LSLRT
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
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
X
1 2 3
1.000 -3.000 2.000
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
X
1 2 3
1.000 -3.000 2.000
