# dsli2.f

          SUBROUTINE DSLI2 (N, B, X, NEL, IEL, JEL, EL)
    C***BEGIN PROLOGUE  DSLI2
    C***PURPOSE  SLAP Lower Triangle Matrix Backsolve.
    C            Routine to solve a system of the form  Lx = b , where L
    C            is a lower triangular matrix.
    C***LIBRARY   SLATEC (SLAP)
    C***CATEGORY  D2A3
    C***TYPE      DOUBLE PRECISION (SSLI2-S, DSLI2-D)
    C***KEYWORDS  ITERATIVE PRECONDITION, LINEAR SYSTEM SOLVE, 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, NEL, IEL(NEL), JEL(NEL)
    C     DOUBLE PRECISION B(N), X(N), EL(NEL)
    C
    C     CALL DSLI2( N, B, X, NEL, IEL, JEL, EL )
    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      :OUT      Double Precision X(N).
    C         Solution to Lx = b.
    C NEL    :IN       Integer.
    C         Number of non-zeros in the EL array.
    C IEL    :IN       Integer IEL(NEL).
    C JEL    :IN       Integer JEL(NEL).
    C EL     :IN       Double Precision EL(NEL).
    C         IEL, JEL, EL contain the unit lower triangular factor   of
    C         the incomplete decomposition   of the A  matrix  stored in
    C         SLAP Row format.  The diagonal of  ones *IS* stored.  This
    C         structure can be  set up by the  DS2LT  routine.  See  the
    C         "Description", below, for more details about the  SLAP Row
    C         format.
    C
    C *Description:
    C       This routine is supplied with the SLAP package  as a routine
    C       to perform the MSOLVE operation in the DIR iteration routine
    C       for the driver routine DSGS.  It must be called via the SLAP
    C       MSOLVE calling sequence convention interface routine DSLI.
    C         **** THIS ROUTINE ITSELF DOES NOT CONFORM TO THE ****
    C               **** SLAP MSOLVE CALLING CONVENTION ****
    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
    C       double precision  array A.  In other words, for each row  in
    C       the matrix  put the diagonal  entry in A.   Then put in  the
    C       other  non-zero elements  going across  the row  (except the
    C       diagonal) in order.  The JA array holds the column index for
    C       each non-zero.  The IA array holds the offsets  into the JA,
    C       A  arrays  for  the   beginning  of  each  row.    That  is,
    C       JA(IA(IROW)),A(IA(IROW)) are the first elements of the IROW-
    C       th row in  JA and A,  and  JA(IA(IROW+1)-1), A(IA(IROW+1)-1)
    C       are  the last elements  of the  IROW-th row.   Note  that we
    C       always have  IA(N+1) = NELT+1, where N is the number of rows
    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 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  Row format  the "inner loop" 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  DSLI
    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  DSLI2