LSLDS
Solves a real symmetric positive definite system of linear equations without iterative refinement .
Required Arguments
A — N by N matrix containing the coefficient matrix of the symmetric positive definite linear system. (Input)
Only the upper triangle of A is referenced.
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).
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).
FORTRAN 90 Interface
Generic: CALL LSLDS (A, B, X [, …])
Specific: The specific interface names are S_LSLDS and D_LSLDS.
FORTRAN 77 Interface
Single: CALL LSLDS (N, A, LDA, B, X)
Double: The double precision name is DLSLDS.
ScaLAPACK Interface
Generic: CALL LSLDS (A0, B0, X0 [, …])
Specific: The specific interface names are S_LSLDS and D_LSLDS.
See the ScaLAPACK Usage Notes below for a description of the arguments for distributed computing.
Description
Routine LSLDS solves a system of linear algebraic equations having a real symmetric positive definite coefficient matrix. 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. LSLDS first uses the routine LFCDS to compute an RTR Cholesky factorization of the coefficient matrix and to estimate the condition number of the matrix. The matrix R is upper triangular. The solution of the linear system is then found using the routine LFSDS. LSLDS fails if any submatrix of R is not positive definite or if R has a zero diagonal element. These errors occur only if A either is very close to a singular matrix or to a matrix which is not positive definite. 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. If the coefficient matrix is ill-conditioned, it is recommended that LSADS be used.
Comments
1. Workspace may be explicitly provided, if desired, by use of L2LDS/DL2LDS. The reference is:
CALL L2LDS (N, A, LDA, B, X, FACT, WK)
The additional arguments are as follows:
FACT — N × N work array containing the RTR factorization of A on output. If A is not needed, A can share the same storage locations as FACT.
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 not positive definite. |
3. Integer Options with Chapter 11 Options Manager
16 This option uses four values to solve memory bank conflict (access inefficiency) problems. In routine L2LDS 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 LSLDS. Additional memory allocation for FACT and option value restoration are done automatically in LSLDS. Users directly calling L2LDS 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 LSLDS or L2LDS. 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 LSLDS temporarily replaces IVAL(2) by IVAL(1). The routine L2CDS computes the condition number if IVAL(2) = 2. Otherwise L2CDS skips this computation. LSLDS restores the option. Default values for the option are IVAL(*) = 1, 2.
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 coefficient matrix of the symmetric positive definite 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 positive definite form and the right-hand-side vector b has three elements.
USE LSLDS_INT
USE WRRRN_INT
! Declare variables
INTEGER LDA, N
PARAMETER (LDA=3, N=3)
REAL A(LDA,LDA), B(N), X(N)
!
! Set values for A and B
!
! A = ( 1.0 -3.0 2.0)
! ( -3.0 10.0 -5.0)
! ( 2.0 -5.0 6.0)
!
! B = ( 27.0 -78.0 64.0)
!
DATA A/1.0, -3.0, 2.0, -3.0, 10.0, -5.0, 2.0, -5.0, 6.0/
DATA B/27.0, -78.0, 64.0/
!
CALL LSLDS (A, B, X)
! Print results
CALL WRRRN (’X’, X, 1, N, 1)
!
END
Output
X
1 2 3
1.000 -4.000 7.000
ScaLAPACK Example
The same system of three linear equations is solved as a distributed computing example. The coefficient matrix has real positive definite 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 LSLDS_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,:) = (/ 1.0, -3.0, 2.0/)
A(2,:) = (/ -3.0, 10.0, -5.0/)
A(3,:) = (/ 2.0, -5.0, 6.0/)
!
B = (/27.0, -78.0, 64.0/)
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 the system of equations
CALL LSLDS (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)
! Exit ScaLAPACK usage
CALL SCALAPACK_EXIT(MP_ICTXT)
! Shut down MPI
MP_NPROCS = MP_SETUP(‘FINAL’)
END
Output
X
1 2 3
1.000 -4.000 7.000
