LINDS

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.

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.

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).

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).

LDAINV — Leading dimension of AINV exactly as specified in the dimension statement of the calling program. (Input)

Default: LDAINV = size (AINV,1).

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 A-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/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.

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 | Description |
---|---|---|

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.

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 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