zhico(l) — SunSoft Performance Library

NAME

zhico - compute the UDU factorization and condition number of a Hermitian matrix A. If the condition number is not needed then xHIFA is slightly faster. It is typical to follow a call to xHICO with a call to xHISL to solve Ax = b or to xHIDI to compute the determinant, inverse, and inertia of A.

SYNOPSIS

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

CALL CHICO (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 = 3)
      PARAMETER (LDA = 3)
C
      REAL       RCOND
      COMPLEX    A(LDA,N), B(N), WORK(N)
      INTEGER    ICOL, IPIVOT(N), IROW
C
      EXTERNAL   CHICO, CHISL
      INTRINSIC CONJG
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          1    1+2i 1+2i         95-180i
C     A = 1+2i   6   -2+6i    b = 545-118i
C         1+2i -2+6i   11         865+ 62i
C
      DATA A / (1.0,0.0), (8E8,8E8), (8E8,8E8),
     $        (1.0,-2.0), (6.0,0.0), (8E8,8E8),
     $        (1.0,-2.0), (6.0,-2.0), (11.0,0.0) /
      DATA B / (95.0,-180.0), (545.0,-118.0), (865.0,62.0) /
C
      PRINT 1000
      DO 100, IROW = 1, N
        PRINT 1010, (CONJG(A(ICOL,IROW)), ICOL = 1, IROW),
     $              (A(IROW,ICOL), ICOL = IROW + 1, N)
100 CONTINUE
      PRINT 1020
      DO 110, IROW = 1, N
        PRINT 1010, (A(IROW,ICOL), ICOL = 1, N)
110 CONTINUE
      PRINT 1030
      PRINT 1040, B
      CALL CHICO (A, LDA, N, IPIVOT, RCOND, WORK)
      PRINT 1050, RCOND
      IF ((RCOND + 1.0) .EQ. 1.0) THEN
        PRINT 1060
      END IF
      CALL CHISL (A, LDA, N, IPIVOT, B)
      PRINT 1070
      PRINT 1040, B
C
1000 FORMAT (1X, ’A in full form:’)
1010 FORMAT (4(: 3X, ’(’, F4.1, ’,’, F4.1, ’)’))
1020 FORMAT (/1X, ’A in Hermitian form: (∗ in unused elements)’)
1030 FORMAT (/1X, ’b:’)
1040 FORMAT (3X, ’(’, F6.1, ’,’, F6.1, ’)’)
1050 FORMAT (/1X, ’Reciprocal condition number of A:’, F6.3)
1060 FORMAT (1X, ’A may be singular to working precision.’)
1070 FORMAT (/1X, ’A∗∗(-1) ∗ b:’)
C
      END

SAMPLE OUTPUT

A in full form:
   ( 1.0, 0.0)   ( 1.0,-2.0)   ( 1.0,-2.0)
   ( 1.0, 2.0)   ( 6.0, 0.0)   ( 6.0,-2.0)
   ( 1.0, 2.0)   ( 6.0, 2.0)   (11.0, 0.0)

A in Hermitian form: (∗ in unused elements)
   ( 1.0, 0.0)   ( 1.0,-2.0)   ( 1.0,-2.0)
   (∗∗∗∗,∗∗∗∗)   ( 6.0, 0.0)   ( 6.0,-2.0)
   (∗∗∗∗,∗∗∗∗)   (∗∗∗∗,∗∗∗∗)   (11.0, 0.0)

b:
   ( 95.0,-180.0)
   ( 545.0,-118.0)
   ( 865.0, 62.0)

Reciprocal condition number of A: 0.001

A∗∗(-1) ∗ b:
   (   5.0,   0.0)
   ( 26.0,   0.0)
   ( 64.0,   0.0)