!DECK DSILUS SUBROUTINE DSILUS (N, NELT, IA, JA, A, ISYM, NL, IL, JL, L, DINV, & NU, IU, JU, U, NROW, NCOL) !***BEGIN PROLOGUE DSILUS !***PURPOSE Incomplete LU Decomposition Preconditioner SLAP Set Up. ! Routine to generate the incomplete LDU decomposition of a ! matrix. The unit lower triangular factor L is stored by ! rows and the unit upper triangular factor U is stored by ! columns. The inverse of the diagonal matrix D is stored. ! No fill in is allowed. !***LIBRARY SLATEC (SLAP) !***CATEGORY D2E !***TYPE DOUBLE PRECISION (SSILUS-S, DSILUS-D) !***KEYWORDS INCOMPLETE LU FACTORIZATION, ITERATIVE PRECONDITION, ! NON-SYMMETRIC LINEAR SYSTEM, SLAP, SPARSE !***AUTHOR Greenbaum, Anne, (Courant Institute) ! Seager, Mark K., (LLNL) ! Lawrence Livermore National Laboratory ! PO BOX 808, L-60 ! Livermore, CA 94550 (510) 423-3141 ! seager@llnl.gov !***DESCRIPTION ! ! *Usage: ! INTEGER N, NELT, IA(NELT), JA(NELT), ISYM ! INTEGER NL, IL(NL), JL(NL), NU, IU(NU), JU(NU) ! INTEGER NROW(N), NCOL(N) ! DOUBLE PRECISION A(NELT), L(NL), DINV(N), U(NU) ! ! CALL DSILUS( N, NELT, IA, JA, A, ISYM, NL, IL, JL, L, ! $ DINV, NU, IU, JU, U, NROW, NCOL ) ! ! *Arguments: ! N :IN Integer ! Order of the Matrix. ! NELT :IN Integer. ! Number of elements in arrays IA, JA, and A. ! IA :IN Integer IA(NELT). ! JA :IN Integer JA(NELT). ! A :IN Double Precision A(NELT). ! These arrays should hold the matrix A in the SLAP Column ! format. See "Description", below. ! ISYM :IN Integer. ! Flag to indicate symmetric storage format. ! If ISYM=0, all non-zero entries of the matrix are stored. ! If ISYM=1, the matrix is symmetric, and only the lower ! triangle of the matrix is stored. ! NL :OUT Integer. ! Number of non-zeros in the L array. ! IL :OUT Integer IL(NL). ! JL :OUT Integer JL(NL). ! L :OUT Double Precision L(NL). ! IL, JL, L contain the unit lower triangular factor of the ! incomplete decomposition of some matrix stored in SLAP ! Row format. The Diagonal of ones *IS* stored. See ! "DESCRIPTION", below for more details about the SLAP format. ! NU :OUT Integer. ! Number of non-zeros in the U array. ! IU :OUT Integer IU(NU). ! JU :OUT Integer JU(NU). ! U :OUT Double Precision U(NU). ! IU, JU, U contain the unit upper triangular factor of the ! incomplete decomposition of some matrix stored in SLAP ! Column format. The Diagonal of ones *IS* stored. See ! "Description", below for more details about the SLAP ! format. ! NROW :WORK Integer NROW(N). ! NROW(I) is the number of non-zero elements in the I-th row ! of L. ! NCOL :WORK Integer NCOL(N). ! NCOL(I) is the number of non-zero elements in the I-th ! column of U. ! ! *Description ! IL, JL, L should contain the unit lower triangular factor of ! the incomplete decomposition of the A matrix stored in SLAP ! Row format. IU, JU, U should contain the unit upper factor ! of the incomplete decomposition of the A matrix stored in ! SLAP Column format This ILU factorization can be computed by ! the DSILUS routine. The diagonals (which are all one's) are ! stored. ! ! =================== S L A P Column format ================== ! ! This routine requires that the matrix A be stored in the ! SLAP Column format. In this format the non-zeros are stored ! counting down columns (except for the diagonal entry, which ! must appear first in each "column") and are stored in the ! double precision array A. In other words, for each column ! in the matrix put the diagonal entry in A. Then put in the ! other non-zero elements going down the column (except the ! diagonal) in order. The IA array holds the row index for ! each non-zero. The JA array holds the offsets into the IA, ! A arrays for the beginning of each column. That is, ! IA(JA(ICOL)), A(JA(ICOL)) points to the beginning of the ! ICOL-th column in IA and A. IA(JA(ICOL+1)-1), ! A(JA(ICOL+1)-1) points to the end of the ICOL-th column. ! Note that we always have JA(N+1) = NELT+1, where N is the ! number of columns in the matrix and NELT is the number of ! non-zeros in the matrix. ! ! Here is an example of the SLAP Column storage format for a ! 