.xi.

Chapter 10: Linear Algebra Operators and Generic Functions

.xi.

Computes the product of a matrix or vector and the inverse of a matrix.

Operator Return Value

Matrix containing the product of A and B^-1. (Output)

Required Operands

A — Right operand matrix or vector. This is an array of rank 1, 2, or 3. It may be real, double, complex, or double complex. (Input)

B — Left operand matrix. This is an array of rank 2, or 3. It may be real, double, complex, double complex, or one of the computational sparse matrix derived types, ?_hbc_sparse. (Input)

Optional Variables, Reserved Names

This operator uses the routines lin_sol_gen or lin_sol_lsq (See Chapter 1, “Linear Systems”).

The option and derived type names are given in the following tables:

Option Names for .xi.	Option Value
use_lin_sol_gen_only	1
use_lin_sol_lsq_only	2
xi_options_for_lin_sol_gen	3
xi_options_for_lin_sol_lsq	4
Skip_error_processing	5

Name of Unallocated Option Array to Use for Setting Options	Use	Derived Type
?_xinv_options(:)	Use when setting options for calls hereafter.	?_options
?_xinv_options_once(:)	Use when setting options for next call only.	?_options

For a description on how to use these options, see Matrix Optional Data Changes. See lin_sol_gen and lin_sol_lsq located in Chapter 1, “Linear Systems” for the specific options for these routines.

FORTRAN 90 Interface

A .xi. B

Description

Computes the product of matrix A and the inverse of matrix B, for square non-singular matrices or the corresponding Moore-Penrose generalized inverse matrix for singular square matrices or rectangular matrices. The operation may be read times generalized inverse. The results are in a precision and data type that matches the most accurate or complex operand.

.xi. can be used with either dense or sparse matrices. It is MPI capable for dense matrices only.

Examples

Dense Matrix Example

use linear_operators

implicit none

integer, parameter :: n=32

real(kind(1e0)) :: one=1.0e0, err

real(kind(1e0)), dimension(n,n) :: A, b, x

! Generate random matrices for A and b:

A = rand(A); b=rand(b)

! Compute the solution matrix of xA = b.

x = b .xi. A

! Check the results.

err = norm(b - (x .x. A))/(norm(A)*norm(x)+norm(b))

if (err <= sqrt(epsilon(one))) &

write (*,*) 'Example for .xi. operator is correct.'

end

Sparse Matrix Example

use wrrrn_int

use linear_operators

type (s_sparse) S

type (s_hbc_sparse) H

integer, parameter :: N=3

real (kind(1.e0)) x(N,N), y(N,N), a(N,N)

real (kind(1.e0)) err

S = s_entry (1, 1, 2.0)

S = s_entry (1, 3, 1.0)

S = s_entry (2, 2, 4.0)

S = s_entry (3, 3, 6.0)

H = S ! sparse

X = H ! dense equivalent of H

A = rand(A)

Y = A .xi. H

call wrrrn ( 'A', A)

call wrrrn ( 'H', X)

call wrrrn ( 'A .xi. H', y)

! Check the results.

err = norm(y - (A .xi. X))

if (err <= sqrt(epsilon(one))) then

write (*,*) 'Sparse example for .xi. operator is correct.'

end if

end

Output

1 2 3

1 0.5926 0.5015 0.5368

2 0.4001 0.9529 0.6988

3 0.0412 0.0633 0.3821

1 2 3

1 2.000 0.000 1.000

2 0.000 4.000 0.000

3 0.000 0.000 6.000

A .xi. H

1 2 3

1 0.2963 0.1254 0.0401

2 0.2001 0.2382 0.0831

3 0.0206 0.0158 0.0602

Sparse example for .xi. operator is correct.

Parallel Example

use linear_operators

use mpi_setup_int

implicit none

! This is the equivalent of Parallel Example 1 for .xi., with box data types

! and functions.

integer, parameter :: n=32, nr=4

real(kind(1e0)) :: one=1e0

real(kind(1e0)), dimension(n,n,nr) :: A, b, x, err(nr)

! Setup for MPI.

MP_NPROCS=MP_SETUP()

! Generate random matrices for A and b:

A = rand(A); b=rand(b)

! Compute the box solution matrix of xA = b.

x = b .xi. A

! Check the results.

err = norm(b - (x .x. A))/(norm(A)*norm(x)+norm(b))

if (ALL(err <= sqrt(epsilon(one))) .and. MP_RANK == 0) &

write (*,*) 'Parallel Example 1 is correct.'

! See to any error messages and quit MPI.

MP_NPROCS=MP_SETUP('Final')

end

Visual Numerics, Inc.
Visual Numerics - Developers of IMSL and PV-WAVE
http://www.vni.com/
PHONE: 713.784.3131
FAX:713.781.9260