# scgs.f

          SUBROUTINE SCGS (N, B, X, NELT, IA, JA, A, ISYM, MATVEC, MSOLVE,
         +   ITOL, TOL, ITMAX, ITER, ERR, IERR, IUNIT, R, R0, P, Q, U, V1,
         +   V2, RWORK, IWORK)
    C***BEGIN PROLOGUE  SCGS
    C***PURPOSE  Preconditioned BiConjugate Gradient Squared Ax=b Solver.
    C            Routine to solve a Non-Symmetric linear system  Ax = b
    C            using the Preconditioned BiConjugate Gradient Squared
    C            method.
    C***LIBRARY   SLATEC (SLAP)
    C***CATEGORY  D2A4, D2B4
    C***TYPE      SINGLE PRECISION (SCGS-S, DCGS-D)
    C***KEYWORDS  BICONJUGATE GRADIENT, ITERATIVE PRECONDITION,
    C             NON-SYMMETRIC LINEAR SYSTEM, 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, ITOL, ITMAX
    C      INTEGER ITER, IERR, IUNIT, IWORK(USER DEFINED)
    C      REAL    B(N), X(N), A(NELT), TOL, ERR, R(N), R0(N), P(N)
    C      REAL    Q(N), U(N), V1(N), V2(N), RWORK(USER DEFINED)
    C      EXTERNAL MATVEC, MSOLVE
    C
    C      CALL SCGS(N, B, X, NELT, IA, JA, A, ISYM, MATVEC,
    C     $     MSOLVE, ITOL, TOL, ITMAX, ITER, ERR, IERR, IUNIT,
    C     $     R, R0, P, Q, U, V1, V2, RWORK, IWORK)
    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     :IN       Integer IA(NELT).
    C JA     :IN       Integer JA(NELT).
    C A      :IN       Real A(NELT).
    C         These arrays contain the matrix data structure for A.
    C         It could take any form.  See "Description", below,
    C         for more 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         operation  Y = A*X  given A and X.  The  name of  the MATVEC
    C         routine must  be declared external  in the  calling program.
    C         The calling sequence of 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 upon
    C         return,  X is an input  vector.  NELT, IA,  JA,  A and  ISYM
    C         define the SLAP matrix data structure: see Description,below.
    C MSOLVE :EXT      External.
    C         Name of a routine which solves a linear system MZ = R  for Z
    C         given R with the preconditioning matrix M (M is supplied via
    C         RWORK  and IWORK arrays).   The name  of  the MSOLVE routine
    C         must be declared  external  in the  calling   program.   The
    C         calling sequence of 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
    C         and  ISYM define the SLAP  matrix  data structure: see
    C         Description, below.  RWORK is a  real array that can be used
    C         to  pass   necessary  preconditioning     information and/or
    C         workspace to MSOLVE.  IWORK is an integer work array for the
    C         same purpose as RWORK.
    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 /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.
    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 => 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 R      :WORK     Real R(N).
    C R0     :WORK     Real R0(N).
    C P      :WORK     Real P(N).
    C Q      :WORK     Real Q(N).
    C U      :WORK     Real U(N).
    C V1     :WORK     Real V1(N).
    C V2     :WORK     Real V2(N).
    C         Real arrays used for workspace.
    C RWORK  :WORK     Real RWORK(USER DEFINED).
    C         Real array that can be used for workspace in 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 SSDBCG and SSLUCS 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
    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 real array A.
    C       In other words,  for  each column    in the matrix   put the
    C       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 JA(N+1)
    C       = NELT+1, where N is the number of columns in the matrix and
    C       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 *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  SSDCGS, SSLUCS
    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               3. 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  ISSCGS, R1MACH, SAXPY, SDOT
    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   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 SSLBLK.  (WRB)
    C   920511  Added complete declaration section.  (WRB)
    C   920929  Corrected format of references.  (FNF)
    C   921019  Changed 500.0 to 500 to reduce SP/DP differences.  (FNF)
    C   921113  Corrected C***CATEGORY line.  (FNF)
    C***END PROLOGUE  SCGS