LFDCB

Computes the determinant of a complex matrix given the LU factorization of the matrix in band storage mode.

Required Arguments

FACT — Complex (2 * NLCA + NUCA + 1) by N array containing the LU factorization of the matrix A as output from routine LFTCB/DLFTCB or LFCCB/DLFCCB. (Input)

NLCA — Number of lower codiagonals in matrix A. (Input)

NUCA — Number of upper codiagonals in matrix A. (Input)

IPVT — Vector of length N containing the pivoting information for the LU factorization as output from routine LFTCB/DLFTCB or LFCCB/DLFCCB. (Input)

DET1 — Complex 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 — Order of the matrix. (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 LFDCB (FACT, NLCA, NUCA, IPVT, DET1, DET2 [, …])

Specific: The specific interface names are S_LFDCB and D_LFDCB.

FORTRAN 77 Interface

Single: CALL LFDCB (N, FACT, LDFACT, NLCA, NUCA, IPVT, DET1, DET2)

Double: The double precision name is DLFDCB.

Description

Routine LFDCB computes the determinant of a complex banded coefficient matrix. To compute the determinant, the coefficient matrix must first undergo an LU factorization. This may be done by calling either LFCCB or LFTCB. The formula det A = det L det U is used to compute the determinant. Since the determinant of a triangular matrix is the product of the diagonal elements,

 

(The matrix U is stored in the upper NUCA + NLCA + 1 rows of FACT as a banded matrix.) Since L is the product of triangular matrices with unit diagonals and of permutation matrices, det L = (-1)k, where k is the number of pivoting interchanges.

LFDCB is based on the LINPACK routine CGBDI; see Dongarra et al. (1979).

Example

The determinant is computed for a complex banded 4 × 4 matrix with one upper and one lower codiagonal.

 

USE LFDCB_INT

USE LFTCB_INT

USE UMACH_INT

! Declare variables

INTEGER LDA, LDFACT, N, NLCA, NUCA, NOUT

PARAMETER (LDA=3, LDFACT=4, N=4, NLCA=1, NUCA=1)

INTEGER IPVT(N)

REAL DET2

COMPLEX A(LDA,N), DET1, FACT(LDFACT,N)

!

! Set values for A in band form

!

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

! ( -2.0-3.0i -0.5+3.0i 3.0-3.0i 1.0-1.0i )

! ( 6.0+1.0i 1.0+1.0i 0.0+2.0i 0.0+0.0i )

!

DATA A/(0.0,0.0), (-2.0,-3.0), (6.0,1.0), (4.0,0.0), (-0.5,3.0),&

(1.0,1.0), (-2.0,2.0), (3.0,-3.0), (0.0,2.0), (-4.0,-1.0),&

(1.0,-1.0), (0.0,0.0)/

!

CALL LFTCB (A, NLCA, NUCA, FACT, IPVT)

! Compute the determinant

CALL LFDCB (FACT, NLCA, NUCA, IPVT, DET1, DET2)

! Print the results

CALL UMACH (2, NOUT)

WRITE (NOUT,99999) DET1, DET2

!

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

F2.0)

END

Output

 

The determinant of A is ( 2.500,-1.500) * 10**1.