*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