*DECK SSICS
SUBROUTINE SSICS (N, NELT, IA, JA, A, ISYM, NEL, IEL, JEL, EL, D,
+ R, IWARN)
C***BEGIN PROLOGUE SSICS
C***PURPOSE Incompl. Cholesky Decomposition Preconditioner SLAP Set Up.
C Routine to generate the Incomplete Cholesky decomposition,
C L*D*L-trans, of a symmetric positive definite matrix, A,
C which is stored in SLAP Column format. The unit lower
C triangular matrix L is stored by rows, and the inverse of
C the diagonal matrix D is stored.
C***LIBRARY SLATEC (SLAP)
C***CATEGORY D2E
C***TYPE SINGLE PRECISION (SSICS-S, DSICS-D)
C***KEYWORDS INCOMPLETE CHOLESKY FACTORIZATION,
C ITERATIVE PRECONDITION, 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 NEL, IEL(NEL), JEL(NEL), IWARN
C REAL A(NELT), EL(NEL), D(N), R(N)
C
C CALL SSICS( N, NELT, IA, JA, A, ISYM, NEL, IEL, JEL, EL, D, R,
C $ IWARN )
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 :INOUT Integer IA(NELT).
C JA :INOUT Integer JA(NELT).
C A :INOUT 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 NEL :OUT Integer.
C Number of non-zeros in the lower triangle of A. Also
C corresponds to the length of the IEL, JEL, EL arrays.
C IEL :OUT Integer IEL(NEL).
C JEL :OUT Integer JEL(NEL).
C EL :OUT Real EL(NEL).
C IEL, JEL, EL contain the unit lower triangular factor of the
C incomplete decomposition of the A matrix stored in SLAP
C Row format. The Diagonal of ones *IS* stored. See
C "Description", below for more details about the SLAP Row fmt.
C D :OUT Real D(N)
C Upon return this array holds D(I) = 1./DIAG(A).
C R :WORK Real R(N).
C Temporary real workspace needed for the factorization.
C IWARN :OUT Integer.
C This is a warning variable and is zero if the IC factoriza-
C tion goes well. It is set to the row index corresponding to
C the last zero pivot found. See "Description", below.
C
C *Description
C =================== S L A P Column format ==================
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 With the SLAP format some of the "inner loops" of this
C routine should vectorize on machines with hardware support
C for vector gather/scatter operations. Your compiler may
C require a compiler directive to convince it that there are
C no implicit vector dependencies. Compiler directives for
C the Alliant FX/Fortran and CRI CFT/CFT77 compilers are
C supplied with the standard SLAP distribution.
C
C The IC factorization does not always exist for SPD matrices.
C In the event that a zero pivot is found it is set to be 1.0
C and the factorization proceeds. The integer variable IWARN
C is set to the last row where the Diagonal was fudged. This
C eventuality hardly ever occurs in practice.
C
C***SEE ALSO SCG, SSICCG
C***REFERENCES 1. Gene Golub and Charles Van Loan, Matrix Computations,
C Johns Hopkins University Press, Baltimore, Maryland,
C 1983.
C***ROUTINES CALLED XERMSG
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 900805 Changed XERRWV calls to calls to XERMSG. (RWC)
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 SSICS
C .. Scalar Arguments ..
INTEGER ISYM, IWARN, N, NEL, NELT
C .. Array Arguments ..
REAL A(NELT), D(N), EL(NEL), R(N)
INTEGER IA(NELT), IEL(NEL), JA(NELT), JEL(NEL)
C .. Local Scalars ..
REAL ELTMP
INTEGER I, IBGN, IC, ICBGN, ICEND, ICOL, IEND, IR, IRBGN, IREND,
+ IROW, IRR, J, JBGN, JELTMP, JEND
CHARACTER XERN1*8
C .. External Subroutines ..
EXTERNAL XERMSG
C***FIRST EXECUTABLE STATEMENT SSICS
C
C Set the lower triangle in IEL, JEL, EL
C
IWARN = 0
C
C All matrix elements stored in IA, JA, A. Pick out the lower
C triangle (making sure that the Diagonal of EL is one) and
C store by rows.
C
NEL = 1
IEL(1) = 1
JEL(1) = 1
EL(1) = 1
D(1) = A(1)
CVD$R NOCONCUR
DO 30 IROW = 2, N
C Put in the Diagonal.
NEL = NEL + 1
IEL(IROW) = NEL
JEL(NEL) = IROW
EL(NEL) = 1
D(IROW) = A(JA(IROW))
C
C Look in all the lower triangle columns for a matching row.
C Since the matrix is symmetric, we can look across the
C ITOW-th row by looking down the IROW-th column (if it is
C stored ISYM=0)...
