LFCCB
Computes the LU factorization of a complex matrix in band storage mode and estimate its L1 condition number.
Required Arguments
A — Complex NLCA + NUCA + 1 by N array containing the N by N matrix in band storage mode to be factored. (Input)
NLCA — Number of lower codiagonals of A. (Input)
NUCA — Number of upper codiagonals of A. (Input)
FACT — Complex 2 * NLCA + NUCA + 1 by N array containing the LU factorization of the matrix A. (Output)
If A is not needed, A can share the first (NLCA + NUCA + 1) * N locations with FACT .
IPVT — Vector of length N containing the pivoting information for the LU factorization. (Output)
RCOND — Scalar containing an estimate of the reciprocal of the L1 condition number of A. (Output)
Optional Arguments
N — Order of the matrix. (Input)
Default: N = size (A,2).
LDA — Leading dimension of A exactly as specified in the dimension statement of the calling program. (Input)
Default: LDA = size (A,1).
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 LFCCB (A, NLCA, NUCA, FACT, IPVT, RCOND [, …])
Specific: The specific interface names are S_LFCCB and D_LFCCB.
FORTRAN 77 Interface
Single: CALL LFCCB (N, A, LDA, NLCA, NUCA, FACT, LDFACT, IPVT, RCOND)
Double: The double precision name is DLFCCB.
Description
Routine LFCCB performs an LU factorization of a complex banded coefficient matrix. It also estimates the condition number of the matrix. The LU factorization is done using scaled partial pivoting. Scaled partial pivoting differs from partial pivoting in that the pivoting strategy is the same as if each row were scaled to have the same ∞-norm.
The L1 condition number of the matrix A is defined to be κ(A) = ∥A∥1∥A-1∥1 Since it is expensive to compute ∥A-1∥1, the condition number is only estimated. The estimation algorithm is the same as used by LINPACK and is described by Cline et al. (1979).
If the estimated condition number is greater than 1/ɛ (where ɛ is machine precision), a warning error is issued. This indicates that very small changes in A can cause very large changes in the solution x. Iterative refinement can sometimes find the solution to such a system.
LFCCB fails if U, the upper triangular part of the factorization, has a zero diagonal element. This can occur only if A is singular or very close to a singular matrix.
The
LU factors are returned in a form that is compatible with IMSL routines
LFICB,
LFSCB and
LFDCB. To solve systems of equations with multiple right-hand-side vectors, use
LFCCB followed by either
LFICB or
LFSCB called once for each right-hand side. The routine
LFDCB can be called to compute the determinant of the coefficient matrix after
LFCCB has performed the factorization.
Let F be the matrix FACT, let ml = NLCA and let mu = NUCA. The first ml + mu + 1 rows of F contain the triangular matrix U in band storage form. The lower ml rows of F contain the multipliers needed to reconstruct L.
LFCCB is based on the LINPACK routine CGBCO; see Dongarra et al. (1979). CGBCO uses unscaled partial pivoting.
Comments
1. Workspace may be explicitly provided, if desired, by use of L2CCB/DL2CCB. The reference is:
CALL L2CCB (N, A, LDA, NLCA, NUCA, FACT, LDFACT, IPVT, RCOND, WK)
The additional argument is
WK — Complex work vector of length N.
2. Informational errors
Type | Code | Description |
|---|
3 | 1 | The input matrix is algorithmically singular. |
4 | 2 | The input matrix is singular. |
Example
The inverse of a 4 × 4 band matrix with one upper and one lower codiagonal is computed. LFCCB is called to factor the matrix and to check for singularity or ill-conditioning. LFICB is called to determine the columns of the inverse.
USE LFCCB_INT
USE UMACH_INT
USE LFICB_INT
USE WRCRN_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 RCOND
COMPLEX A(LDA,N), AINV(N,N), FACT(LDFACT,N), RJ(N), RES(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 )
! ( 0.0-3.0i -0.5+3.0i 3.0-3.0i 1.0-1.0i )
! ( 6.0+1.0i 4.0+1.0i 0.0+2.0i 0.0+0.0i )
!
DATA A/(0.0,0.0), (0.0,-3.0), (6.0,1.0), (4.0,0.0), (-0.5,3.0),&
(4.0,1.0), (-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 LFCCB (A, NLCA, NUCA, FACT, IPVT, RCOND)
! Print the reciprocal condition number
! and the L1 condition number
CALL UMACH (2, NOUT)
WRITE (NOUT,99999) RCOND, 1.0E0/RCOND
! Set up the columns of the identity
! matrix one at a time in RJ
RJ = (0.0E0,0.0E0)
DO 10 J=1, N
RJ(J) = (1.0E0,0.0E0)
! RJ is the J-th column of the identity
! matrix so the following LFICB
! reference places the J-th column of
! the inverse of A in the J-th column
! of AINV
CALL LFICB (A, NLCA, NUCA, FACT, IPVT, RJ, AINV(:,J), RES)
RJ(J) = (0.0E0,0.0E0)
10 CONTINUE
! Print results
CALL WRCRN (’AINV’, AINV)
!
99999 FORMAT (’ RCOND = ’,F5.3,/,’ L1 condition number = ’,F6.3)
END
Output
RCOND = 0.022
L1 condition number = 45.933
AINV
1 2 3 4
1 ( 0.562, 0.170) ( 0.125, 0.260) (-0.385,-0.135) (-0.239,-1.165)
2 ( 0.122, 0.421) (-0.195, 0.094) ( 0.101,-0.289) ( 0.874,-0.179)
3 ( 0.034, 0.904) (-0.437, 0.090) (-0.153,-0.527) ( 1.087,-1.172)
4 ( 0.938, 0.870) (-0.347, 0.527) (-0.679,-0.374) ( 0.415,-1.759)
