# dsics.f

          SUBROUTINE DSICS (N, NELT, IA, JA, A, ISYM, NEL, IEL, JEL, EL, D,
         +   R, IWARN)
    C***BEGIN PROLOGUE  DSICS
    C***PURPOSE  Incompl. Cholesky Decomposition Preconditioner SLAP Set Up.
    C            Routine to generate the Incomplete Cholesky decomposition,
    C            L*D*L-trans, of a symmetric positive definite matrix, A,
    C            which is stored in SLAP Column format.  The unit lower
    C            triangular matrix L is stored by rows, and the inverse of
    C            the diagonal matrix D is stored.
    C***LIBRARY   SLATEC (SLAP)
    C***CATEGORY  D2E
    C***TYPE      DOUBLE PRECISION (SSICS-S, DSICS-D)
    C***KEYWORDS  INCOMPLETE CHOLESKY FACTORIZATION,
    C             ITERATIVE PRECONDITION, 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
    C     INTEGER NEL, IEL(NEL), JEL(NEL), IWARN
    C     DOUBLE PRECISION A(NELT), EL(NEL), D(N), R(N)
    C
    C     CALL DSICS( N, NELT, IA, JA, A, ISYM, NEL, IEL, JEL, EL, D, R,
    C    $    IWARN )
    C
    C *Arguments:
    C N      :IN       Integer.
    C         Order of the Matrix.
    C NELT   :IN       Integer.
    C         Number of elements in arrays IA, JA, and A.
    C IA     :INOUT    Integer IA(NELT).
    C JA     :INOUT    Integer JA(NELT).
    C A      :INOUT    Double Precision A(NELT).
    C         These arrays should hold the matrix A in the SLAP Column
    C         format.  See "Description", below.
    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 lower
    C         triangle of the matrix is stored.
    C NEL    :OUT      Integer.
    C         Number of non-zeros in the lower triangle of A.   Also
    C         corresponds to the length of the IEL, JEL, EL arrays.
    C IEL    :OUT      Integer IEL(NEL).
    C JEL    :OUT      Integer JEL(NEL).
    C EL     :OUT      Double Precision EL(NEL).
    C         IEL, JEL, EL contain the unit lower triangular factor  of the
    C         incomplete decomposition   of the A  matrix  stored  in  SLAP
    C         Row format.   The Diagonal of   ones   *IS*   stored.     See
    C         "Description", below for more details about the SLAP Row fmt.
    C D      :OUT      Double Precision D(N)
    C         Upon return this array holds D(I) = 1./DIAG(A).
    C R      :WORK     Double Precision R(N).
    C         Temporary double precision workspace needed for the
    C         factorization.
    C IWARN  :OUT      Integer.
    C         This is a warning variable and is zero if the IC factoriza-
    C         tion goes well.  It is set to the row index corresponding to
    C         the last zero pivot found.  See "Description", below.
    C
    C *Description
    C       =================== S L A P Column format ==================
    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       double precision array A.   In other words,  for each column
    C       in the matrix 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)) points   to the beginning  of the
    C       ICOL-th   column    in    IA and   A.      IA(JA(ICOL+1)-1),
    C       A(JA(ICOL+1)-1) points to  the  end of the   ICOL-th column.
    C       Note that we always have  JA(N+1) = NELT+1,  where N is  the
    C       number of columns in  the matrix and NELT  is the number  of
    C       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
    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  format some  of  the   "inner  loops" of this
    C       routine should vectorize  on  machines with hardware support
    C       for vector   gather/scatter  operations.  Your compiler  may
    C       require a compiler directive to  convince it that  there are
    C       no  implicit  vector  dependencies.  Compiler directives for
    C       the Alliant    FX/Fortran and CRI   CFT/CFT77 compilers  are
    C       supplied with the standard SLAP distribution.
    C
    C       The IC factorization does not always exist for SPD matrices.
    C       In the event that a zero pivot is found it is set  to be 1.0
    C       and the factorization proceeds.   The integer variable IWARN
    C       is set to the last row where the Diagonal was fudged.  This
    C       eventuality hardly ever occurs in practice.
    C
    C***SEE ALSO  DCG, DSICCG
    C***REFERENCES  1. Gene Golub and Charles Van Loan, Matrix Computations,
    C                  Johns Hopkins University Press, Baltimore, Maryland,
    C                  1983.
    C***ROUTINES CALLED  XERMSG
    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   900805  Changed XERRWV calls to calls to XERMSG.  (RWC)
    C   910411  Prologue converted to Version 4.0 format.  (BAB)
    C   920511  Added complete declaration section.  (WRB)
    C   920929  Corrected format of reference.  (FNF)
    C   930701  Updated CATEGORY section.  (FNF, WRB)
    C***END PROLOGUE  DSICS