*DECK SSMMI2 SUBROUTINE SSMMI2 (N, B, X, IL, JL, L, DINV, IU, JU, U) C***BEGIN PROLOGUE SSMMI2 C***PURPOSE SLAP Backsolve for LDU Factorization of Normal Equations. C To solve a system of the form (L*D*U)*(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 (SSMMI2-S, DSMMI2-D) C***KEYWORDS 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, IL(NL), JL(NL), IU(NU), JU(NU) C REAL B(N), X(N), L(NL), DINV(N), U(NU) C C CALL SSMMI2( 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)(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 MSOLVE operation in the SBCGN iteration C routine for the driver SSLUCN. It must be called via the C SLAP MSOLVE calling sequence convention interface routine C SSMMTI. 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 SSMMI2