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 ≤ ǀDET1ǀ < 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 * 10^DET2.

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.

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