*DECK SSILUS
SUBROUTINE SSILUS (N, NELT, IA, JA, A, ISYM, NL, IL, JL, L, DINV,
+ NU, IU, JU, U, NROW, NCOL)
C***BEGIN PROLOGUE SSILUS
C***PURPOSE Incomplete LU Decomposition Preconditioner SLAP Set Up.
C Routine to generate the incomplete LDU decomposition of a
C matrix. The unit lower triangular factor L is stored by
C rows and the unit upper triangular factor U is stored by
C columns. The inverse of the diagonal matrix D is stored.
C No fill in is allowed.
C***LIBRARY SLATEC (SLAP)
C***CATEGORY D2E
C***TYPE SINGLE PRECISION (SSILUS-S, DSILUS-D)
C***KEYWORDS INCOMPLETE LU FACTORIZATION, ITERATIVE PRECONDITION,
C NON-SYMMETRIC 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, NELT, IA(NELT), JA(NELT), ISYM
C INTEGER NL, IL(NL), JL(NL), NU, IU(NU), JU(NU)
C INTEGER NROW(N), NCOL(N)
C REAL A(NELT), L(NL), DINV(N), U(NU)
C
C CALL SSILUS( N, NELT, IA, JA, A, ISYM, NL, IL, JL, L,
C $ DINV, NU, IU, JU, U, NROW, NCOL )
C
C *Arguments:
C N :IN Integer
C Order of the Matrix.
C NELT :IN Integer.
C Number of elements in arrays IA, JA, and A.
C IA :IN Integer IA(NELT).
C JA :IN Integer JA(NELT).
C A :IN Real A(NELT).
C These arrays should hold the matrix A in the SLAP Column
C format. See "Description", below.
C ISYM :IN Integer.
C Flag to indicate symmetric storage format.
C If ISYM=0, all non-zero entries of the matrix are stored.
C If ISYM=1, the matrix is symmetric, and only the lower
C triangle of the matrix is stored.
C NL :OUT Integer.
C Number of non-zeros in the L array.
C IL :OUT Integer IL(NL).
C JL :OUT Integer JL(NL).
C L :OUT Real L(NL).
C IL, JL, L contain the unit lower triangular factor of the
C incomplete decomposition of some matrix stored in SLAP
C Row format. The Diagonal of ones *IS* stored. See
C "DESCRIPTION", below for more details about the SLAP format.
C NU :OUT Integer.
C Number of non-zeros in the U array.
C IU :OUT Integer IU(NU).
C JU :OUT Integer JU(NU).
C U :OUT 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. See
C "Description", below for more details about the SLAP
C format.
C NROW :WORK Integer NROW(N).
C NROW(I) is the number of non-zero elements in the I-th row
C of L.
C NCOL :WORK Integer NCOL(N).
C NCOL(I) is the number of non-zero elements in the I-th
C column of U.
C
C *Description
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***SEE ALSO SILUR
C***REFERENCES 1. Gene Golub and Charles Van Loan, Matrix Computations,
C Johns Hopkins University Press, Baltimore, Maryland,
C 1983.
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 920929 Corrected format of reference. (FNF)
C 930701 Updated CATEGORY section. (FNF, WRB)
C***END PROLOGUE SSILUS
C .. Scalar Arguments ..
INTEGER ISYM, N, NELT, NL, NU
C .. Array Arguments ..
REAL A(NELT), DINV(N), L(NL), U(NU)
INTEGER IA(NELT), IL(NL), IU(NU), JA(NELT), JL(NL), JU(NU),
+ NCOL(N), NROW(N)
C .. Local Scalars ..
REAL TEMP
INTEGER I, IBGN, ICOL, IEND, INDX, INDX1, INDX2, INDXC1, INDXC2,
+ INDXR1, INDXR2, IROW, ITEMP, J, JBGN, JEND, JTEMP, K, KC,
+ KR
C***FIRST EXECUTABLE STATEMENT SSILUS
C
C Count number of elements in each row of the lower triangle.
C
DO 10 I=1,N
NROW(I) = 0
NCOL(I) = 0
10 CONTINUE
CVD$R NOCONCUR
CVD$R NOVECTOR
DO 30 ICOL = 1, N
JBGN = JA(ICOL)+1
JEND = JA(ICOL+1)-1
IF( JBGN.LE.JEND ) THEN
DO 20 J = JBGN, JEND
IF( IA(J).LT.ICOL ) THEN
NCOL(ICOL) = NCOL(ICOL) + 1
ELSE
NROW(IA(J)) = NROW(IA(J)) + 1
IF( ISYM.NE.0 ) NCOL(IA(J)) = NCOL(IA(J)) + 1
ENDIF
20 CONTINUE
ENDIF
30 CONTINUE
JU(1) = 1
IL(1) = 1
DO 40 ICOL = 1, N
IL(ICOL+1) = IL(ICOL) + NROW(ICOL)
JU(ICOL+1) = JU(ICOL) + NCOL(ICOL)
NROW(ICOL) = IL(ICOL)
NCOL(ICOL) = JU(ICOL)
40 CONTINUE
C
C Copy the matrix A into the L and U structures.
DO 60 ICOL = 1, N
DINV(ICOL) = A(JA(ICOL))
JBGN = JA(ICOL)+1
JEND = JA(ICOL+1)-1
IF( JBGN.LE.JEND ) THEN
DO 50 J = JBGN, JEND
IROW = IA(J)
IF( IROW.LT.ICOL ) THEN
C Part of the upper triangle.
IU(NCOL(ICOL)) = IROW
U(NCOL(ICOL)) = A(J)
NCOL(ICOL) = NCOL(ICOL) + 1
ELSE
C Part of the lower triangle (stored by row).