IF( ISYM.EQ.0 ) THEN
ICBGN = JA(IROW)
ICEND = JA(IROW+1)-1
ELSE
ICBGN = 1
ICEND = IROW-1
ENDIF
DO 20 IC = ICBGN, ICEND
IF( ISYM.EQ.0 ) THEN
ICOL = IA(IC)
IF( ICOL.GE.IROW ) GOTO 20
ELSE
ICOL = IC
ENDIF
JBGN = JA(ICOL)+1
JEND = JA(ICOL+1)-1
IF( JBGN.LE.JEND .AND. IA(JEND).GE.IROW ) THEN
CVD$ NOVECTOR
DO 10 J = JBGN, JEND
IF( IA(J).EQ.IROW ) THEN
NEL = NEL + 1
JEL(NEL) = ICOL
EL(NEL) = A(J)
GOTO 20
ENDIF
10 CONTINUE
ENDIF
20 CONTINUE
30 CONTINUE
IEL(N+1) = NEL+1
C
C Sort ROWS of lower triangle into descending order (count out
C along rows out from Diagonal).
C
DO 60 IROW = 2, N
IBGN = IEL(IROW)+1
IEND = IEL(IROW+1)-1
IF( IBGN.LT.IEND ) THEN
DO 50 I = IBGN, IEND-1
CVD$ NOVECTOR
DO 40 J = I+1, IEND
IF( JEL(I).GT.JEL(J) ) THEN
JELTMP = JEL(J)
JEL(J) = JEL(I)
JEL(I) = JELTMP
ELTMP = EL(J)
EL(J) = EL(I)
EL(I) = ELTMP
ENDIF
40 CONTINUE
50 CONTINUE
ENDIF
60 CONTINUE
C
C Perform the Incomplete Cholesky decomposition by looping
C over the rows.
C Scale the first column. Use the structure of A to pick out
C the rows with something in column 1.
C
IRBGN = JA(1)+1
IREND = JA(2)-1
DO 65 IRR = IRBGN, IREND
IR = IA(IRR)
C Find the index into EL for EL(1,IR).
C Hint: it's the second entry.
I = IEL(IR)+1
EL(I) = EL(I)/D(1)
65 CONTINUE
C
DO 110 IROW = 2, N
C
C Update the IROW-th diagonal.
C
DO 66 I = 1, IROW-1
R(I) = 0
66 CONTINUE
IBGN = IEL(IROW)+1
IEND = IEL(IROW+1)-1
IF( IBGN.LE.IEND ) THEN
CLLL. OPTION ASSERT (NOHAZARD)
CDIR$ IVDEP
CVD$ NODEPCHK
DO 70 I = IBGN, IEND
R(JEL(I)) = EL(I)*D(JEL(I))
D(IROW) = D(IROW) - EL(I)*R(JEL(I))
70 CONTINUE
C
C Check to see if we have a problem with the diagonal.
C
IF( D(IROW).LE.0.0E0 ) THEN
IF( IWARN.EQ.0 ) IWARN = IROW
D(IROW) = 1
ENDIF
ENDIF
C
C Update each EL(IROW+1:N,IROW), if there are any.
C Use the structure of A to determine the Non-zero elements
C of the IROW-th column of EL.
C
IRBGN = JA(IROW)
IREND = JA(IROW+1)-1
DO 100 IRR = IRBGN, IREND
IR = IA(IRR)
IF( IR.LE.IROW ) GOTO 100
C Find the index into EL for EL(IR,IROW)
IBGN = IEL(IR)+1
IEND = IEL(IR+1)-1
IF( JEL(IBGN).GT.IROW ) GOTO 100
DO 90 I = IBGN, IEND
IF( JEL(I).EQ.IROW ) THEN
ICEND = IEND
91 IF( JEL(ICEND).GE.IROW ) THEN
ICEND = ICEND - 1
GOTO 91
ENDIF
C Sum up the EL(IR,1:IROW-1)*R(1:IROW-1) contributions.
CLLL. OPTION ASSERT (NOHAZARD)
CDIR$ IVDEP
CVD$ NODEPCHK
DO 80 IC = IBGN, ICEND
EL(I) = EL(I) - EL(IC)*R(JEL(IC))
80 CONTINUE
EL(I) = EL(I)/D(IROW)
GOTO 100
ENDIF
90 CONTINUE
C
C If we get here, we have real problems...
WRITE (XERN1, '(I8)') IROW
CALL XERMSG ('SLATEC', 'SSICS',
$ 'A and EL data structure mismatch in row '// XERN1, 1, 2)
100 CONTINUE
110 CONTINUE
C
C Replace diagonals by their inverses.
C
CVD$ CONCUR
DO 120 I =1, N
D(I) = 1.0E0/D(I)
120 CONTINUE
RETURN
C------------- LAST LINE OF SSICS FOLLOWS ----------------------------
END