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SUBROUTINE DSLUI4 (N, B, X, IL, JL, L, DINV, IU, JU, U)
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C***BEGIN PROLOGUE DSLUI4
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C***PURPOSE SLAP Backsolve for LDU Factorization.
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C Routine to solve a system of the form (L*D*U)' X = B,
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C where L is a unit lower triangular matrix, D is a diagonal
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C matrix, and U is a unit upper triangular matrix and '
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C***LIBRARY SLATEC (SLAP)
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C***TYPE DOUBLE PRECISION (SSLUI4-S, DSLUI4-D)
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C***KEYWORDS ITERATIVE PRECONDITION, NON-SYMMETRIC LINEAR SYSTEM SOLVE,
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C***AUTHOR Greenbaum, Anne, (Courant Institute)
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C Seager, Mark K., (LLNL)
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C Lawrence Livermore National Laboratory
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C Livermore, CA 94550 (510) 423-3141
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C INTEGER N, IL(NL), JL(NL), IU(NU), JU(NU)
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C DOUBLE PRECISION B(N), X(N), L(NL), DINV(N), U(NU)
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C CALL DSLUI4( N, B, X, IL, JL, L, DINV, IU, JU, U )
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C Order of the Matrix.
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C B :IN Double Precision B(N).
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C X :OUT Double Precision X(N).
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C Solution of (L*D*U)trans x = b.
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C IL :IN Integer IL(NL).
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C JL :IN Integer JL(NL).
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C L :IN Double Precision L(NL).
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C IL, JL, L contain the unit lower triangular factor of the
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C incomplete decomposition of some matrix stored in SLAP Row
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C format. The diagonal of ones *IS* stored. This structure
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C can be set up by the DSILUS routine. See the
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C "Description", below for more details about the SLAP
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C format. (NL is the number of non-zeros in the L array.)
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C DINV :IN Double Precision DINV(N).
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C Inverse of the diagonal matrix D.
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C IU :IN Integer IU(NU).
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C JU :IN Integer JU(NU).
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C U :IN Double Precision U(NU).
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C IU, JU, U contain the unit upper triangular factor of the
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C incomplete decomposition of some matrix stored in SLAP
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C Column format. The diagonal of ones *IS* stored. This
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C structure can be set up by the DSILUS routine. See the
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C "Description", below for more details about the SLAP
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C format. (NU is the number of non-zeros in the U array.)
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C This routine is supplied with the SLAP package as a routine
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C to perform the MTSOLV operation in the SBCG iteration
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C routine for the driver DSLUBC. It must be called via the
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C SLAP MTSOLV calling sequence convention interface routine
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C **** THIS ROUTINE ITSELF DOES NOT CONFORM TO THE ****
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C **** SLAP MSOLVE CALLING CONVENTION ****
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C IL, JL, L should contain the unit lower triangular factor of
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C the incomplete decomposition of the A matrix stored in SLAP
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C Row format. IU, JU, U should contain the unit upper factor
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C of the incomplete decomposition of the A matrix stored in
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C SLAP Column format This ILU factorization can be computed by
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C the DSILUS routine. The diagonals (which are all one's) are
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C =================== S L A P Column format ==================
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C This routine requires that the matrix A be stored in the
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C SLAP Column format. In this format the non-zeros are stored
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C counting down columns (except for the diagonal entry, which
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C must appear first in each "column") and are stored in the
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C double precision array A. In other words, for each column
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C in the matrix put the diagonal entry in A. Then put in the
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C other non-zero elements going down the column (except the
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C diagonal) in order. The IA array holds the row index for
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C each non-zero. The JA array holds the offsets into the IA,
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C A arrays for the beginning of each column. That is,
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C IA(JA(ICOL)), A(JA(ICOL)) points to the beginning of the
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C ICOL-th column in IA and A. IA(JA(ICOL+1)-1),
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C A(JA(ICOL+1)-1) points to the end of the ICOL-th column.
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C Note that we always have JA(N+1) = NELT+1, where N is the
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C number of columns in the matrix and NELT is the number of
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C non-zeros in the matrix.
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C Here is an example of the SLAP Column storage format for a
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C 5x5 Matrix (in the A and IA arrays '|' denotes the end of a
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C 5x5 Matrix SLAP Column format for 5x5 matrix on left.
