Computes the inverse of a complex general matrix.
Required Arguments
A — Complex N by N matrix containing the matrix to be inverted. (Input)
AINV — Complex N by N matrix containing
the inverse of A.
(Output)
If A is not needed, A and AINV 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).
LDAINV — Leading dimension of AINV exactly as
specified in the dimension statement of the calling program.
(Input)
Default: LDAINV = size (AINV,1).
FORTRAN 90 Interface
Generic: CALL LINCG (A, AINV [,…])
Specific: The specific interface names are S_LINCG and D_LINCG.
FORTRAN 77 Interface
Single: CALL LINCG (N, A, LDA, AINV, LDAINV)
Double: The double precision name is DLINCG.
ScaLAPACK Interface
Generic: CALL LINCG (A0, AINV0 [,…])
Specific: The specific interface names are S_LINCG and D_LINCG.
See the ScaLAPACK Usage Notes below for a description of the arguments for distributed computing.
Description
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. Finally A is computed using A=UL.
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
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.
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
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)
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