LFDQH

Computes the determinant of a complex Hermitian positive definite matrix given the RHR Cholesky factorization in band Hermitian storage mode.

Required Arguments

FACT — Complex NCODA + 1 by N array containing the RHR factorization of the Hermitian positive definite band matrix A. (Input)
FACT is obtained as output from routine LFCQH/DLFCQH or LFTQH/DLFTQH.

NCODA — Number of upper or lower codiagonals of A. (Input)

DET1 — Scalar containing the mantissa of the determinant. (Output)
The value DET1 is normalized so that 1.0 ≤ ∣DET1∣ < 10.0 or DET1 = 0.0.

DET2 — Scalar containing the exponent of the determinant. (Output)
The determinant is returned in the form det (A) = DET1 * 10DET2.

Optional Arguments

N — Number of equations. (Input)
Default: N = size (FACT,2).

LDFACT — Leading dimension of FACT exactly as specified in the dimension statement of the calling program. (Input)
Default: LDFACT = size (FACT,1).

FORTRAN 90 Interface

Generic: CALL LFDQH (FACT, NCODA, DET1, DET2 [, …])

Specific: The specific interface names are S_LFDQH and D_LFDQH.

FORTRAN 77 Interface

Single: CALL LFDQH (N, FACT, LDFACT, NCODA, DET1, DET2)

Double: The double precision name is DLFDQH.

Description

Routine LFDQH computes the determinant of a complex Hermitian positive definite band coefficient matrix. To compute the determinant, the coefficient matrix must first undergo an RH R factorization. This may be done by calling either LFCQH or LFTQH. The formula det A = det RH det R = (det R)2 is used to compute the determinant. Since the determinant of a triangular matrix is the product of the diagonal elements,

LFDQH is based on the LINPACK routine CPBDI; see Dongarra et al. (1979).

Example

The determinant is computed for a 5 × 5 complex Hermitian positive definite band matrix with one codiagonal.

USE LFDQH_INT

USE LFTQH_INT

USE UMACH_INT

! Declare variables

INTEGER LDA, LDFACT, N, NCODA, NOUT

PARAMETER (LDA=2, N=5, LDFACT=2, NCODA=1)

REAL DET1, DET2

COMPLEX A(LDA,N), FACT(LDFACT,N)

! Set values for A in band Hermitian form

! A = ( 0.0+0.0i -1.0+1.0i 1.0+2.0i 0.0+4.0i 1.0+1.0i )

! ( 2.0+0.0i 4.0+0.0i 10.0+0.0i 6.0+0.0i 9.0+0.0i )

DATA A/(0.0,0.0), (2.0,0.0), (-1.0,1.0), (4.0, 0.0), (1.0,2.0),&

(10.0,0.0), (0.0,4.0), (6.0,0.0), (1.0,1.0), (9.0,0.0)/

! Factor the matrix

CALL LFTQH (A, NCODA, FACT)

! Compute the determinant

CALL LFDQH (FACT, NCODA, DET1, DET2)

! Print results

CALL UMACH (2, NOUT)

WRITE (NOUT,99999) DET1, DET2

99999 FORMAT (’ The determinant of A is ’,F6.3,’ * 10**’,F2.0)

END

Output

The determinant of A is 1.736 * 10**3.