LFCDH

Computes the RH R factorization of a complex Hermitian positive definite matrix and estimate its L1 condition number.

Required Arguments

A — Complex N by N Hermitian positive definite matrix to be factored. (Input) Only the upper triangle of A is referenced.

FACT — Complex N by N matrix containing the upper triangular matrix R of the factorization of A in the upper triangle. (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 LFCDH (A, FACT, RCOND [, …])

Specific: The specific interface names are S_LFCDH and D_LFCDH.

FORTRAN 77 Interface

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

Double: The double precision name is DLFCDH.

ScaLAPACK Interface

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

Specific: The specific interface names are S_LFCDH and D_LFCDH.

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

Description

Routine LFCDH computes an RH R Cholesky factorization and estimates the condition number of a complex Hermitian positive definite coefficient matrix. The matrix R is upper triangular.

The L1 condition 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.

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

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.

Comments

1. Workspace may be explicitly provided, if desired, by use of L2CDH/DL2CDH. The reference is:

CALL L2CDH (N, A, LDA, FACT, LDFACT, RCOND, WK)

The additional argument is

WK — Complex work vector of length N.

2. Informational errors

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

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

3 | 4 | The input matrix is not Hermitian. It has a diagonal entry with a small imaginary part. |

4 | 4 | The input matrix is not Hermitian. |

4 | 2 | The input matrix is not positive definite. It has a diagonal entry with an imaginary part |

ScaLAPACK Usage Notes

The arguments which differ from the standard version of this routine are:

A0 — Complex MXLDA by MXCOL local matrix containing the local portions of the distributed matrix A. A contains the Hermitian positive definite matrix to be factored. (Input)

Only the upper triangle of A is referenced.

Only the upper triangle of A is referenced.

FACT0 — Complex 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 triangle. (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 5 × 5 Hermitian positive definite matrix is computed. LFCDH is called to factor the matrix and to check for nonpositive definiteness or ill-conditioning. LFIDH is called to determine the columns of the inverse.

USE LFCDH_INT

USE LFIDH_INT

USE UMACH_INT

USE WRCRN_INT

! Declare variables

INTEGER LDA, LDFACT, N, NOUT

PARAMETER (LDA=5, LDFACT=5, N=5)

REAL RCOND

COMPLEX A(LDA,LDA), AINV(LDA,LDA), FACT(LDFACT,LDFACT),&

RES(N), RJ(N)

!

! Set values for A

!

! A = ( 2.0+0.0i -1.0+1.0i 0.0+0.0i 0.0+0.0i 0.0+0.0i )

! ( 4.0+0.0i 1.0+2.0i 0.0+0.0i 0.0+0.0i )

! ( 10.0+0.0i 0.0+4.0i 0.0+0.0i )

! ( 6.0+0.0i 1.0+1.0i )

! ( 9.0+0.0i )

!

DATA A /(2.0,0.0), 4*(0.0,0.0), (-1.0,1.0), (4.0,0.0),&

4*(0.0,0.0), (1.0,2.0), (10.0,0.0), 4*(0.0,0.0),&

(0.0,4.0), (6.0,0.0), 4*(0.0,0.0), (1.0,1.0), (9.0,0.0)/

! Factor the matrix A

CALL LFCDH (A, FACT, RCOND)

! Set up the columns of the identity

! matrix one at a time in RJ

RJ = (0.0E0, 0.0E0)

DO 10 J=1, N

RJ(J) = (1.0E0,0.0E0)

! RJ is the J-th column of the identity

! matrix so the following LFIDH

! reference places the J-th column of

! the inverse of A in the J-th column

! of AINV

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

RJ(J) = (0.0E0,0.0E0)

10 CONTINUE

! Print the results

CALL UMACH (2, NOUT)

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

CALL WRCRN (’AINV’, AINV)

!

