*DECK DSILUS SUBROUTINE DSILUS (N, NELT, IA, JA, A, ISYM, NL, IL, JL, L, DINV, + NU, IU, JU, U, NROW, NCOL) C***BEGIN PROLOGUE DSILUS 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 DOUBLE 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 DOUBLE PRECISION A(NELT), L(NL), DINV(N), U(NU) C C CALL DSILUS( 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 Double Precision 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 Double Precision 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 Double Precision 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 DSILUS 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 double precision array A. In other words, for each column C in the matrix put the diagonal entry in A. Then put in the C other non-zero elements going down the column (except the C diagonal) in order. The IA array holds the row index for C each non-zero. The JA array holds the offsets into the IA, C A arrays for the beginning of each column. That is, C IA(JA(ICOL)), A(JA(ICOL)) points to the beginning of the C ICOL-th column in IA and A. IA(JA(ICOL+1)-1), C A(JA(ICOL+1)-1) points to the end of the ICOL-th column. C Note that we always have JA(N+1) = NELT+1, where N is the C number of columns in the matrix and NELT is the number of C 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 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***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 890404 DATE WRITTEN C 890404 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 DSILUS C .. Scalar Arguments .. INTEGER ISYM, N, NELT, NL, NU C .. Array Arguments .. DOUBLE PRECISION 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 .. DOUBLE PRECISION 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 DSILUS 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) = 1.0D0/DINV(I) 430 CONTINUE C RETURN C------------- LAST LINE OF DSILUS FOLLOWS ---------------------------- END