Computes the inverse of a real symmetric positive definite matrix.
Required Arguments
A — N by N matrix containing
the symmetric positive definite matrix to be inverted. (Input)
Only the upper triangle of A is referenced.
AINV — 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 — Order of the matrix A.
(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 LINDS (A, AINV [,…])
Specific: The specific interface names are S_LINDS and D_LINDS.
FORTRAN 77 Interface
Single: CALL LINDS (N, A, LDA, AINV, LDAINV)
Double: The double precision name is DLINDS.
ScaLAPACK Interface
Generic: CALL LINDS (A0, AINV0 [,…])
Specific: The specific interface names are S_LINDS and D_LINDS.
See the ScaLAPACK Usage Notes below for a description of the arguments for distributed computing.
Description
Routine LINDS
computes the inverse of a real symmetric positive definite 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. LINDS
first uses the routine LFCDS to compute an RTR factorization
of the coefficient matrix and to estimate the condition number of the matrix.
LINRT is then used to compute
R-1. Finally A-1
is computed using R-1 = R-1
R-T.
LINDS fails if any submatrix of R is not positive definite or if R has a zero diagonal element. These errors occur only if A 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/ε (where ε is machine precision), a warning error is issued. This indicates that very small changes in A can cause very large changes in A.
Comments
1. Workspace may be explicitly provided, if desired, by use of L2NDS/DL2NDS. The reference is:
CALL L2NDS (N, A, LDA, AINV, LDAINV, WK)
The additional argument is:
WK — Work vector of length N.
2. Informational errors
Type Code
3 1 The input matrix is too ill-conditioned. The solution might not be accurate.
4 2 The input matrix is not positive definite.
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 symmetric positive definite matrix to be inverted. (Input)
AINV0 — MXLDA by MXCOL 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 real positive definite 3 × 3 matrix.
USE
LINDS_INT
USE WRRRN_INT
! Declare variables
INTEGER LDA, LDAINV
PARAMETER (LDA=3, LDAINV=3)
REAL A(LDA,LDA), AINV(LDAINV,LDAINV)
!
! Set values for A
! A = ( 1.0 -3.0 2.0)
! ( -3.0 10.0 -5.0)
! ( 2.0 -5.0 6.0)
!
DATA A/1.0, -3.0, 2.0, -3.0, 10.0, -5.0, 2.0, -5.0, 6.0/
!
CALL LINDS (A, AINV)
! Print results
CALL WRRRN ('AINV', AINV)
!
END
Output
AINV
1 2
3
1 35.00 8.00 -5.00
2
8.00 2.00 -1.00
3
-5.00 -1.00 1.00
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 LINDS_INT
USE WRRRN_INT
USE
SCALAPACK_SUPPORT
IMPLICIT
NONE
INCLUDE ‘mpif.h'
! Declare variables
INTEGER
J, LDA, LDFACT, N, DESCA(9)
INTEGER INFO, MXCOL,
MXLDA
REAL, ALLOCATABLE
:: A(:,:),
AINV(:,:)
REAL, ALLOCATABLE
:: A0(:,:), AINV0(:,:)
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, -3.0,
2.0/)
A(2,:) = (/
-3.0, 10.0,
-5.0/)
A(3,:) =
(/ 2.0, -5.0, 6.0/)
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)
!
Allocate space for the local arrays
ALLOCATE(A0(MXLDA,MXCOL),
AINV0(MXLDA,MXCOL))
!
Map input arrays to the processor grid
CALL
SCALAPACK_MAP(A, DESCA, A0)
! Call the routine to get the inverse
CALL LINDS (A0, AINV0)
! Unmap the results from the distributed
! arrays back to a nondistributed 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, AINV.
IF(MP_RANK.EQ.0) CALL WRRRN
('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 35.00 8.00 -5.00
2 8.00 2.00
-1.00
3 -5.00 -1.00 1.00
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