LSADH

Solves a Hermitian positive definite system of linear equations with iterative refinement.

Required Arguments

A — Complex N by N matrix containing the coefficient matrix of the Hermitian positive definite linear system. (Input)

Only the upper triangle of A is referenced.

Only the upper triangle of A is referenced.

B — Complex vector of length N containing the right-hand side of the linear system. (Input)

X — Complex vector of length N containing the solution of the linear system. (Output)

Optional Arguments

N — Number of equations. (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).

FORTRAN 90 Interface

Generic: CALL LSADH (A, B, X [, …])

Specific: The specific interface names are S_LSADH and D_LSADH.

FORTRAN 77 Interface

Single: CALL LSADH (N, A, LDA, B, X)

Double: The double precision name is DLSADH.

ScaLAPACK Interface

Generic: CALL LSADH (A0, B0, X0 [, …])

Specific: The specific interface names are S_LSADH and D_LSADH.

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

Description

Routine LSADH solves a system of linear algebraic equations having a complex Hermitian positive definite coefficient matrix. It first uses the routine LFCDH to compute an RH R Cholesky factorization of the coefficient matrix and to estimate the condition number of the matrix. The matrix R is upper triangular. The solution of the linear system is then found using the iterative refinement routine LFIDH.

LSADH fails if any submatrix of R is not positive definite, if R has a zero diagonal element or if the iterative refinement algorithm fails to converge. These errors occur only if A either is very close to a singular matrix or is a matrix that 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 the solution x. Iterative refinement can sometimes find the solution to such a system. LSADH solves the problem that is represented in the computer; however, this problem may differ from the problem whose solution is desired.

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 L2ADH/DL2ADH. The reference is:

CALL L2ADH (N, A, LDA, B, X, FACT, WK)

The additional arguments are as follows:

FACT — N × N work array containing the RH R factorization of A on output.

WK — Complex 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. |

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. |

3. Integer Options with Chapter 11 Options Manager

16 This option uses four values to solve memory bank conflict (access inefficiency) problems. In routine L2ADH the leading dimension of FACT is increased by IVAL(3) when N is a multiple of IVAL(4). The values IVAL(3) and IVAL(4) are temporarily replaced by IVAL(1) and IVAL(2), respectively, in LSADH. Additional memory allocation for FACT and option value restoration are done automatically in LSADH. Users directly calling L2ADH can allocate additional space for FACT and set IVAL(3) and IVAL(4) so that memory bank conflicts no longer cause inefficiencies. There is no requirement that users change existing applications that use LSADH or L2ADH. Default values for the option are IVAL(*) = 1, 16, 0, 1.

17 This option has two values that determine if the L1condition number is to be computed. Routine LSADH temporarily replaces IVAL(2) by IVAL(1). The routine L2CDH computes the condition number if IVAL(2) = 2. Otherwise L2CDH skips this computation. LSADH restores the option. Default values for the option are IVAL(*) = 1, 2.

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 coefficient matrix of the Hermitian positive definite linear system. (Input)

Only the upper triangle of A is referenced.

Only the upper triangle of A is referenced.

B0 — Complex local vector of length MXLDA containing the local portions of the distributed vector B. B contains the right-hand side of the linear system. (Input)

X0 — Complex local vector of length MXLDA containing the local portions of the distributed vector X. X contains the solution to the linear system. (Output)

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

A system of five linear equations is solved. The coefficient matrix has complex positive definite form and the right-hand-side vector b has five elements.

USE LSADH_INT

USE WRCRN_INT

! Declare variables

INTEGER LDA, N

PARAMETER (LDA=5, N=5)

COMPLEX A(LDA,LDA), B(N), X(N)

!

! Set values for A and B

!

! 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 )

!

! B = ( 1.0+5.0i 12.0-6.0i 1.0-16.0i -3.0-3.0i 25.0+16.0i )

!

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)/

DATA B /(1.0,5.0), (12.0,-6.0), (1.0,-16.0), (-3.0,-3.0),&

(25.0,16.0)/

!

CALL LSADH (A, B, X)

! Print results

CALL WRCRN (’X’, X, 1, N, 1)

!

END

Output

X

1 2 3 4

( 2.000, 1.000) ( 3.000, 0.000) (-1.000,-1.000) ( 0.000,-2.000)

5

( 3.000, 2.000)

ScaLAPACK Example

The same system of five linear equations is solved as a distributed computing example. The coefficient matrix has complex positive definite form and the right-hand-side vector b has five elements. 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 LSADH_INT

USE WRCRN_INT

USE SCALAPACK_SUPPORT

IMPLICIT NONE

INCLUDE ‘mpif.h’

! Declare variables

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

INTEGER INFO, MXCOL, MXLDA

COMPLEX, ALLOCATABLE :: A(:,:), B(:), X(:)

COMPLEX, ALLOCATABLE :: A0(:,:), B0(:), X0(:)

PARAMETER (LDA=5, N=5)

! Set up for MPI

MP_NPROCS = MP_SETUP()

IF(MP_RANK .EQ. 0) THEN

ALLOCATE (A(LDA,N), B(N), X(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)/)

!

B = (/(1.0, 5.0),(12.0, -6.0),(1.0, -16.0),(-3.0, -3.0),(25.0, 16.0)/)

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), B0(MXLDA), X0(MXLDA))

! Map input arrays to the processor grid

CALL SCALAPACK_MAP(A, DESCA, A0)

CALL SCALAPACK_MAP(B, DESCX, B0)

! Solve the system of equations

CALL LSADH (A0, B0, X0)

! Unmap the results from the distributed

! arrays back to a non-distributed array.

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

! array.

CALL SCALAPACK_UNMAP(X0, DESCX, X)

! Print results.

! Only Rank=0 has the solution, X.

IF(MP_RANK .EQ. 0)CALL WRCRN (’X’, X, 1, N, 1)

IF (MP_RANK .EQ. 0) DEALLOCATE(A, B, X)

DEALLOCATE(A0, B0, X0)

! Exit ScaLAPACK usage

CALL SCALAPACK_EXIT(MP_ICTXT)

! Shut down MPI

MP_NPROCS = MP_SETUP(‘FINAL’)

END

Output

X

1 2 3 4

( 2.000, 1.000) ( 3.000, 0.000) (-1.000,-1.000) ( 0.000,-2.000)

5

( 3.000, 2.000)