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

Calling Sequence

CALL ZHIDI	(ZA, LDA, N, IPIVOT, DDET, INERT, ZWORK, JOB)
CALL CHIDI	(CA, LDA, N, IPIVOT, SDET, INERT, CWORK, JOB)
void zhidi	(doublecomplex *za, int lda, int n, int *ipivot, double *ddet, int *inert, int job)
void chidi	(complex *ca, int lda, int n, int *ipivot, float *sdet, int *inert, int job)

Arguments

xA	On 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.
LDA	Leading dimension of the array A as specified in a dimension or type statement. LDA max(1,N).
N	Order of the original matrix A. N 0.
IPIVOT	Pivot vector as computed by xHICO or xHIFA.
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 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: 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)