For each  non-zero
    C       the  user puts   the row  and  column index   of that matrix
    C       element in the IA and JA arrays.  The  value of the non-zero
    C       matrix  element is  placed in  the corresponding location of
    C       the A  array.  This is  an extremely easy data  structure to
    C       generate.  On  the other hand it  is  not too  efficient  on
    C       vector  computers   for the  iterative  solution  of  linear
    C       systems.  Hence, SLAP  changes this input  data structure to
    C       the SLAP   Column  format for the  iteration (but   does not
    C       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       In  this format   the non-zeros are    stored counting  down
    C       columns (except  for the diagonal  entry, which must  appear
    C       first  in each "column") and are  stored in the  double pre-
    C       cision array  A. In  other  words,  for each  column  in the
    C       matrix  first 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)) are the first elements of the ICOL-
    C       th column in IA and A, and IA(JA(ICOL+1)-1), A(JA(ICOL+1)-1)
    C       are  the last elements of the ICOL-th column.   Note that we
    C       always have JA(N+1)=NELT+1, where N is the number of columns
    C       in the matrix  and NELT  is the number  of non-zeros  in the
    C       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 *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  DSDOMN, DSLUOM, ISDOMN
    C***REFERENCES  1. Mark K. Seager, A SLAP for the Masses, in
    C                  G. F. Carey, Ed., Parallel Supercomputing: Methods,
    C                  Algorithms and Applications, Wiley, 1989, pp.135-155.
    C***ROUTINES CALLED  D1MACH, DAXPY, DCOPY, DDOT, ISDOMN
    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   891004  Added new reference.
    C   910411  Prologue converted to Version 4.0 format.  (BAB)
    C   910502  Removed MATVEC and MSOLVE from ROUTINES CALLED list.  (FNF)
    C   920407  COMMON BLOCK renamed DSLBLK.  (WRB)
    C   920511  Added complete declaration section.  (WRB)
    C   920929  Corrected format of reference.  (FNF)
    C   921019  Changed 500.0 to 500 to reduce SP/DP differences.  (FNF)
    C   921113  Corrected C***CATEGORY line.  (FNF)
    C   930326  Removed unused variable.  (FNF)
    C***END PROLOGUE  DOMN