text: *DECK SCGS
        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
  C     .. Scalar Arguments ..
        REAL ERR, TOL
        INTEGER IERR, ISYM, ITER, ITMAX, ITOL, IUNIT, N, NELT
  C     .. Array Arguments ..
        REAL A(NELT), B(N), P(N), Q(N), R(N), R0(N), RWORK(*), U(N),
       +     V1(N), V2(N), X(N)
        INTEGER IA(NELT), IWORK(*), JA(NELT)
  C     .. Subroutine Arguments ..
        EXTERNAL MATVEC, MSOLVE
  C     .. Local Scalars ..
        REAL AK, AKM, BK, BNRM, FUZZ, RHON, RHONM1, SIGMA, SOLNRM, TOLMIN
        INTEGER I, K
  C     .. External Functions ..
        REAL R1MACH, SDOT
        INTEGER ISSCGS
        EXTERNAL R1MACH, SDOT, ISSCGS
  C     .. External Subroutines ..
        EXTERNAL SAXPY
  C     .. Intrinsic Functions ..
        INTRINSIC ABS
  C***FIRST EXECUTABLE STATEMENT  SCGS
  C
  C         Check some of the input data.
  C
        ITER = 0
        IERR = 0
        IF( N.LT.1 ) THEN
           IERR = 3
           RETURN
        ENDIF
        TOLMIN = 500*R1MACH(3)
        IF( TOL.LT.TOLMIN ) THEN
           TOL = TOLMIN
           IERR = 4
        ENDIF
  C
  C         Calculate initial residual and pseudo-residual, and check
  C         stopping criterion.
        CALL MATVEC(N, X, R, NELT, IA, JA, A, ISYM)
        DO 10 I = 1, N
           V1(I)  = R(I) - B(I)
   10   CONTINUE
        CALL MSOLVE(N, V1, R, NELT, IA, JA, A, ISYM, RWORK, IWORK)
  C
        IF( ISSCGS(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, AK, BK, BNRM, SOLNRM) .NE. 0 )
       $     GO TO 200
        IF( IERR.NE.0 ) RETURN
  C
  C         Set initial values.
  C
        FUZZ = R1MACH(3)**2
        DO 20 I = 1, N
           R0(I) = R(I)
   20   CONTINUE
        RHONM1 = 1
  C
  C         ***** ITERATION LOOP *****
  C
        DO 100 K=1,ITMAX
           ITER = K
  C
  C         Calculate coefficient BK and direction vectors U, V and P.
           RHON = SDOT(N, R0, 1, R, 1)
           IF( ABS(RHONM1).LT.FUZZ ) GOTO 998
           BK = RHON/RHONM1
           IF( ITER.EQ.1 ) THEN
              DO 30 I = 1, N
                 U(I) = R(I)
                 P(I) = R(I)
   30         CONTINUE
           ELSE
              DO 40 I = 1, N
                 U(I) = R(I) + BK*Q(I)
                 V1(I) = Q(I) + BK*P(I)
   40         CONTINUE
              DO 50 I = 1, N
                 P(I) = U(I) + BK*V1(I)
   50         CONTINUE
           ENDIF
  C
  C         Calculate coefficient AK, new iterate X, Q
           CALL MATVEC(N, P, V2, NELT, IA, JA, A, ISYM)
           CALL MSOLVE(N, V2, V1, NELT, IA, JA, A, ISYM, RWORK, IWORK)
           SIGMA = SDOT(N, R0, 1, V1, 1)
           IF( ABS(SIGMA).LT.FUZZ ) GOTO 999
           AK = RHON/SIGMA
           AKM = -AK
           DO 60 I = 1, N
              Q(I) = U(I) + AKM*V1(I)
   60      CONTINUE
           DO 70 I = 1, N
              V1(I) = U(I) + Q(I)
   70      CONTINUE
  C         X = X - ak*V1.
           CALL SAXPY( N, AKM, V1, 1, X, 1 )
  C                     -1
  C         R = R - ak*M  *A*V1
           CALL MATVEC(N, V1, V2, NELT, IA, JA, A, ISYM)
           CALL MSOLVE(N, V2, V1, NELT, IA, JA, A, ISYM, RWORK, IWORK)
           CALL SAXPY( N, AKM, V1, 1, R, 1 )
  C
  C         check stopping criterion.
           IF( ISSCGS(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, AK, BK, BNRM, SOLNRM) .NE. 0 )
       $        GO TO 200
  C
  C         Update RHO.
           RHONM1 = RHON
   100  CONTINUE
  C
  C         *****   end of loop  *****
  C         Stopping criterion not satisfied.
        ITER = ITMAX + 1
        IERR = 2
   200  RETURN
  C
  C         Breakdown of method detected.
   998  IERR = 5
        RETURN
  C
  C         Stagnation of method detected.
   999  IERR = 6
        RETURN
  C------------- LAST LINE OF SCGS FOLLOWS ----------------------------
        END
  