DSMMI2(3)

SYNOPSIS
SUBROUTINE DSMMI2 (N, B, X, IL, JL, L, DINV, IU, JU, U)

NAME
dsmmi2 - SLAP Backsolve for LDU Factorization of Normal Equations. To solve a system of the form (L*D*U)*(L*D*U)' X = B, where L is a unit lower triangular matrix, D is a diagonal matrix, and U is a unit upper triangular matrix and ' denotes transpose.

LIBRARY
SLATEC (SLAP)

CATEGORY
D2E

TYPE
DOUBLE PRECISION (SSMMI2-S, DSMMI2-D)

KEYWORDS
ITERATIVE PRECONDITION, LINEAR SYSTEM, SLAP, SPARSE

AUTHOR
Greenbaum, Anne, (Courant Institute) Seager, Mark K., (LLNL) Lawrence Livermore National Laboratory PO BOX 808, L-60 Livermore, CA 94550 (510) 423-3141 seager@llnl.gov

DESCRIPTION
*Usage: INTEGER N, IL(NL), JL(NL), IU(NU), JU(NU) DOUBLE PRECISION B(N), X(N), L(NL), DINV(N), U(NU) CALL DSMMI2( N, B, X, IL, JL, L, DINV, IU, JU, U ) *Arguments: N :IN Integer Order of the Matrix. B :IN Double Precision B(N). Right hand side. X :OUT Double Precision X(N). Solution of (L*D*U)(L*D*U)trans x = b. IL :IN Integer IL(NL). JL :IN Integer JL(NL). L :IN Double Precision L(NL). IL, JL, L contain the unit lower triangular factor of the incomplete decomposition of some matrix stored in SLAP Row format. The diagonal of ones *IS* stored. This structure can be set up by the DSILUS routine. See the "Description", below for more details about the SLAP format. (NL is the number of non-zeros in the L array.) DINV :IN Double Precision DINV(N). Inverse of the diagonal matrix D. IU :IN Integer IU(NU). JU :IN Integer JU(NU). U :IN Double Precision U(NU). IU, JU, U contain the unit upper triangular factor of the incomplete decomposition of some matrix stored in SLAP Column format. The diagonal of ones *IS* stored. This structure can be set up by the DSILUS routine. See the "Description", below for more details about the SLAP format. (NU is the number of non-zeros in the U array.) *Description: This routine is supplied with the SLAP package as a routine to perform the MSOLVE operation in the SBCGN iteration routine for the driver DSLUCN. It must be called via the SLAP MSOLVE calling sequence convention interface routine DSMMTI. **** THIS ROUTINE ITSELF DOES NOT CONFORM TO THE **** **** SLAP MSOLVE CALLING CONVENTION **** IL, JL, L should contain the unit lower triangular factor of the incomplete decomposition of the A matrix stored in SLAP Row format. IU, JU, U should contain the unit upper factor of the incomplete decomposition of the A matrix stored in SLAP Column format This ILU factorization can be computed by the DSILUS routine. The diagonals (which are all one's) are stored. =================== S L A P Column format ================== This routine requires that the matrix A be stored in the SLAP Column format. In this format the non-zeros are stored counting down columns (except for the diagonal entry, which must appear first in each "column") and are stored in the double precision array A. In other words, for each column in the matrix put the diagonal entry in A. Then put in the other non-zero elements going down the column (except the diagonal) in order. The IA array holds the row index for each non-zero. The JA array holds the offsets into the IA, A arrays for the beginning of each column. That is, IA(JA(ICOL)), A(JA(ICOL)) points to the beginning of the ICOL-th column in IA and A. IA(JA(ICOL+1)-1), A(JA(ICOL+1)-1) points to the end of the ICOL-th column. Note that we always have JA(N+1) = NELT+1, where N is the number of columns in the matrix and NELT is the number of non-zeros in the matrix. Here is an example of the SLAP Column storage format for a 5x5 Matrix (in the A and IA arrays '|' denotes the end of a column): 5x5 Matrix SLAP Column format for 5x5 matrix on left. 1 2 3 4 5 6 7 8 9 10 11 |11 12 0 0 15| A: 11 21 51 | 22 12 | 33 53 | 44 | 55 15 35 |21 22 0 0 0| IA: 1 2 5 | 2 1 | 3 5 | 4 | 5 1 3 | 0 0 33 0 35| JA: 1 4 6 8 9 12 | 0 0 0 44 0| |51 0 53 0 55| ==================== S L A P Row format ==================== This routine requires that the matrix A be stored in the SLAP Row format. In this format the non-zeros are stored counting across rows (except for the diagonal entry, which must appear first in each "row") and are stored in the double precision array A. In other words, for each row in the matrix put the diagonal entry in A. Then put in the other non-zero elements going across the row (except the diagonal) in order. The JA array holds the column index for each non-zero. The IA array holds the offsets into the JA, A arrays for the beginning of each row. That is, JA(IA(IROW)),A(IA(IROW)) are the first elements of the IROW- th row in JA and A, and JA(IA(IROW+1)-1), A(IA(IROW+1)-1) are the last elements of the IROW-th row. Note that we always have IA(N+1) = NELT+1, where N is the number of rows in the matrix and NELT is the number of non-zeros in the matrix. Here is an example of the SLAP Row storage format for a 5x5 Matrix (in the A and JA arrays '|' denotes the end of a row): 5x5 Matrix SLAP Row format for 5x5 matrix on left. 1 2 3 4 5 6 7 8 9 10 11 |11 12 0 0 15| A: 11 12 15 | 22 21 | 33 35 | 44 | 55 51 53 |21 22 0 0 0| JA: 1 2 5 | 2 1 | 3 5 | 4 | 5 1 3 | 0 0 33 0 35| IA: 1 4 6 8 9 12 | 0 0 0 44 0| |51 0 53 0 55| With the SLAP format the "inner loops" of this routine should vectorize on machines with hardware support for vector gather/scatter operations. Your compiler may require a compiler directive to convince it that there are no implicit vector dependencies. Compiler directives for the Alliant FX/Fortran and CRI CFT/CFT77 compilers are supplied with the standard SLAP distribution.

SEE ALSO
DSILUS

REFERENCES
(NONE)

ROUTINES CALLED
(NONE)

REVISION HISTORY (YYMMDD)
871119 DATE WRITTEN
881213 Previous REVISION DATE
890915 Made changes requested at July 1989 CML Meeting. (MKS)
890922 Numerous changes to prologue to make closer to SLATEC standard. (FNF)
890929 Numerous changes to reduce SP/DP differences. (FNF)
910411 Prologue converted to Version 4.0 format. (BAB)
920511 Added complete declaration section. (WRB)
921113 Corrected C***CATEGORY line. (FNF)
930701 Updated CATEGORY section. (FNF, WRB)