LSACB
Solves a complex system of linear equations in band storage mode with iterative refinement.
Required Arguments
A — Complex NLCA + NUCA + 1 by N array containing the N by N banded coefficient matrix in band storage mode. (Input)
NLCA — Number of lower codiagonals of A. (Input)
NUCA — Number of upper codiagonals of A. (Input)
B — Complex vector of length N containing the right-hand side of the linear system. (Input)
X — Complex vector of length N containing the solution to the linear system. (Output)
Optional Arguments
N — Number of equations. (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).
IPATH — Path indicator. (Input)
IPATH = 1 means the system AX = B is solved.
IPATH = 2 means the system AHX = B is solved.
Default: IPATH = 1.
FORTRAN 90 Interface
Generic: CALL LSACB (A, NLCA, NUCA, B, X [, …])
Specific: The specific interface names are S_LSACB and D_LSACB.
FORTRAN 77 Interface
Single: CALL LSACB (N, A, LDA, NLCA, NUCA, B, IPATH, X)
Double: The double precision name is DLSACB.
Description
Routine
LSACB solves a system of linear algebraic equations having a complex banded coefficient matrix. It first uses the routine
LFCCB to compute an
LU factorization of the coefficient matrix and to estimate the condition number of the matrix. The solution of the linear system is then found using the iterative refinement routine
LFICB.
LSACB fails if U, the upper triangular part of the factorization, has a zero diagonal element or if the iterative refinement algorithm fails to converge. These errors occur only if A is singular or very close to a singular matrix.
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. LSACB solves the problem that is represented in the computer; however, this problem may differ from the problem whose solution is desired.
Comments
1. Workspace may be explicitly provided, if desired, by use of L2ACB/DL2ACB. The reference is:
CALL L2ACB (N, A, LDA, NLCA, NUCA, B, IPATH, X, FACT, IPVT, WK)
The additional arguments are as follows:
FACT — Complex work vector of length (2 * NLCA + NUCA + 1) * N containing the LU factorization of A on output.
IPVT — Integer work vector of length N containing the pivoting information for the LU factorization of A on output.
WK — Complex work vector of length N.
2. Informational errors
Type | Code | Description |
|---|
3 | 3 | The input matrix is too ill-conditioned. The solution might not be accurate. |
4 | 2 | The input matrix is singular. |
3.
Integer Options with
Chapter 11 Options Manager
16 This option uses four values to solve memory bank conflict (access inefficiency) problems. In routine L2ACB the leading dimension of FACT is increased by IVAL(3) when N is a multiple of IVAL(4). The values IVAL(3) and IVAL(4) are temporarily replaced by IVAL(1) and IVAL(2), respectively, in LSACB. Additional memory allocation for FACT and option value restoration are done automatically in LSACB. Users directly calling L2ACB can allocate additional space for FACT and set IVAL(3) and IVAL(4) so that memory bank conflicts no longer cause inefficiencies. There is no requirement that users change existing applications that use LSACB or L2ACB. Default values for the option are IVAL(*) = 1,16,0,1.
17 This option has two values that determine if the L1 condition number is to be computed. Routine LSACB temporarily replaces IVAL(2) by IVAL(1). The routine L2CCB computes the condition number if IVAL(2) = 2. Otherwise L2CCB skips this computation. LSACB restores the option. Default values for the option are IVAL(*) = 1,2.
Example
A system of four linear equations is solved. The coefficient matrix has complex banded form with one upper and one lower codiagonal. The right-hand-side vector b has four elements.
USE LSACB_INT
USE WRCRN_INT
! Declare variables
INTEGER LDA, N, NLCA, NUCA
PARAMETER (LDA=3, N=4, NLCA=1, NUCA=1)
COMPLEX A(LDA,N), B(N), X(N)
!
! Set values for A in band form, and B
!
! 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 )
!
! B = ( -10.0-5.0i 9.5+5.5i 12.0-12.0i 0.0+8.0i )
!
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)/
DATA B/(-10.0,-5.0), (9.5,5.5), (12.0,-12.0), (0.0,8.0)/
! Solve A*X = B
CALL LSACB (A, NLCA, NUCA, B, X)
! Print results
CALL WRCRN (’X’, X, 1, N, 1)
!
END
Output
X
1 2 3 4
( 3.000, 0.000) (-1.000, 1.000) ( 3.000, 0.000) (-1.000, 1.000)
