Determinant, Inertia, and Inverse of a UDU-Factored Hermitian Matrix in Packed Storage

Calling Sequence

CALL ZHPDI	(ZA, N, IPIVOT, DDET, INERT, ZWORK, JOB)
CALL CHPDI	(CA, N, IPIVOT, SDET, INERT, CWORK, JOB)
void zhpdi	(doublecomplex *za, int n, int *ipivot, double *ddet, int *inert, int job)
void chpdi	(complex *ca, int n, int *ipivot, float *sdet, int *inert, int job)

Arguments

xA	On entry, the UDU factorization of the matrix, as computed by xHPCO or xHPFA. On exit, if the c digit of JOB ¼ 0, then A contains the upper triangle of the inverse of the original matrix A; otherwise unchanged.
N	Order of the original matrix A. N 0.
IPIVOT	Pivot vector as computed by xHPCO or xHPFA.
xDET	On exit, if the b digit of JOB 0, then DET contains the determinant of the matrix A. The determinant is stored as b × 10expon 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.
INERT	On 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.
xWORK	Scratch array with a dimension of N.
JOB	Integer 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: 1.000
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)