5x5 Matrix (in the A and IA arrays '|' denotes the end of a ! column): ! ! 5x5 Matrix SLAP Column format for 5x5 matrix on left. ! 1 2 3 4 5 6 7 8 9 10 11 ! |11 12 0 0 15| A: 11 21 51 | 22 12 | 33 53 | 44 | 55 15 35 ! |21 22 0 0 0| IA: 1 2 5 | 2 1 | 3 5 | 4 | 5 1 3 ! | 0 0 33 0 35| JA: 1 4 6 8 9 12 ! | 0 0 0 44 0| ! |51 0 53 0 55| ! ! ==================== S L A P Row format ==================== ! ! This routine requires that the matrix A be stored in the ! SLAP Row format. In this format the non-zeros are stored ! counting across rows (except for the diagonal entry, which ! must appear first in each "row") and are stored in the ! double precision array A. In other words, for each row in ! the matrix put the diagonal entry in A. Then put in the ! other non-zero elements going across the row (except the ! diagonal) in order. The JA array holds the column index for ! each non-zero. The IA array holds the offsets into the JA, ! A arrays for the beginning of each row. That is, ! JA(IA(IROW)),A(IA(IROW)) are the first elements of the IROW- ! th row in JA and A, and JA(IA(IROW+1)-1), A(IA(IROW+1)-1) ! are the last elements of the IROW-th row. Note that we ! always have IA(N+1) = NELT+1, where N is the number of rows ! in the matrix and NELT is the number of non-zeros in the ! matrix. ! ! Here is an example of the SLAP Row storage format for a 5x5 ! Matrix (in the A and JA arrays '|' denotes the end of a row): ! ! 5x5 Matrix SLAP Row format for 5x5 matrix on left. ! 1 2 3 4 5 6 7 8 9 10 11 ! |11 12 0 0 15| A: 11 12 15 | 22 21 | 33 35 | 44 | 55 51 53 ! |21 22 0 0 0| JA: 1 2 5 | 2 1 | 3 5 | 4 | 5 1 3 ! | 0 0 33 0 35| IA: 1 4 6 8 9 12 ! | 0 0 0 44 0| ! |51 0 53 0 55| ! !***SEE ALSO SILUR !***REFERENCES 1. Gene Golub and Charles Van Loan, Matrix Computations, ! Johns Hopkins University Press, Baltimore, Maryland, ! 1983. !***ROUTINES CALLED (NONE) !***REVISION HISTORY (YYMMDD) ! 890404 DATE WRITTEN ! 890404 Previous REVISION DATE ! 890915 Made changes requested at July 1989 CML Meeting. (MKS) ! 890922 Numerous changes to prologue to make closer to SLATEC ! standard. (FNF) ! 890929 Numerous changes to reduce SP/DP differences. (FNF) ! 910411 Prologue converted to Version 4.0 format. (BAB) ! 920511 Added complete declaration section. (WRB) ! 920929 Corrected format of reference. (FNF) ! 930701 Updated CATEGORY section. (FNF, WRB) !***END PROLOGUE DSILUS ! .. Scalar Arguments .. INTEGER ISYM, N, NELT, NL, NU ! .. 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) ! .. 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 !***FIRST EXECUTABLE STATEMENT DSILUS ! ! Count number of elements in each row of the lower triangle. ! DO 10 I=1,N NROW(I) = 0 NCOL(I) = 0 10 CONTINUE !VD$R NOCONCUR !VD$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 ! ! 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 ! Part of the upper triangle. IU(NCOL(ICOL)) = IROW U(NCOL(ICOL)) = A(J) NCOL(ICOL) = NCOL(ICOL) + 1 ELSE ! 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 ! 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 ! ! 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 ! ! Perform the incomplete LDU decomposition. DO 300 I=2,N ! ! 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 ! ! 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 ! ! 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 ! 300 CONTINUE ! ! Replace diagonal elements by their inverses. !VD$ VECTOR DO 430 I=1,N DINV(I) = 1.0D0/DINV(I) 430 CONTINUE ! RETURN !------------- LAST LINE OF DSILUS FOLLOWS ---------------------------- END