# sslui4.f

          SUBROUTINE SSLUI4 (N, B, X, IL, JL, L, DINV, IU, JU, U)
    C***BEGIN PROLOGUE  SSLUI4
    C***PURPOSE  SLAP Backsolve for LDU Factorization.
    C            Routine to solve a system of the form  (L*D*U)' X = B,
    C            where L is a unit lower triangular matrix, D is a diagonal
    C            matrix, and U is a unit upper triangular matrix and '
    C            denotes transpose.
    C***LIBRARY   SLATEC (SLAP)
    C***CATEGORY  D2E
    C***TYPE      SINGLE PRECISION (SSLUI4-S, DSLUI4-D)
    C***KEYWORDS  ITERATIVE PRECONDITION, NON-SYMMETRIC LINEAR SYSTEM SOLVE,
    C             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, IL(NL), JL(NL), IU(NU), JU(NU)
    C     REAL    B(N), X(N), L(NL), DINV(N), U(NU)
    C
    C     CALL SSLUI4( N, B, X, IL, JL, L, DINV, IU, JU, U )
    C
    C *Arguments:
    C N      :IN       Integer
    C         Order of the Matrix.
    C B      :IN       Real B(N).
    C         Right hand side.
    C X      :OUT      Real X(N).
    C         Solution of (L*D*U)trans x = b.
    C IL     :IN       Integer IL(NL).
    C JL     :IN       Integer JL(NL).
    C L      :IN       Real     L(NL).
    C         IL, JL, L contain the unit lower triangular  factor of the
    C         incomplete decomposition of some matrix stored in SLAP Row
    C         format.  The diagonal of ones *IS* stored.  This structure
    C         can    be  set  up  by   the  SSILUS  routine.   See   the
    C         "Description",  below for  more  details about  the   SLAP
    C         format.  (NL is the number of non-zeros in the L array.)
    C DINV   :IN       Real DINV(N).
    C         Inverse of the diagonal matrix D.
    C IU     :IN       Integer IU(NU).
    C JU     :IN       Integer JU(NU).
    C U      :IN       Real     U(NU).
    C         IU, JU, U contain the  unit upper triangular factor of the
    C         incomplete  decomposition of some  matrix stored  in  SLAP
    C         Column  format.   The diagonal of  ones *IS* stored.  This
    C         structure can be set up by the  SSILUS routine.  See   the
    C         "Description",  below for  more  details  about  the  SLAP
    C         format.  (NU is the number of non-zeros in the U array.)
    C
    C *Description:
    C       This routine is supplied with the SLAP package as  a routine
    C       to  perform  the  MTSOLV  operation  in  the SBCG  iteration
    C       routine for the  driver SSLUBC.   It must  be called via the
    C       SLAP  MTSOLV calling  sequence convention interface  routine
    C       SSLUTI.
    C         **** THIS ROUTINE ITSELF DOES NOT CONFORM TO THE ****
    C               **** SLAP MSOLVE CALLING CONVENTION ****
    C
    C       IL, JL, L should contain the unit lower triangular factor of
    C       the incomplete decomposition of the A matrix  stored in SLAP
    C       Row format.  IU, JU, U should contain  the unit upper factor
    C       of the  incomplete decomposition of  the A matrix  stored in
    C       SLAP Column format This ILU factorization can be computed by
    C       the SSILUS routine. The diagonals (which are all one's) are
    C       stored.
    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       ==================== S L A P Row format ====================
    C
    C       This routine requires  that the matrix A  be  stored  in the
    C       SLAP  Row format.   In this format  the non-zeros are stored
    C       counting across  rows (except for the diagonal  entry, which
    C       must appear first in each "row") and  are stored in the real
    C       array A.  In other words, for each row in the matrix put the
    C       diagonal entry in  A.   Then   put  in the   other  non-zero
    C       elements   going  across the  row (except   the diagonal) in
    C       order.   The  JA array  holds   the column   index for  each
    C       non-zero.   The IA  array holds the  offsets into  the JA, A
    C       arrays  for   the   beginning  of   each  row.   That    is,
    C       JA(IA(IROW)),  A(IA(IROW)) points  to  the beginning  of the
    C       IROW-th row in JA and A.   JA(IA(IROW+1)-1), A(IA(IROW+1)-1)
    C       points to the  end of the  IROW-th row.  Note that we always
    C       have IA(N+1) =  NELT+1, where  N  is  the number of rows  in
    C       the matrix  and NELT  is the  number   of  non-zeros in  the
    C       matrix.
    C
    C       Here is an example of the SLAP Row storage format for a  5x5
    C       Matrix (in the A and JA arrays '|' denotes the end of a row):
    C
    C           5x5 Matrix         SLAP Row 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 12 15 | 22 21 | 33 35 | 44 | 55 51 53
    C       |21 22  0  0  0|  JA:  1  2  5 |  2  1 |  3  5 |  4 |  5  1  3
    C       | 0  0 33  0 35|  IA:  1  4  6    8  9   12
    C       | 0  0  0 44  0|
    C       |51  0 53  0 55|
    C
    C       With  the SLAP  format  the "inner  loops" of  this  routine
    C       should vectorize   on machines with   hardware  support  for
    C       vector gather/scatter operations.  Your compiler may require
    C       a  compiler directive  to  convince   it that there  are  no
    C       implicit vector  dependencies.  Compiler directives  for the
    C       Alliant FX/Fortran and CRI CFT/CFT77 compilers  are supplied
    C       with the standard SLAP distribution.
    C
    C***SEE ALSO  SSILUS
    C***REFERENCES  (NONE)
    C***ROUTINES CALLED  (NONE)
    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   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   920511  Added complete declaration section.  (WRB)
    C   921113  Corrected C***CATEGORY line.  (FNF)
    C   930701  Updated CATEGORY section.  (FNF, WRB)
    C***END PROLOGUE  SSLUI4