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 ≤ ǀDET1 ǀ < 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 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.

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