LFDRG

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

Required Arguments

FACT — N by N matrix containing the LU factorization of the matrix A as output from routine LFTRG/DLFTRG or LFCRG/DLFCRG. (Input)

IPVT — Vector of length N containing the pivoting information for the LU factorization as output from routine LFTRG/DLFTRG or LFCRG/DLFCRG. (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.

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.

The determinant is returned in the form det(A) = DET1 * 10DET2.

Optional Arguments

N — Order of the matrix. (Input)

Default: N = size (FACT,2).

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).

Default: LDFACT = size (FACT,1).

FORTRAN 90 Interface

Generic: CALL LFDRG (FACT, IPVT, DET1, DET2 [, …])

Specific: The specific interface names are S_LFDRG and D_LFDRG.

FORTRAN 77 Interface

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

Double: The double precision name is DLFDRG.

Description

Routine LFDRG computes the determinant of a real general coefficient matrix. To compute the determinant, the coefficient matrix must first undergo an LU factorization. This may be done by calling either LFCRG or LFTRG. 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.

Routine LFDRG is based on the LINPACK routine SGEDI; see Dongarra et al. (1979)

Example

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

USE LFDRG_INT

USE LFTRG_INT

USE UMACH_INT

! Declare variables

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

INTEGER IPVT(N), NOUT

REAL A(LDA,LDA), DET1, DET2, FACT(LDFACT,LDFACT)

!

! Set values for A

! A = ( 33.0 16.0 72.0)

! (-24.0 -10.0 -57.0)

! ( 18.0 -11.0 7.0)

!

DATA A/33.0, -24.0, 18.0, 16.0, -10.0, -11.0, 72.0, -57.0, 7.0/

!

CALL LFTRG (A, FACT, IPVT)

! Compute the determinant

CALL LFDRG (FACT, 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 -4.761 * 10**3.