LSARB

Solves a real system of linear equations in band storage mode with iterative refinement.

Required Arguments

A — (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 — Vector of length N containing the right-hand side of the linear system. (Input)

X — Vector of length N containing the solution to the linear system. (Output)

Optional Arguments

N — Number of equations. (Input)

Default: N = size (A,2).

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).

Default: LDA = size (A,1).

IPATH — Path indicator. (Input)

IPATH = 1 means the system AX = B is solved.

IPATH = 2 means the system ATX = B is solved.

Default: IPATH =1.

IPATH = 1 means the system AX = B is solved.

IPATH = 2 means the system ATX = B is solved.

Default: IPATH =1.

FORTRAN 90 Interface

Generic: CALL LSARB (A, NLCA, NUCA, B, X [, …])

Specific: The specific interface names are S_LSARB and D_LSARB.

FORTRAN 77 Interface

Single: CALL LSARB (N, A, LDA, NLCA, NUCA, B, IPATH, X)

Double: The double precision name is DLSARB.

Description

Routine LSARB solves a system of linear algebraic equations having a real banded coefficient matrix. It first uses the routine LFCRB 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 LFIRB.

LSARB 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. LSARB 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 L2ARB/DL2ARB. The reference is:

CALL L2ARB (N, A, LDA, NLCA, NUCA, B, IPATH, X, FACT, IPVT, WK)

The additional arguments are as follows:

FACT — Work vector of length (2 * NLCA + NUCA + 1) × N containing the LU factorization of A on output.

IPVT — Work vector of length N containing the pivoting information for the LU factorization of A on output.

WK — Work vector of length N.

2. Informational errors

Type | Code | Description |
---|---|---|

3 | 1 | 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 L2ARB 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 LSARB. Additional memory allocation for FACT and option value restoration are done automatically in LSARB. Users directly calling L2ARB 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 LSARB or L2ARB. Default values for the option are

IVAL(*) = 1, 16, 0, 1.

IVAL(*) = 1, 16, 0, 1.

17 This option has two values that determine if the L1 condition number is to be computed. Routine LSARB temporarily replaces IVAL(2) by IVAL(1). The routine L2CRB computes the condition number if IVAL(2) = 2. Otherwise L2CRB skips this computation. LSARB 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 real banded form with 1 upper and 1 lower codiagonal. The right-hand-side vector b has four elements.

USE LSARB_INT

USE WRRRN_INT

! Declare variables

INTEGER LDA, N, NLCA, NUCA

PARAMETER (LDA=3, N=4, NLCA=1, NUCA=1)

REAL A(LDA,N), B(N), X(N)

! Set values for A in band form, and B

!

! A = ( 0.0 -1.0 -2.0 2.0)

! ( 2.0 1.0 -1.0 1.0)

! ( -3.0 0.0 2.0 0.0)

!

! B = ( 3.0 1.0 11.0 -2.0)

!

DATA A/0.0, 2.0, -3.0, -1.0, 1.0, 0.0, -2.0, -1.0, 2.0,&

2.0, 1.0, 0.0/

DATA B/3.0, 1.0, 11.0, -2.0/

!

CALL LSARB (A, NLCA, NUCA, B, X)

! Print results

CALL WRRRN (’X’, X, 1, N, 1)

!

END

Output

X

1 2 3 4

2.000 1.000 -3.000 4.000