99999 FORMAT (’ RCOND = ’,F5.3,/,’ L1 Condition number = ’,F6.3)

END

Output

RCOND < 0.075

L1 Condition number < 25.0

AINV

1 2 3 4

1 ( 0.7166, 0.0000) ( 0.2166,-0.2166) (-0.0899,-0.0300) (-0.0207, 0.0622)

2 ( 0.2166, 0.2166) ( 0.4332, 0.0000) (-0.0599,-0.1198) (-0.0829, 0.0415)

3 (-0.0899, 0.0300) (-0.0599, 0.1198) ( 0.1797, 0.0000) ( 0.0000,-0.1244)

4 (-0.0207,-0.0622) (-0.0829,-0.0415) ( 0.0000, 0.1244) ( 0.2592, 0.0000)

5 ( 0.0092, 0.0046) ( 0.0138,-0.0046) (-0.0138,-0.0138) (-0.0288, 0.0288)

5

1 ( 0.0092,-0.0046)

2 ( 0.0138, 0.0046)

3 (-0.0138, 0.0138)

4 (-0.0288,-0.0288)

5 ( 0.1175, 0.0000)

ScaLAPACK Example

The inverse of the same 5 × 5 Hermitian positive definite matrix in the preceding example is computed as a distributed computing example. LFCDH is called to factor the matrix and to check for nonpositive definiteness or ill-conditioning. LFIDH is called to determine the columns of the inverse. SCALAPACK_MAP and SCALAPACK_UNMAP are IMSL utility routines (see 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 LFCDH_INT

USE LFIDH_INT

USE WRCRN_INT

USE SCALAPACK_SUPPORT

IMPLICIT NONE

INCLUDE ‘mpif.h’

! Declare variables

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

INTEGER INFO, MXCOL, MXLDA

REAL RCOND

COMPLEX, ALLOCATABLE :: A(:,:), AINV(:,:), RJ(:), RJ0(:)

COMPLEX, ALLOCATABLE :: A0(:,:), FACT0(:,:), RES0(:), X0(:)

PARAMETER (LDA=5, N=5)

! 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 and B

A(1,:) = (/(2.0, 0.0),(-1.0, 1.0),( 0.0, 0.0),(0.0, 0.0),(0.0, 0.0)/)

A(2,:) = (/(0.0, 0.0),( 4.0, 0.0),( 1.0, 2.0),(0.0, 0.0),(0.0, 0.0)/)

A(3,:) = (/(0.0, 0.0),( 0.0, 0.0),(10.0, 0.0),(0.0, 4.0),(0.0, 0.0)/)

A(4,:) = (/(0.0, 0.0),( 0.0, 0.0),( 0.0, 0.0),(6.0, 0.0),(1.0, 1.0)/)

A(5,:) = (/(0.0, 0.0),( 0.0, 0.0),( 0.0, 0.0),(0.0, 0.0),(9.0, 0.0)/)

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(DESCX, 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 arrays to the processor grid

CALL SCALAPACK_MAP(A, DESCA, A0)

! Factor the matrix A

CALL LFCDH (A0, FACT0, RCOND)

! Set up the columns of the identity

! matrix one at a time in RJ

RJ = (0.0E0, 0.0E0)

DO 10 J=1, N

RJ(J) = (1.0E0,0.0E0)

CALL SCALAPACK_MAP(RJ, DESCX, RJ0)

! RJ is the J-th column of the identity

! matrix so the following LFIDH

! reference solves for the J-th column of

! the inverse of A

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

! Unmap the results from the distributed

! array back to a non-distributed array

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

RJ(J) = (0.0E0,0.0E0)

10 CONTINUE

! Print the results.

! After the unmap, only Rank=0 has the full

! array.

IF(MP_RANK .EQ. 0) THEN

CALL UMACH (2, NOUT)

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

CALL WRCRN (’AINV’, AINV)

ENDIF

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’)

99999 FORMAT (’ RCOND = ’,F5.3,/,’ L1 Condition number = ’,F6.3)

END

Output

RCOND < 0.075

L1 Condition number < 25.0

AINV

1 2 3 4

1 ( 0.7166, 0.0000) ( 0.2166,-0.2166) (-0.0899,-0.0300) (-0.0207, 0.0622)

2 ( 0.2166, 0.2166) ( 0.4332, 0.0000) (-0.0599,-0.1198) (-0.0829, 0.0415)

3 (-0.0899, 0.0300) (-0.0599, 0.1198) ( 0.1797, 0.0000) ( 0.0000,-0.1244)

4 (-0.0207,-0.0622) (-0.0829,-0.0415) ( 0.0000, 0.1244) ( 0.2592, 0.0000)

5 ( 0.0092, 0.0046) ( 0.0138,-0.0046) (-0.0138,-0.0138) (-0.0288, 0.0288)

5

1 ( 0.0092,-0.0046)

2 ( 0.0138, 0.0046)

3 (-0.0138, 0.0138)

4 (-0.0288,-0.0288)

5 ( 0.1175, 0.0000)