LFCDS

Computes the RTR Cholesky factorization of a real symmetric positive definite matrix and estimate its L1 condition number.

Required Arguments

A — N by N symmetric positive definite matrix to be factored. (Input)

Only the upper triangle of A is referenced.

Only the upper triangle of A is referenced.

FACT — N by N matrix containing the upper triangular matrix R of the factorization of A in the upper triangular part. (Output)

Only the upper triangle of FACT will be used. If A is not needed, A and FACT can share the same storage locations.

Only the upper triangle of FACT will be used. If A is not needed, A and FACT can share the same storage locations.

RCOND — Scalar containing an estimate of the reciprocal of the L1 condition number of A. (Output)

Optional Arguments

N — Order of the matrix. (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).

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

Default: LDFACT = size (FACT,1).

Default: LDFACT = size (FACT,1).

FORTRAN 90 Interface

Generic: CALL LFCDS (A, FACT, RCOND [, …])

Specific: The specific interface names are S_LFCDS and D_LFCDS.

FORTRAN 77 Interface

Single: CALL LFCDS (N, A, LDA, FACT, LDFACT, RCOND)

Double: The double precision name is DLFCDS.

ScaLAPACK Interface

Generic: CALL LFCDS (A0, FACT0, RCOND [, …])

Specific: The specific interface names are S_LFCDS and D_LFCDS.

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

Description

Routine LFCDS computes an RTR Cholesky factorization and estimates the condition number of a real symmetric positive definite coefficient matrix. The matrix R is upper triangular.

The L1condition number of the matrix A is defined to be κ(A) = ∥A∥1∥A-1∥1. Since it is expensive to compute ∥A-1∥1, the condition number is only estimated. The estimation algorithm is the same as used by LINPACK and is described by Cline et al. (1979).

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.

LFCDS 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.

The RTR factors are returned in a form that is compatible with routines LFIDS, LFSDS and LFDDS. To solve systems of equations with multiple right-hand-side vectors, use LFCDS followed by either LFIDS or LFSDS called once for each right-hand side. The routine LFDDS can be called to compute the determinant of the coefficient matrix after LFCDS has performed the factorization.

Comments

1. Workspace may be explicitly provided, if desired, by use of L2CDS/DL2CDS. The reference is:

CALL L2CDS (N, A, LDA, FACT, LDFACT, RCOND, WK)

The additional argument is:

WK — Work vector of length N.

2. Informational errors

Type | Code | Description |
---|---|---|

3 | 1 | The input matrix is algorithmically singular. |

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 factored. (Input)

FACT0 — MXLDA by MXCOL local matrix containing the local portions of the distributed matrix FACT. FACT contains the upper triangular matrix R of the factorization of A in the upper triangular part. (Output)

Only the upper triangle of FACT will be used. If A is not needed, A and FACT can share the same storage locations.

Only the upper triangle of FACT will be used. If A is not needed, A and FACT 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 of a 3 × 3 matrix is computed. LFCDS is called to factor the matrix and to check for nonpositive definiteness or ill-conditioning. LFIDS is called to determine the columns of the inverse.

USE LFCDS_INT

USE UMACH_INT

USE WRRRN_INT

USE LFIDS_INT

! Declare variables

INTEGER LDA, LDFACT, N, NOUT

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

REAL A(LDA,LDA), AINV(LDA,LDA), RCOND, FACT(LDFACT,LDFACT),&

RES(N), RJ(N)

!

! 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/

! Factor the matrix A

CALL LFCDS (A, FACT, RCOND)

! Set up the columns of the identity

! matrix one at a time in RJ

RJ = 0.0E0

DO 10 J=1, N

RJ(J) = 1.0E0

! RJ is the J-th column of the identity

! matrix so the following LFIDS

! reference places the J-th column of

! the inverse of A in the J-th column

! of AINV

CALL LFIDS (A, FACT, RJ, AINV(:,J), RES)

RJ(J) = 0.0E0

10 CONTINUE

! Print the results

CALL UMACH (2, NOUT)

WRITE (NOUT,99999) RCOND, 1.0E0/RCOND

CALL WRRRN (’AINV’, AINV)

99999 FORMAT (’ RCOND = ’,F5.3,/,’ L1 Condition number = ’,F9.3)

END

Output

RCOND < 0.005

L1 Condition number < 875.0

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. LFCDS is called to factor the matrix and to check for singularity or ill-conditioning. LFIDS is called to determine the columns of the inverse. 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 LFCDS_INT

USE UMACH_INT

USE LFIDS_INT

USE WRRRN_INT

USE SCALAPACK_SUPPORT

IMPLICIT NONE

INCLUDE ‘mpif.h’

! Declare variables

INTEGER J, LDA, N, NOUT, DESCA(9), DESCL(9)

INTEGER INFO, MXCOL, MXLDA

REAL, ALLOCATABLE :: A(:,:), AINV(:,:), X0(:), RJ(:)

REAL, ALLOCATABLE :: A0(:,:), FACT0(:,:), RES0(:), RJ0(:)

REAL RCOND

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)

CALL DESCINIT(DESCL, N, 1, MP_MB, 1, 0, 0, MP_ICTXT, MXLDA, INFO)

! Allocate space for the local arrays

ALLOCATE(A0(MXLDA,MXCOL), X0(MXLDA),FACT0(MXLDA,MXCOL), RJ(N), &

RJ0(MXLDA), RES0(MXLDA))

! Map input array to the processor grid

CALL SCALAPACK_MAP(A, DESCA, A0)

! Call the factorization routine

CALL LFCDS (A0, FACT0, RCOND)

! Print the reciprocal condition number

! and the L1 condition number

IF(MP_RANK .EQ. 0) THEN

CALL UMACH (2, NOUT)

WRITE (NOUT,99998) RCOND, 1.0E0/RCOND

ENDIF

! Set up the columns of the identity

! matrix one at a time in RJ

RJ = 0.0E0

DO 10 J=1, N

RJ(J) = 1.0

! Map input array to the processor grid

CALL SCALAPACK_MAP(RJ, DESCL, RJ0)

! RJ is the J-th column of the identity

! matrix so the following LFIDS

! reference computes the J-th column of

! the inverse of A

CALL LFIDS (A0, FACT0, RJ0, X0, RES0)

RJ(J) = 0.0

CALL SCALAPACK_UNMAP(X0, DESCL, AINV(:,J))

10 CONTINUE

! 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, FACT0, RJ, RJ0, RES0, X0)

! Exit ScaLAPACK usage

CALL SCALAPACK_EXIT(MP_ICTXT)

! Shut down MPI

MP_NPROCS = MP_SETUP(‘FINAL’)

99998 FORMAT (’ RCOND = ’,F5.3,/,’ L1 Condition number = ’,F9.3)

END

Output

RCOND < 0.005

L1 Condition number < 875.0

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