LINCG

Routine LINCG computes the inverse of a complex general 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.

LINCG first uses the routine LFCCG to compute an LU factorization of the coefficient matrix and to estimate the condition number of the matrix. LFCCG computes U and the information needed to compute L. LINCT is then used to compute U-1, i.e. use the inverse of U. Finally A-1 is computed using A-1 = U-1L-1.

LINCG fails if U, the upper triangular part of the factorization, has a zero diagonal element or if the iterative refinement algorithm fails to converge. This errors occurs 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 A-1.

Comments

1. Workspace may be explicitly provided, if desired, by use of L2NCG/DL2NCG. The reference is:

CALL L2NCG (N, A, LDA, AINV, LDAINV, WK, IWK)

The additional arguments are as follows:

WK — Complex work vector of length N + N(N − 1)/2.

IWK — Integer work vector of length N.

2. Informational errors

Type	Code	Description
3	1	The input matrix is too ill-conditioned. The inverse might not be accurate.
4	2	The input matrix is singular.

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 matrix to be inverted. (Input)

AINV0 — MXLDA by MXCOL complex local matrix containing the local portions of the distributed matrix AINV. AINV contains the inverse of the matrix A. (Output)
If A is not needed, A and AINV 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

The inverse is computed for a complex general 3 × 3 matrix.

USE LINCG_INT

USE WRCRN_INT

USE CSSCAL_INT

! Declare variables

PARAMETER (LDA=3, LDAINV=3, N=3)

REAL THIRD

COMPLEX A(LDA,LDA), AINV(LDAINV,LDAINV)

! Set values for A

! A = ( 1.0+1.0i 2.0+3.0i 3.0+3.0i)

! ( 2.0+1.0i 5.0+3.0i 7.0+4.0i)

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

DATA A/(1.0,1.0), (2.0,1.0), (-2.0,1.0), (2.0,3.0), (5.0,3.0),&

(-4.0,4.0), (3.0,3.0), (7.0,4.0), (-5.0,3.0)/

! Scale A by dividing by three

THIRD = 1.0/3.0

DO 10 I=1, N

CALL CSSCAL (N, THIRD, A(:,I), 1)

10 CONTINUE

! Calculate the inverse of A

CALL LINCG (A, AINV)

! Print results

CALL WRCRN (’AINV’, AINV)

END

Output

AINV

1 2 3

1 ( 6.400,-2.800) (-3.800, 2.600) (-2.600, 1.200)

2 (-1.600,-1.800) ( 0.200, 0.600) ( 0.400,-0.800)

3 (-0.600, 2.200) ( 1.200,-1.400) ( 0.400, 0.200)

ScaLAPACK Example

The inverse of the same 3 × 3 matrix is computed as a distributed example. 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 LINCG_INT

USE WRCRN_INT

USE SCALAPACK_SUPPORT

IMPLICIT NONE

INCLUDE ‘mpif.h’

! Declare variables

INTEGER J, LDA, N, DESCA(9)

INTEGER INFO, MXCOL, MXLDA, NPROW, NPCOL

COMPLEX, ALLOCATABLE :: A(:,:), AINV(:,:)

COMPLEX, ALLOCATABLE :: A0(:,:), AINV0(:,:)

REAL THIRD

PARAMETER (LDA=3, N=3)

! Set up for MPI

MP_NPROCS = MP_SETUP()

IF(MP_RANK .EQ. 0) THEN

ALLOCATE (A(LDA,N), AINV(LDA,N))

! Set values for A

A(1,:) = (/ ( 1.0, 1.0), ( 2.0, 3.0), ( 3.0, 3.0)/)

A(2,:) = (/ ( 2.0, 1.0), ( 5.0, 3.0), ( 7.0, 4.0)/)

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

! Scale A by dividing by three

THIRD = 1.0/3.0

A = A * THIRD

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)

! Allocate space for the local arrays

ALLOCATE(A0(MXLDA,MXCOL), AINV0(MXLDA,MXCOL))

! Map input array to the processor grid

CALL SCALAPACK_MAP(A, DESCA, A0)

! Factor A

CALL LINCG (A0, AINV0)

! 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(AINV0, DESCA, AINV)

! Print results.

! Only Rank=0 has the solution, X.

IF(MP_RANK.EQ.0) CALL WRCRN (’AINV’, AINV)

IF (MP_RANK .EQ. 0) DEALLOCATE(A, AINV)

DEALLOCATE(A0, AINV0)

! Exit ScaLAPACK usage

CALL SCALAPACK_EXIT(MP_ICTXT)

! Shut down MPI

MP_NPROCS = MP_SETUP(‘FINAL’)

END