zhpdi(3P)

NAME

zhpdi - compute the determinant, inertia, and inverse of a Hermitian matrix A in packed storage, which has been UDU-factored by xHPCO or xHPFA.

SYNOPSIS

CALL ZHPDI (ZA, N, IPIVOT, DDET, INERT, ZWORK, JOB)

CALL CHPDI (CA, N, IPIVOT, SDET, INERT, CWORK, JOB)

void zhpdi(doublecomplex ∗za, long int n, long int ∗ipivot,
double ∗det, long int ∗inert, long int job)

void chpdi(complex ∗ca, long int n, long int ∗ipivot,
float ∗det, long int ∗inert, long int job)

ARGUMENTS

xAOn entry, the UDU factorization of the matrix, as computed by
xHPCO or xHPFA. On exit, if the c digit of JOB <> 0, then A contains the upper triangle of the inverse of the original matrix A; otherwise unchanged.

NOrder of the original matrix A. N >= 0.

IPIVOTPivot vector as computed by xHPCO or xHPFA.

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 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, LENGTA, N
      PARAMETER (IDODET = 10)
      PARAMETER (IDOINR = 100)
      PARAMETER (IDOINV = 1)
      PARAMETER (N = 3)
      PARAMETER (LENGTA = (N ∗ N + N) / 2)
C
      REAL       DET(2), RCOND
      COMPLEX    A(LENGTA), WORK(N)
      INTEGER    INERT(3), IPIVOT(N), JOB
C
      EXTERNAL   CHPCO, CHPDI
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), (1.0,-2.0), (6.0,0.0),
     $         (1.0,-2.0), (6.0,-2.0), (11.0,0.0) /
C
      PRINT 1000
      PRINT 1010, A(1), A(2), A(4)
      PRINT 1010, CONJG(A(2)), A(3), A(5)
      PRINT 1010, CONJG(A(4)), CONJG(A(5)), A(6)
      CALL CHPCO (A, N, IPIVOT, RCOND, WORK)
      PRINT 1020, RCOND
      IF ((RCOND + 1.0) .EQ. 1.0) THEN
        PRINT 1030
      END IF
      JOB = IDOINR + IDODET + IDOINV
      CALL CHPDI (A, N, IPIVOT, DET, INERT, WORK, JOB)
      PRINT 1040, DET(1) ∗ (10.0D0 ∗∗ DET(2))
      PRINT 1050, INERT
      PRINT 1060
      PRINT 1010, A(1), A(2), A(4)
      PRINT 1010, CONJG(A(2)), A(3), A(5)
      PRINT 1010, CONJG(A(4)), CONJG(A(5)), A(6)
C
1000 FORMAT (1X, ’A in full form:’)
1010 FORMAT (4(: 3X, ’(’, F5.1, ’,’, F5.1, ’)’))
1020 FORMAT (/1X, ’Reciprocal condition number of A:’, F6.3)
1030 FORMAT (1X, ’A may be singular to working precision.’)
1040 FORMAT (/1X, ’Determinant of A: ’, F6.3)
1050 FORMAT (1X, ’Inertia of A: <’, I1, ’,’, I1, ’,’, I1, ’>’)
1060 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)

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)