LSASF
Solves a real symmetric system of linear equations with iterative refinement.
Required Arguments
A — N by N matrix containing the coefficient matrix of the symmetric linear system. (Input)
Only the upper triangle of A is referenced.
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).
LDA — Leading dimension of A exactly as specified in the dimension statement of the calling program. (Input)
Default: LDA = size (A,1).
FORTRAN 90 Interface
Generic: CALL LSASF (A, B, X [, …])
Specific: The specific interface names are S_LSASF and D_LSASF.
FORTRAN 77 Interface
Single: CALL LSASF (N, A, LDA, B, X)
Double: The double precision name is DLSASF.
Description
Routine LSASF solves systems of linear algebraic equations having a real symmetric indefinite coefficient matrix. It first uses the routine LFCSF to compute a U DUT factorization of the coefficient matrix and to estimate the condition number of the matrix. D is a block diagonal matrix with blocks of order 1 or 2, and U is a matrix composed of the product of a permutation matrix and a unit upper triangular matrix. The solution of the linear system is then found using the iterative refinement routine LFISF.
LSASF fails if a block in D is singular 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. LSASF 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 L2ASF/DL2ASF. The reference is
CALL L2ASF (N, A, LDA, B, X, FACT, IPVT, WK)
The additional arguments are as follows:
FACT — N × N work array containing information about the U DUT factorization of A on output. If A is not needed, A and FACT can share the same storage location.
IPVT — Integer work vector of length N containing the pivoting information for the 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 L2ASF 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 LSASF. Additional memory allocation for FACT and option value restoration are done automatically in LSASF. Users directly calling L2ASF 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 LSASF or L2ASF. 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 LSASF temporarily replaces IVAL(2) by IVAL(1). The routine L2CSF computes the condition number if IVAL(2) = 2. Otherwise L2CSF skips this computation. LSASF restores the option. Default values for the option are IVAL(*) = 1, 2.
Example
A system of three linear equations is solved. The coefficient matrix has real symmetric form and the right-hand-side vector b has three elements.
USE LSASF_INT
USE WRRRN_INT
! Declare variables
PARAMETER (LDA=3, N=3)
REAL A(LDA,LDA), B(N), X(N)
!
! Set values for A and B
!
! A = ( 1.0 -2.0 1.0)
! ( -2.0 3.0 -2.0)
! ( 1.0 -2.0 3.0)
!
! B = ( 4.1 -4.7 6.5)
!
DATA A/1.0, -2.0, 1.0, -2.0, 3.0, -2.0, 1.0, -2.0, 3.0/
DATA B/4.1, -4.7, 6.5/
!
CALL LSASF (A, B, X)
! Print results
CALL WRRRN (’X’, X, 1, N, 1)
END
X
1 2 3
-4.100 -3.500 1.200