JL(NROW(IROW)) = ICOL
L(NROW(IROW)) = A(J)
NROW(IROW) = NROW(IROW) + 1
IF( ISYM.NE.0 ) THEN
C Symmetric...Copy lower triangle into upper triangle as well.
IU(NCOL(IROW)) = ICOL
U(NCOL(IROW)) = A(J)
NCOL(IROW) = NCOL(IROW) + 1
ENDIF
ENDIF
50 CONTINUE
ENDIF
60 CONTINUE
C
C Sort the rows of L and the columns of U.
DO 110 K = 2, N
JBGN = JU(K)
JEND = JU(K+1)-1
IF( JBGN.LT.JEND ) THEN
DO 80 J = JBGN, JEND-1
DO 70 I = J+1, JEND
IF( IU(J).GT.IU(I) ) THEN
ITEMP = IU(J)
IU(J) = IU(I)
IU(I) = ITEMP
TEMP = U(J)
U(J) = U(I)
U(I) = TEMP
ENDIF
70 CONTINUE
80 CONTINUE
ENDIF
IBGN = IL(K)
IEND = IL(K+1)-1
IF( IBGN.LT.IEND ) THEN
DO 100 I = IBGN, IEND-1
DO 90 J = I+1, IEND
IF( JL(I).GT.JL(J) ) THEN
JTEMP = JU(I)
JU(I) = JU(J)
JU(J) = JTEMP
TEMP = L(I)
L(I) = L(J)
L(J) = TEMP
ENDIF
90 CONTINUE
100 CONTINUE
ENDIF
110 CONTINUE
C
C Perform the incomplete LDU decomposition.
DO 300 I=2,N
C
C I-th row of L
INDX1 = IL(I)
INDX2 = IL(I+1) - 1
IF(INDX1 .GT. INDX2) GO TO 200
DO 190 INDX=INDX1,INDX2
IF(INDX .EQ. INDX1) GO TO 180
INDXR1 = INDX1
INDXR2 = INDX - 1
INDXC1 = JU(JL(INDX))
INDXC2 = JU(JL(INDX)+1) - 1
IF(INDXC1 .GT. INDXC2) GO TO 180
160 KR = JL(INDXR1)
170 KC = IU(INDXC1)
IF(KR .GT. KC) THEN
INDXC1 = INDXC1 + 1
IF(INDXC1 .LE. INDXC2) GO TO 170
ELSEIF(KR .LT. KC) THEN
INDXR1 = INDXR1 + 1
IF(INDXR1 .LE. INDXR2) GO TO 160
ELSEIF(KR .EQ. KC) THEN
L(INDX) = L(INDX) - L(INDXR1)*DINV(KC)*U(INDXC1)
INDXR1 = INDXR1 + 1
INDXC1 = INDXC1 + 1
IF(INDXR1 .LE. INDXR2 .AND. INDXC1 .LE. INDXC2) GO TO 160
ENDIF
180 L(INDX) = L(INDX)/DINV(JL(INDX))
190 CONTINUE
C
C I-th column of U
200 INDX1 = JU(I)
INDX2 = JU(I+1) - 1
IF(INDX1 .GT. INDX2) GO TO 260
DO 250 INDX=INDX1,INDX2
IF(INDX .EQ. INDX1) GO TO 240
INDXC1 = INDX1
INDXC2 = INDX - 1
INDXR1 = IL(IU(INDX))
INDXR2 = IL(IU(INDX)+1) - 1
IF(INDXR1 .GT. INDXR2) GO TO 240
210 KR = JL(INDXR1)
220 KC = IU(INDXC1)
IF(KR .GT. KC) THEN
INDXC1 = INDXC1 + 1
IF(INDXC1 .LE. INDXC2) GO TO 220
ELSEIF(KR .LT. KC) THEN
INDXR1 = INDXR1 + 1
IF(INDXR1 .LE. INDXR2) GO TO 210
ELSEIF(KR .EQ. KC) THEN
U(INDX) = U(INDX) - L(INDXR1)*DINV(KC)*U(INDXC1)
INDXR1 = INDXR1 + 1
INDXC1 = INDXC1 + 1
IF(INDXR1 .LE. INDXR2 .AND. INDXC1 .LE. INDXC2) GO TO 210
ENDIF
240 U(INDX) = U(INDX)/DINV(IU(INDX))
250 CONTINUE
C
C I-th diagonal element
260 INDXR1 = IL(I)
INDXR2 = IL(I+1) - 1
IF(INDXR1 .GT. INDXR2) GO TO 300
INDXC1 = JU(I)
INDXC2 = JU(I+1) - 1
IF(INDXC1 .GT. INDXC2) GO TO 300
270 KR = JL(INDXR1)
280 KC = IU(INDXC1)
IF(KR .GT. KC) THEN
INDXC1 = INDXC1 + 1
IF(INDXC1 .LE. INDXC2) GO TO 280
ELSEIF(KR .LT. KC) THEN
INDXR1 = INDXR1 + 1
IF(INDXR1 .LE. INDXR2) GO TO 270
ELSEIF(KR .EQ. KC) THEN
DINV(I) = DINV(I) - L(INDXR1)*DINV(KC)*U(INDXC1)
INDXR1 = INDXR1 + 1
INDXC1 = INDXC1 + 1
IF(INDXR1 .LE. INDXR2 .AND. INDXC1 .LE. INDXC2) GO TO 270
ENDIF
C
300 CONTINUE
C
C Replace diagonal elements by their inverses.
CVD$ VECTOR
DO 430 I=1,N
DINV(I) = 1.0E0/DINV(I)
430 CONTINUE
C
RETURN
C------------- LAST LINE OF SSILUS FOLLOWS ----------------------------
END