*DECK SLLTI2 SUBROUTINE SLLTI2 (N, B, X, NEL, IEL, JEL, EL, DINV) C***BEGIN PROLOGUE SLLTI2 C***PURPOSE SLAP Backsolve routine for LDL' Factorization. C Routine to solve a system of the form L*D*L' X = B, C where L is a unit lower triangular matrix and D is a C diagonal matrix and ' means transpose. C***LIBRARY SLATEC (SLAP) C***CATEGORY D2E C***TYPE SINGLE PRECISION (SLLTI2-S, DLLTI2-D) C***KEYWORDS INCOMPLETE FACTORIZATION, ITERATIVE PRECONDITION, SLAP, C SPARSE, SYMMETRIC LINEAR SYSTEM SOLVE 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 REAL B(N), X(N), EL(NEL), DINV(N) C C CALL SLLTI2( N, B, X, NEL, IEL, JEL, EL, DINV ) C C *Arguments: C N :IN Integer C Order of the Matrix. C B :IN Real B(N). C Right hand side vector. C X :OUT Real X(N). C Solution to L*D*L' x = 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 Real 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 SS2LT routine. See the C "Description", below for more details about the SLAP Row C format. C DINV :IN Real DINV(N). C Inverse of the diagonal matrix D. C C *Description: C This routine is supplied with the SLAP package as a routine C to perform the MSOLVE operation in the SCG iteration routine C for the driver routine SSICCG. It must be called via the C SLAP MSOLVE calling sequence convention interface routine C SSLLI. C **** THIS ROUTINE ITSELF DOES NOT CONFORM TO THE **** C **** SLAP MSOLVE CALLING CONVENTION **** C C IEL, JEL, EL should contain the unit lower triangular factor C of the incomplete decomposition of the A matrix stored in C SLAP Row format. This IC factorization can be computed by C the SSICS routine. The diagonal (which is all one's) is C stored. 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 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 SSICCG, SSICS 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 SLLTI2 C .. Scalar Arguments .. INTEGER N, NEL C .. Array Arguments .. REAL B(N), DINV(N), EL(NEL), X(N) INTEGER IEL(NEL), JEL(NEL) C .. Local Scalars .. INTEGER I, IBGN, IEND, IROW C***FIRST EXECUTABLE STATEMENT SLLTI2 C C Solve L*y = b, storing result in x. C DO 10 I=1,N X(I) = B(I) 10 CONTINUE DO 30 IROW = 1, N IBGN = IEL(IROW) + 1 IEND = IEL(IROW+1) - 1 IF( IBGN.LE.IEND ) THEN CLLL. OPTION ASSERT (NOHAZARD) CDIR\$ IVDEP CVD\$ NOCONCUR CVD\$ NODEPCHK DO 20 I = IBGN, IEND X(IROW) = X(IROW) - EL(I)*X(JEL(I)) 20 CONTINUE ENDIF 30 CONTINUE C C Solve D*Z = Y, storing result in X. C DO 40 I=1,N X(I) = X(I)*DINV(I) 40 CONTINUE C C Solve L-trans*X = Z. C DO 60 IROW = N, 2, -1 IBGN = IEL(IROW) + 1 IEND = IEL(IROW+1) - 1 IF( IBGN.LE.IEND ) THEN CLLL. OPTION ASSERT (NOHAZARD) CDIR\$ IVDEP CVD\$ NOCONCUR CVD\$ NODEPCHK DO 50 I = IBGN, IEND X(JEL(I)) = X(JEL(I)) - EL(I)*X(IROW) 50 CONTINUE ENDIF 60 CONTINUE C RETURN C------------- LAST LINE OF SLLTI2 FOLLOWS ---------------------------- END