LSARG

Routine LSARG solves a system of linear algebraic equations having a real general coefficient matrix. It first uses routine LFCRG 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 LFIRG. The underlying code is based on either LINPACK , LAPACK, or ScaLAPACK code depending upon which supporting libraries are used during linking. For a detailed explanation see Using ScaLAPACK, LAPACK, LINPACK, and EISPACK in the Introduction section of this manual.

LSARG 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. LSARG 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 L2ARG/DL2ARG. The reference is:

CALL L2ARG (N, A, LDA, B, IPATH, X, FACT, IPVT, WK)

The additional arguments are as follows:

FACT — Work vector of length N2 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 — 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.

ScaLAPACK Usage Notes

The arguments which differ from the standard version of this routine are:

A0 — MXLDA by MXCOL local matrix containing the local portions of the distributed matrix A. A contains the coefficients of the linear system. (Input)

B0 — Local vector of length MXLDA containing the local portions of the distributed vector B. B contains the right-hand side of the linear system. (Input)

X0 — Local vector of length MXLDA containing the local portions of the distributed vector X. X contains the solution to the linear system. (Output)

All other arguments are global and are the same as described for the standard version of the routine. In the argument descriptions above, MXLDA and MXCOL can be obtained through a call to SCALAPACK_GETDIM (see Utilities) after a call to SCALAPACK_SETUP (see Utilities) has been made. See the ScaLAPACK Example below.

Examples

Example

A system of three linear equations is solved. The coefficient matrix has real general form and the right-hand-side vector b has three elements.

USE LSARG_INT

USE WRRRN_INT

IMPLICIT NONE

! Declare variables

INTEGER LDA, N

PARAMETER (LDA=3, N=3)

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

! Set values for A and B

A(1,:) = (/ 33.0, 16.0, 72.0/)

A(2,:) = (/-24.0, -10.0, -57.0/)

A(3,:) = (/ 18.0, -11.0, 7.0/)

B = (/129.0, -96.0, 8.5/)

! Solve the system of equations

CALL LSARG (A, B, X)

! Print results

CALL WRRRN (’X’, X, 1, N, 1)

END

Output

1 2 3

1.000 1.500 1.000

ScaLAPACK Example

The same system of three linear equations is solved as a distributed computing example. The coefficient matrix has real general form and the right-hand-side vector b has three elements. SCALAPACK_MAP and SCALAPACK_UNMAP are IMSL utility routines (see Utilities) used to map and unmap arrays to and from the processor grid. They are used here for brevity. DESCINIT is a ScaLAPACK tools routine which initializes the descriptors for the local arrays.

USE MPI_SETUP_INT

USE LSARG_INT

USE WRRRN_INT

USE SCALAPACK_SUPPORT

IMPLICIT NONE

INCLUDE ‘mpif.h’

! Declare variables

INTEGER N, DESCA(9), DESCX(9)

INTEGER INFO, MXLDA, MXCOL

REAL, ALLOCATABLE :: A(:,:), B(:), X(:)

REAL, ALLOCATABLE :: A0(:,:), B0(:), X0(:)

PARAMETER (N=3)

! Set up for MPI

MP_NPROCS = MP_SETUP()

IF (MP_RANK .EQ. 0) THEN

ALLOCATE (A(N,N), B(N), X(N))

! Set values for A and B

A(1,:) = (/ 33.0, 16.0, 72.0/)

A(2,:) = (/-24.0, -10.0, -57.0/)

A(3,:) = (/ 18.0, -11.0, 7.0/)

B = (/129.0, -96.0, 8.5/)

ENDIF

! Set up a 1D processor grid and define

! its context id, MP_ICTXT

CALL SCALAPACK_SETUP(N, N, .TRUE., .TRUE.)

! Get the array descriptor entities MXLDA,

! AND MXCOL

CALL SCALAPACK_GETDIM(N, N, MP_MB, MP_NB, MXLDA, MXCOL)

! Set up the array descriptors

CALL DESCINIT(DESCA, N, N, MP_MB, MP_NB, 0, 0, MP_ICTXT, MXLDA, INFO)

CALL DESCINIT(DESCX, N, 1, MP_MB, 1, 0, 0, MP_ICTXT, MXLDA, INFO)

! Allocate space for the local arrays

ALLOCATE (A0(MXLDA,MXCOL), B0(MXLDA), X0(MXLDA))

! Map input arrays to the processor grid

CALL SCALAPACK_MAP(A, DESCA, A0)

CALL SCALAPACK_MAP(B, DESCX, B0)

! Solve the system of equations

CALL LSARG (A0, B0, X0)

! Unmap the results from the distributed

! arrays back to a non-distributed array.

! After the unmap, only Rank=0 has the full

! array.

CALL SCALAPACK_UNMAP(X0, DESCX, X)

! Print results.

! Only Rank=0 has the solution, X.

IF (MP_RANK .EQ. 0) CALL WRRRN (’X’, X, 1, N, 1)

IF (MP_RANK .EQ. 0) DEALLOCATE(A, B, X)

DEALLOCATE(A0, B0, X0)

! Exit ScaLAPACK usage

CALL SCALAPACK_EXIT(MP_ICTXT)

! Shut down MPI

MP_NPROCS = MP_SETUP(‘FINAL’)

END

Output

1 2 3

1.000 1.500 1.000