LFTDH

Computes the RHR  factorization of  a complex Hermitian positive definite matrix.

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.

Optional Arguments

N — Order of the matrix.      (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).

LDFACT — Leading dimension of FACT exactly as specified in the dimension statement of the calling program.   (Input)
Default: LDFACT = size (FACT,1).

FORTRAN 90 Interface

Generic:                              CALL LFTDH (A, FACT [,…])

Specific:                             The specific interface names are S_LFTDH and D_LFTDH.

FORTRAN 77 Interface

Single:                                CALL LFTDH (N, A, LDA, FACT, LDFACT)

Double:                              The double precision name is DLFTDH.

ScaLAPACK Interface

Generic:                              CALL LFTDH (A0, FACT0 [,…])

Specific:                             The specific interface names are S_LFTDH and D_LFTDH.

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

Description

Routine LFTDH computes an RH R Cholesky factorization of a complex Hermitian positive definite coefficient matrix. The matrix R is upper triangular.

LFTDH 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 IMSL 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

Informational errors

Type       Code

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

4                   2          The input matrix is not positive definite.

4                   4          The input matrix is not Hermitian. 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.

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.

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 of a 5 Χ 5 matrix is computed. LFTDH is called to factor the matrix and to check for nonpositive definiteness. LFSDH is called to determine the columns of the inverse.

 

      USE LFTDH_INT
      USE LFSDH_INT
      USE WRCRN_INT
!                                 Declare variables

      INTEGER    LDA, LDFACT, N

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

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

!                                 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 LFSDH

!                                 reference places the J-th column of

!                                 the inverse of A in the J-th column

!                                 of AINV

         CALL LFSDH (FACT, RJ, AINV(:,J))

         RJ(J) = (0.0E0,0.0E0)

   10 CONTINUE

!                                 Print the results

 

 

      CALL WRCRN ('AINV', AINV, ITRING=1)

!

      END

Output

 

                                     AINV
                  1                  2                  3                 4
1 ( 0.7166, 0.0000) ( 0.2166,-0.2166)  (-0.0899,-0.0300)  (-0.0207, 0.0622)
2                   ( 0.4332, 0.0000)  (-0.0599,-0.1198)  (-0.0829, 0.0415)
3                                      ( 0.1797, 0.0000)  ( 0.0000,-0.1244)
4                                                         ( 0.2592, 0.0000)
                   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 Χ 5 Hermitian positive definite matrix in the preceding example is computed as a distributed computing example. LFTDH is called to factor the matrix and to check for nonpositive definiteness. LFSDH (page 192) 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 LFTDH_INT
      USE LFSDH_INT
      USE WRCRN_INT
      USE SCALAPACK_SUPPORT
      IMPLICIT NONE
      INCLUDE ‘mpif.h'

!                                 Declare variables

      INTEGER        J, LDA, N, DESCA(9), DESCX(9)
      INTEGER       INFO, MXCOL, MXLDA
      COMPLEX, ALLOCATABLE ::        A(:,:), AINV(:,:), RJ(:), RJ0(:)
      COMPLEX, ALLOCATABLE ::        A0(:,:), FACT0(:,:), 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))

!                                 Map input arrays to the processor grid

      CALL SCALAPACK_MAP(A, DESCA, A0)
!                                 Factor the matrix A

      CALL LFTDH (A0, FACT0)

!                                 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 LFSDH (FACT0, RJ0, X0)

!                                 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) CALL WRCRN ('AINV', AINV)

      IF (MP_RANK .EQ. 0) DEALLOCATE(A, AINV)

      DEALLOCATE(A0, FACT0, RJ, RJ0, X0)

!                                Exit ScaLAPACK usage

      CALL SCALAPACK_EXIT(MP_ICTXT)

!                                Shut down MPI

      MP_NPROCS = MP_SETUP(‘FINAL')
      END

Output

 

                                   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)

6  (-0.0288,-0.0288)

7  ( 0.1175, 0.0000)

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