LFDRB

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

Required Arguments

FACT — (2 * NLCA + NUCA + 1) by N array containing the LU factorization of the matrix A as output from routine LFTRB/DLFTRB or LFCRB/DLFCRB. (Input)

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

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

IPVT — Vector of length N containing the pivoting information for the LU factorization as output from routine LFTRB/DLFTRB or LFCRB/DLFCRB. (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 — 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 LFDRB (FACT, NLCA, NUCA, IPVT, DET1, DET2 [, …])

Specific: The specific interface names are S_LFDRB and D_LFDRB.

FORTRAN 77 Interface

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

Double: The double precision name is DLFDRB.

Description

Routine LFDRB computes the determinant of a real banded coefficient matrix. To compute the determinant, the coefficient matrix must first undergo an LU factorization. This may be done by calling either LFCRB or LFTRB. 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.

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

Example

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

 

USE LFDRB_INT

USE LFTRB_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 A(LDA,N), DET1, DET2, FACT(LDFACT,N)

! Set values for A in band form

! A = ( 0.0 -1.0 -2.0 2.0)

! ( 2.0 1.0 -1.0 1.0)

! ( -3.0 0.0 2.0 0.0)

!

DATA A/0.0, 2.0, -3.0, -1.0, 1.0, 0.0, -2.0, -1.0, 2.0,&

2.0, 1.0, 0.0/

!

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

! Compute the determinant

CALL LFDRB (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, ’ * 10**’, F2.0)

END

Output

 

The determinant of A is 5.000 * 10**0.