*DECK SSDSCL
SUBROUTINE SSDSCL (N, NELT, IA, JA, A, ISYM, X, B, DINV, JOB,
+ ITOL)
C***BEGIN PROLOGUE SSDSCL
C***PURPOSE Diagonal Scaling of system Ax = b.
C This routine scales (and unscales) the system Ax = b
C by symmetric diagonal scaling.
C***LIBRARY SLATEC (SLAP)
C***CATEGORY D2E
C***TYPE SINGLE PRECISION (SSDSCL-S, DSDSCL-D)
C***KEYWORDS DIAGONAL, 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 This routine scales (and unscales) the system Ax = b by symmetric
C diagonal scaling. The new system is:
C -1/2 -1/2 1/2 -1/2
C D AD (D x) = D b
C when scaling is selected with the JOB parameter. When unscaling
C is selected this process is reversed. The true solution is also
C scaled or unscaled if ITOL is set appropriately, see below.
C
C *Usage:
C INTEGER N, NELT, IA(NELT), JA(NELT), ISYM, JOB, ITOL
C REAL A(NELT), X(N), B(N), DINV(N)
C
C CALL SSDSCL( N, NELT, IA, JA, A, ISYM, X, B, DINV, JOB, ITOL )
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 upper
C or lower triangle of the matrix is stored.
C X :INOUT Real X(N).
C Initial guess that will be later used in the iterative
C solution.
C of the scaled system.
C B :INOUT Real B(N).
C Right hand side vector.
C DINV :INOUT Real DINV(N).
C Upon return this array holds 1./DIAG(A).
C This is an input if JOB = 0.
C JOB :IN Integer.
C Flag indicating whether to scale or not.
C JOB non-zero means do scaling.
C JOB = 0 means do unscaling.
C ITOL :IN Integer.
C Flag indicating what type of error estimation to do in the
C iterative method. When ITOL = 11 the exact solution from
C common block SSLBLK will be used. When the system is scaled
C then the true solution must also be scaled. If ITOL is not
C 11 then this vector is not referenced.
C
C *Common Blocks:
C SOLN :INOUT Real SOLN(N). COMMON BLOCK /SSLBLK/
C The true solution, SOLN, is scaled (or unscaled) if ITOL is
C set to 11, see above.
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 With the SLAP format all 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
C *Cautions:
C This routine assumes that the diagonal of A is all non-zero
C and that the operation DINV = 1.0/DIAG(A) will not under-
C flow or overflow. This is done so that the loop vectorizes.
C Matrices with zero or near zero or very large entries will
C have numerical difficulties and must be fixed before this
C routine is called.
C
C***SEE ALSO SSDCG
C***REFERENCES (NONE)
C***ROUTINES CALLED (NONE)
C***COMMON BLOCKS SSLBLK
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 910502 Added C***FIRST EXECUTABLE STATEMENT line. (FNF)
C 920407 COMMON BLOCK renamed SSLBLK. (WRB)
C 920511 Added complete declaration section. (WRB)
C 921113 Corrected C***CATEGORY line. (FNF)
C 930701 Updated CATEGORY section. (FNF, WRB)
C***END PROLOGUE SSDSCL
C .. Scalar Arguments ..
INTEGER ISYM, ITOL, JOB, N, NELT
C .. Array Arguments ..
REAL A(NELT), B(N), DINV(N), X(N)
INTEGER IA(NELT), JA(NELT)
C .. Arrays in Common ..
REAL SOLN(1)
C .. Local Scalars ..
REAL DI
INTEGER ICOL, J, JBGN, JEND
C .. Intrinsic Functions ..
INTRINSIC SQRT
C .. Common blocks ..
COMMON /SSLBLK/ SOLN
C***FIRST EXECUTABLE STATEMENT SSDSCL
C
C SCALING...
C
IF( JOB.NE.0 ) THEN
DO 10 ICOL = 1, N
DINV(ICOL) = 1.0E0/SQRT( A(JA(ICOL)) )
10 CONTINUE
ELSE
C
C UNSCALING...
C
DO 15 ICOL = 1, N
DINV(ICOL) = 1.0E0/DINV(ICOL)
15 CONTINUE
ENDIF
C
DO 30 ICOL = 1, N
JBGN = JA(ICOL)
JEND = JA(ICOL+1)-1
DI = DINV(ICOL)
DO 20 J = JBGN, JEND
A(J) = DINV(IA(J))*A(J)*DI
20 CONTINUE
30 CONTINUE
C
DO 40 ICOL = 1, N
B(ICOL) = B(ICOL)*DINV(ICOL)
X(ICOL) = X(ICOL)/DINV(ICOL)
40 CONTINUE
C
C Check to see if we need to scale the "true solution" as well.
C
IF( ITOL.EQ.11 ) THEN
DO 50 ICOL = 1, N
SOLN(ICOL) = SOLN(ICOL)/DINV(ICOL)
50 CONTINUE
ENDIF
C
RETURN
C------------- LAST LINE OF SSDSCL FOLLOWS ----------------------------
END