Computes the determinant of a complex general matrix given the LU factorization of the matrix.

Required Arguments

FACT — Complex N by N matrix containing the LU factorization of the coefficient matrix A as output from routine LFCCG/DLFCCG or LFTCG/DLFTCG.   (Input)

IPVT — Vector of length N containing the pivoting information for the LU factorization of A as output from routine LFCCG/DLFCCG or LFTCG/DLFTCG.   (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 * 10^DET.

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 LFDCG (FACT, IPVT, DET1, DET2  [,…])

Specific:                             The specific interface names are S_LFDCG and D_LFDCG.

FORTRAN 77 Interface

Single:                                CALL LFDCG (N, FACT, LDFACT, IPVT, DET1, DET2)

Double:                              The double precision name is DLFDCG.

Description

Routine LFDCG computes the determinant of a complex general coefficient matrix. To compute the determinant the coefficient matrix must first undergo an LU factorization. This may be done by calling either LFCCG or LFTCG. 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 triangle of FACT.) 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.

LFDCG is based on the LINPACK routine CGEDI; see Dongarra et al. (1979).

Example

The determinant is computed for a complex general 3 × 3 matrix.

 

      USE LFDCG_INT
      USE LFTCG_INT
      USE UMACH_INT

!                                 Declare variables

      PARAMETER  (LDA=3, LDFACT=3, N=3)

      INTEGER    IPVT(N), NOUT

      REAL       DET2

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

!                                 Set values for  A

!

!                                 A = (  3.0-2.0i  2.0+4.0i  0.0-3.0i)

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

!                                     (  4.0+0.0i -5.0+1.0i  3.0-2.0i)

!

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

            (-5.0,1.0), (0.0,-3.0), (1.0,2.0), (3.0,-2.0)/

!

!                                 Factor A

      CALL LFTCG (A, FACT, IPVT)

!                                 Compute the determinant for the

!                                 factored matrix

      CALL LFDCG (FACT, IPVT, DET1, DET2)

!                                 Print results

      CALL UMACH (2, NOUT)

      WRITE (NOUT,99999) DET1, DET2

!

99999 FORMAT (' The determinant of A is',3X,'(',F6.3,',',F6.3,&

             ') * 10**',F2.0)

      END

Output

 

The determinant of A is ( 0.700, 1.100) * 10**1.

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