*DECK DLLTI2
SUBROUTINE DLLTI2 (N, B, X, NEL, IEL, JEL, EL, DINV)
C***BEGIN PROLOGUE DLLTI2
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 DOUBLE 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 DOUBLE PRECISION B(N), X(N), EL(NEL), DINV(N)
C
C CALL DLLTI2( N, B, X, NEL, IEL, JEL, EL, DINV )
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 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 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 DINV :IN Double Precision 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 DSICCG. It must be called via the
C SLAP MSOLVE calling sequence convention interface routine
C DSLLI.
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 DSICS 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
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 DSICCG, DSICS
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 DLLTI2
C .. Scalar Arguments ..
INTEGER N, NEL
C .. Array Arguments ..
DOUBLE PRECISION 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 DLLTI2
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 DLLTI2 FOLLOWS ----------------------------
END