zhidi(l) — SunSoft Performance Library

NAME

zhidi - compute the determinant, inertia, and inverse of a Hermitian matrix A, which has been UDU-factored by xHICO or xHIFA.

SYNOPSIS

CALL ZHIDI (ZA, LDA, N, IPIVOT, DDET, INERT, ZWORK, JOB)

CALL CHIDI (CA, LDA, N, IPIVOT, SDET, INERT, CWORK, JOB)

ARGUMENTS

xAOn entry, the UDU factorization of the matrix A, as computed by
xHICO or xHIFA. On exit, if the c digit of JOB <> 0, then the upper triangle of A contains the upper triangle of the inverse of the original matrix A if the inverse was requested, otherwise unchanged. 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 original matrix A. N >= 0.

IPIVOTPivot vector as computed by xHICO or xHIFA.

xDETOn exit, if the b digit of JOB >= 0, then DET contains the
determinant of the matrix A. The determinant is stored as b ∗ (10 ∗∗ expon) where b is stored in DET(1) and expon is stored in DET(2). 1.0 <= |DET(1)| <= 10.0 or DET(1) = 0.0. If the b digit of JOB <> 0, DET is not referenced.

INERTOn exit, if the a digit of JOB <> 0, then INERT contains an integer triplet where:
INERT(1) = number of positive eigenvalues
INERT(2) = number of negative eigenvalues
INERT(3) = number of zero eigenvalues
If the a digit of JOB = 0 then INERT is not referenced.

xWORKScratch array with a dimension of N.

JOBInteger in the form abc; determines operation the subroutine will perform:
a <> 0 Compute the inertia.
b <> 0 Compute the determinant.
c <> 0 Compute the inverse.

SAMPLE PROGRAM

      PROGRAM TEST
      IMPLICIT NONE
C
      INTEGER    IDODET, IDOINR, IDOINV, LDA, N
      PARAMETER (IDODET = 10)
      PARAMETER (IDOINR = 100)
      PARAMETER (IDOINV = 1)
      PARAMETER (N = 3)
      PARAMETER (LDA = 3)
C
      REAL       DET(2), RCOND
      COMPLEX    A(LDA,N), WORK(N)
      INTEGER    ICOL, INERT(3), IPIVOT(N), IROW, JOB
C
      EXTERNAL   CHICO, CHIDI
C
C     Initialize the array A to store the matrix A shown below.
C
C          1    1+2i 1+2i
C     A = 1+2i   6   -2+6i
C         1+2i -2+6i   11
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) /
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
      CALL CHICO (A, LDA, N, IPIVOT, RCOND, WORK)
      PRINT 1030, RCOND
      IF ((RCOND + 1.0) .EQ. 1.0) THEN
        PRINT 1040
      END IF
      JOB = IDOINR + IDODET + IDOINV
      CALL CHIDI (A, LDA, N, IPIVOT, DET, INERT, WORK, JOB)
      PRINT 1050, DET(1) ∗ (10.0D0 ∗∗ DET(2))
      PRINT 1060, INERT
      PRINT 1070
      DO 120, IROW = 1, N
        PRINT 1010, (CONJG(A(ICOL,IROW)), ICOL = 1, IROW),
     $              (A(IROW,ICOL), ICOL = IROW + 1, N)
120 CONTINUE
C
1000 FORMAT (1X, ’A in full form:’)
1010 FORMAT (4(: 3X, ’(’, F5.1, ’,’, F5.1, ’)’))
1020 FORMAT (/1X, ’A in Hermitian form: (∗ in unused elements)’)
1030 FORMAT (/1X, ’Reciprocal condition number of A:’, F6.3)
1040 FORMAT (1X, ’A may be singular to working precision.’)
1050 FORMAT (/1X, ’Determinant of A: ’, F6.3)
1060 FORMAT (1X, ’Inertia of A: <’, I1, ’,’, I1, ’,’, I1, ’>’)
1070 FORMAT (/1X, ’A∗∗(-1):’)
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)

Reciprocal condition number of A: 0.001

Determinant of A: 0.008
Inertia of A: <3,0,0>

A∗∗(-1):
   ( 26.0, 0.0)   ( -1.0, 12.0)   ( -4.0, -2.0)
   ( -1.0,-12.0)   ( 6.0, 0.0)   ( -1.0, 2.0)
   ( -4.0, 2.0)   ( -1.0, -2.0)   ( 1.0, 0.0)