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C 1 2 3 4 5 6 7 8 9 10 11
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C |11 12 0 0 15| A: 11 21 51 | 22 12 | 33 53 | 44 | 55 15 35
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C |21 22 0 0 0| IA: 1 2 5 | 2 1 | 3 5 | 4 | 5 1 3
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C | 0 0 33 0 35| JA: 1 4 6 8 9 12
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C ==================== S L A P Row format ====================
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C This routine requires that the matrix A be stored in the
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C SLAP Row format. In this format the non-zeros are stored
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C counting across rows (except for the diagonal entry, which
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C must appear first in each "row") and are stored in the
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C double precision array A. In other words, for each row in
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C the matrix put the diagonal entry in A. Then put in the
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C other non-zero elements going across the row (except the
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C diagonal) in order. The JA array holds the column index for
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C each non-zero. The IA array holds the offsets into the JA,
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C A arrays for the beginning of each row. That is,
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C JA(IA(IROW)),A(IA(IROW)) are the first elements of the IROW-
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C th row in JA and A, and JA(IA(IROW+1)-1), A(IA(IROW+1)-1)
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C are the last elements of the IROW-th row. Note that we
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C always have IA(N+1) = NELT+1, where N is the number of rows
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C in the matrix and NELT is the number of non-zeros in the
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C Here is an example of the SLAP Row storage format for a 5x5
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C Matrix (in the A and JA arrays '|' denotes the end of a row):
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C 5x5 Matrix SLAP Row format for 5x5 matrix on left.
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C 1 2 3 4 5 6 7 8 9 10 11
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C |11 12 0 0 15| A: 11 12 15 | 22 21 | 33 35 | 44 | 55 51 53
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C |21 22 0 0 0| JA: 1 2 5 | 2 1 | 3 5 | 4 | 5 1 3
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C | 0 0 33 0 35| IA: 1 4 6 8 9 12
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C With the SLAP format the "inner loops" of this routine
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C should vectorize on machines with hardware support for
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C vector gather/scatter operations. Your compiler may require
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C a compiler directive to convince it that there are no
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C implicit vector dependencies. Compiler directives for the
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C Alliant FX/Fortran and CRI CFT/CFT77 compilers are supplied
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C with the standard SLAP distribution.
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C***REFERENCES (NONE)
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C***ROUTINES CALLED (NONE)
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C***REVISION HISTORY (YYMMDD)
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C 871119 DATE WRITTEN
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C 881213 Previous REVISION DATE
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C 890915 Made changes requested at July 1989 CML Meeting. (MKS)
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C 890922 Numerous changes to prologue to make closer to SLATEC
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C 890929 Numerous changes to reduce SP/DP differences. (FNF)
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C 910411 Prologue converted to Version 4.0 format. (BAB)
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C 920511 Added complete declaration section. (WRB)
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C 921113 Corrected C***CATEGORY line. (FNF)
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C 930701 Updated CATEGORY section. (FNF, WRB)
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C***END PROLOGUE DSLUI4
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C .. Scalar Arguments ..
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C .. Array Arguments ..
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DOUBLE PRECISION B(N), DINV(N), L(*), U(*), X(N)
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INTEGER IL(*), IU(*), JL(*), JU(*)
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C .. Local Scalars ..
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INTEGER I, ICOL, IROW, J, JBGN, JEND
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C***FIRST EXECUTABLE STATEMENT DSLUI4
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C Solve U'*Y = X, storing result in X, U stored by columns.
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JEND = JU(IROW+1) - 1
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IF( JBGN.LE.JEND ) THEN
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CLLL. OPTION ASSERT (NOHAZARD)
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X(IROW) = X(IROW) - U(J)*X(IU(J))
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C Solve D*Z = Y, storing result in X.
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C Solve L'*X = Z, L stored by rows.
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DO 110 ICOL = N, 2, -1
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JEND = IL(ICOL+1) - 1
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IF( JBGN.LE.JEND ) THEN
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CLLL. OPTION ASSERT (NOHAZARD)
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DO 100 J = JBGN, JEND
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X(JL(J)) = X(JL(J)) - L(J)*X(ICOL)
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C------------- LAST LINE OF DSLUI4 FOLLOWS ----------------------------
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deps/openlibm/slatec/dslui4.f
