LSACG

Solves a complex general system of linear equations with 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)

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 = 2 means the system AHX = B is solved.

Default: IPATH = 1

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 LSACG (A, B, X [, …])

Specific: The specific interface names are S_LSACG and D_LSACG.

FORTRAN 77 Interface

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

Double: The double precision name is DLSACG.

ScaLAPACK Interface

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

Specific: The specific interface names are S_LSACG and D_LSACG.

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

Description

Routine LSACG 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. LSACG 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 the iterative refinement routine LFICG.

LSACG 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. LSACG 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 L2ACG/DL2ACG. The reference is:

CALL L2ACG (N, A, LDA, B, IPATH, X, FACT, IPVT, WK)

The additional arguments are as follows:

FACT — Complex work vector of length N2containing 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 — 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 L2ACG 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 LSACG. Additional memory allocation for FACT and option value restoration are done automatically in LSACG. Users directly calling L2ACG 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 LSACG or L2ACG. Default values for the option are

IVAL(*) = 1, 16, 0, 1.

IVAL(*) = 1, 16, 0, 1.

17 This option has two values that determine if the L1condition number is to be computed. Routine LSACG temporarily replaces IVAL(2) by IVAL(1). The routine L2CCG computes the condition number if IVAL(2) = 2. Otherwise L2CCG skips this computation. LSACG 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 LSACG_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 LSACG (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 LSACG_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 LSACG (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)