LSLCG

Solves a complex general system of linear equations without iterative refinement.

Required Arguments

A — Complex N by N matrix containing the coefficients of the linear system. (Input)

B — Complex vector of length N containing the right-hand side of the linear system. (Input)

X — Complex 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)

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).

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).

IPATH — Path indicator. (Input)

IPATH = 1 means the system AX = B is solved.

IPATH = 1 means the system AX = B is solved.

IPATH = 2 means the system AHX = B is solved.

Default: IPATH = 1

FORTRAN 90 Interface

Generic: CALL LSLCG (A, B, X [, …])

Specific: The specific interface names are S_LSLCG and D_LSLCG.

FORTRAN 77 Interface

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

Double: The double precision name is DLSLCG.

ScaLAPACK Interface

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

Specific: The specific interface names are S_LSLCG and D_LSLCG.

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

Description

Routine LSLCG solves a system of linear algebraic equations with a complex general 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. LSLCG first uses the routine LFCCG 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 LFSCG.

LSLCG fails if U, the upper triangular part of the factorization, has a zero diagonal element. This occurs only if A either is a singular matrix or is 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. If the coefficient matrix is ill-conditioned or poorly scaled, it is recommended that LSACG be used.

Comments

1. Workspace may be explicitly provided, if desired, by use of L2LCG/DL2LCG. The reference is:

CALL L2LCG (N, A, LDA, B, IPATH, X, FACT, IPVT, WK)

The additional arguments are as follows:

FACT — N × N work array containing the LU factorization of A on output. If A is not needed, A and FACT can share the same storage locations.

IPVT — Integer work vector of length N containing the pivoting information for the LU factorization of A on output.

WK — Complex 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 L2LCG 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 LSLCG. Additional memory allocation for FACT and option value restoration are done automatically in LSLCG. Users directly calling L2LCG 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 LSLCG or L2LCG. 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 LSLCG temporarily replaces IVAL(2) by IVAL(1). The routine L2CCG computes the condition number if IVAL(2) = 2. Otherwise L2CCG skips this computation. LSLCG 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 complex local matrix containing the local portions of the distributed matrix A. A contains the coefficients of the linear system. (Input)

B0 — Complex 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 — Complex 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 1

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

USE LSLCG_INT

USE WRCRN_INT

! Declare variables

PARAMETER (LDA=3, N=3)

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

! Set values for A and B

!

! A = ( 3.0-2.0i 2.0+4.0i 0.0-3.0i)

! ( 1.0+1.0i 2.0-6.0i 1.0+2.0i)

! ( 4.0+0.0i -5.0+1.0i 3.0-2.0i)

!

! B = (10.0+5.0i 6.0-7.0i -1.0+2.0i)

!

DATA A/(3.0,-2.0), (1.0,1.0), (4.0,0.0), (2.0,4.0), (2.0,-6.0),&

(-5.0,1.0), (0.0,-3.0), (1.0,2.0), (3.0,-2.0)/

DATA B/(10.0,5.0), (6.0,-7.0), (-1.0,2.0)/

! Solve AX = B (IPATH = 1)

CALL LSLCG (A, B, X)

! Print results

CALL WRCRN (’X’, X, 1, N, 1)

END

Output

X

1 2 3

( 1.000,-1.000) ( 2.000, 1.000) ( 0.000, 3.000)

ScaLAPACK Example

The same system of three linear equations is solved as a distributed computing example. The coefficient matrix has complex general 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 LSLCG_INT

USE WRCRN_INT

USE SCALAPACK_SUPPORT

IMPLICIT NONE

INCLUDE ‘mpif.h’

! Declare variables

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

INTEGER INFO, MXCOL, MXLDA

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

COMPLEX, 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,:) = (/ (3.0, -2.0), (2.0, 4.0), (0.0, -3.0)/)

A(2,:) = (/ (1.0, 1.0), (2.0, -6.0), (1.0, 2.0)/)

A(3,:) = (/ (4.0, 0.0), (-5.0, 1.0), (3.0, -2.0)/)

!

B = (/(10.0, 5.0), (6.0, -7.0), (-1.0, 2.0)/)

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 LSLCG (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 WRCRN (’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

X

1 2 3

( 1.000,-1.000) ( 2.000, 1.000) ( 0.000, 3.000)