LSLRT

FNLMath : Linear Systems : LSLRT

LSLRT

Required Arguments

ScaLAPACK Usage Notes

Examples

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Solves a real triangular system of linear equations.

Required Arguments

A — N by N matrix containing the coefficient matrix for the triangular linear system. (Input)
For a lower triangular system, only the lower triangular part and diagonal of A are referenced. For an upper triangular system, only the upper triangular part and diagonal of A are 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)
If B is not needed, B and X can share the same storage locations.

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 solve AX = B, A lower triangular.
IPATH = 2 means solve AX = B, A upper triangular.
IPATH = 3 means solve ATX = B, A lower triangular.
IPATH = 4 means solve ATX = B, A upper triangular.
Default: IPATH = 1.

FORTRAN 90 Interface

Generic: CALL LSLRT (A, B, X [, …])

Specific: The specific interface names are S_LSLRT and D_LSLRT.

FORTRAN 77 Interface

Single: CALL LSLRT (N, A, LDA, B, IPATH, X)

Double: The double precision name is DLSLRT.

ScaLAPACK Interface

Generic: CALL LSLRT (A0, B0, X0 [, …])

Specific: The specific interface names are S_LSLRT and D_LSLRT.

See the ScaLAPACK Usage Notes below for a description of the arguments for distributed computing.

Description

Routine LSLRT solves a system of linear algebraic equations with a real triangular coefficient matrix. LSLRT fails if the matrix A has a zero diagonal element, in which case A is singular. 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.

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)
For a lower triangular system, only the lower triangular part and diagonal of A are referenced. For an upper triangular system, only the upper triangular part and diagonal of A are referenced.

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)
If B is not needed, B and X can share the same storage locations.

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 lower triangular form and the right-hand-side vector, b, has three elements.

USE LSLRT_INT

USE WRRRN_INT

! Declare variables

PARAMETER (LDA=3)

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

! Set values for A and B

! A = ( 2.0 )

! ( 2.0 -1.0 )

! ( -4.0 2.0 5.0)

! B = ( 2.0 5.0 0.0)

DATA A/2.0, 2.0, -4.0, 0.0, -1.0, 2.0, 0.0, 0.0, 5.0/

DATA B/2.0, 5.0, 0.0/

! Solve AX = B (IPATH = 1)

CALL LSLRT (A, B, X)

! Print results

CALL WRRRN (’X’, X, 1, 3, 1)

END

Output

1 2 3

1.000 -3.000 2.000

ScaLAPACK Example

The same system of three linear equations is solved as a distributed computing example. The coefficient matrix has lower triangular form and the right-hand-side vector b has three elements. SCALAPACK_MAP and SCALAPACK_UNMAP are IMSL utility routines (see Chapter 11, “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 LSLRT_INT

USE WRRRN_INT

USE SCALAPACK_SUPPORT

IMPLICIT NONE

INCLUDE ‘mpif.h’

! Declare variables

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

INTEGER INFO, MXCOL, MXLDA

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

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

PARAMETER (LDA=3, N=3)

! Set up for MPI

MP_NPROCS = MP_SETUP()

IF(MP_RANK .EQ. 0) THEN

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

! Set values for A and B

A(1,:) = (/ 2.0, 0.0, 0.0/)

A(2,:) = (/ 2.0, -1.0, 0.0/)

A(3,:) = (/-4.0, 2.0, 5.0/)

B = (/ 2.0, 5.0, 0.0/)

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 AX = B (IPATH = 1)

CALL LSLRT (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 -3.000 2.000