zsico(l) — SunSoft Performance Library

NAME

zsico - compute the UDU factorization and condition number of a symmetric matrix A. If the condition number is not needed then xSIFA is slightly faster. It is typical to follow a call to xSICO with a call to xSISL to solve Ax = b or to xSIDI to compute the determinant, inverse, and inertia of A.

SYNOPSIS

CALL DSICO (DA, LDA, N, IPIVOT, DRCOND, DWORK)

CALL SSICO (SA, LDA, N, IPIVOT, SRCOND, SWORK)

CALL ZSICO (ZA, LDA, N, IPIVOT, DRCOND, ZWORK)

CALL CSICO (CA, LDA, N, IPIVOT, SRCOND, CWORK)

ARGUMENTS

xAOn entry, the upper triangle of the matrix A.
On exit, a UDU factorization of the matrix A. The strict lower triangle of A is not referenced.

LDALeading dimension of the array A as specified in a dimension or type statement. LDA >= max(1,N).

NOrder of the matrix A. N >= 0.

IPIVOTOn exit, a vector of pivot indices.

xRCONDOn exit, an estimate of the reciprocal condition number of
A. 0.0 <= RCOND <= 1.0. As the value of RCOND gets smaller, operations with A such as solving Ax = b may become less stable. If RCOND satisfies RCOND + 1.0 = 1.0 then A may be singular to working precision.

xWORKScratch array with a dimension of N.

SAMPLE PROGRAM

      PROGRAM TEST
      IMPLICIT NONE
C
      INTEGER           LDA, N
      PARAMETER        (N = 4)
      PARAMETER        (LDA = 5)
C
      DOUBLE PRECISION A(LDA,N), B(N), RCOND, WORK(N)
      INTEGER           ICOL, IPIVOT(N), IROW
C
      EXTERNAL          DSICO, DSISL
C
C     Initialize the array A to store the matrix A shown below.
C     Initialize the array B to store the vector b shown below.
C
C         -.5   -.5   -.5   -.5        12
C     A = -.5 -1.5 -1.5 -1.5    b = 6
C         -.5 -1.5 -2.5 -2.5         6
C         -.5 -1.5 -2.5 -3.5        12
C
      DATA A / -5.0D-1, 4∗8D8, -5.0D-1, -1.5D0, 3∗8D8, -5.0D-1,
     $         -1.5D0, -2.5D0, 2∗8D8, -5.0D-1, -1.5D0, -2.5D0,
     $         -3.5D0, 8D8 /
      DATA B / 1.2D1, 6.0D0, 6.0D0, 1.2D1 /
C
      PRINT 1000
      DO 100, IROW = 1, N
        PRINT 1010, (A(ICOL,IROW), ICOL = 1, IROW),
     $              (A(IROW,ICOL), ICOL = IROW + 1, N)
100 CONTINUE
      PRINT 1020
      PRINT 1010, ((A(IROW,ICOL), ICOL = 1, N), IROW = 1, N)
      PRINT 1030
      PRINT 1040, B
      CALL DSICO (A, LDA, N, IPIVOT, RCOND, WORK)
      PRINT 1050, RCOND
      IF ((RCOND + 1.0D0) .EQ. RCOND) THEN
        PRINT 1060
      END IF
      PRINT 1070, 1.0D0 / RCOND
      CALL DSISL (A, LDA, N, IPIVOT, B)
      PRINT 1080
      PRINT 1040, B
C
1000 FORMAT (1X, ’A in full form:’)
1010 FORMAT (4(3X, F5.1))
1020 FORMAT (/1X, ’A in symmetric form: (∗ in unused elements)’)
1030 FORMAT (/1X, ’b:’)
1040 FORMAT (3X, F5.1)
1050 FORMAT (/1X, ’Reciprocal condition number of A:’, F6.3)
1060 FORMAT (1X, ’A may be singular to working precision.’)
1070 FORMAT (1X, ’Condition number of A: ’, F6.3)
1080 FORMAT (/1X, ’A∗∗(-1) ∗ b:’)
C
      END

SAMPLE OUTPUT

A in full form:
    -0.5    -0.5    -0.5    -0.5
    -0.5    -1.5    -1.5    -1.5
    -0.5    -1.5    -2.5    -2.5
    -0.5    -1.5    -2.5    -3.5

A in symmetric form: (∗ in unused elements)
    -0.5    -0.5    -0.5    -0.5
   ∗∗∗∗∗    -1.5    -1.5    -1.5
   ∗∗∗∗∗   ∗∗∗∗∗    -2.5    -2.5
   ∗∗∗∗∗   ∗∗∗∗∗   ∗∗∗∗∗    -3.5

b:
    12.0
     6.0
     6.0
    12.0

Reciprocal condition number of A: 0.031
Condition number of A: 32.000

A∗∗(-1) ∗ b:
   -30.0
     6.0
     6.0
    -6.0