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NAG Toolbox: nag_lapack_zsptri (f07qw)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_zsptri (f07qw) computes the inverse of a complex symmetric matrix A, where A has been factorized by nag_lapack_zsptrf (f07qr), using packed storage.

Syntax

[ap, info] = f07qw(uplo, ap, ipiv, 'n', n)
[ap, info] = nag_lapack_zsptri(uplo, ap, ipiv, 'n', n)

Description

nag_lapack_zsptri (f07qw) is used to compute the inverse of a complex symmetric matrix A, the function must be preceded by a call to nag_lapack_zsptrf (f07qr), which computes the Bunch–Kaufman factorization of A, using packed storage.
If uplo='U', A=PUDUTPT and A-1 is computed by solving UTPTXPU=D-1.
If uplo='L', A=PLDLTPT and A-1 is computed by solving LTPTXPL=D-1.

References

Du Croz J J and Higham N J (1992) Stability of methods for matrix inversion IMA J. Numer. Anal. 12 1–19

Parameters

Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
Specifies how A has been factorized.
uplo='U'
A=PUDUTPT, where U is upper triangular.
uplo='L'
A=PLDLTPT, where L is lower triangular.
Constraint: uplo='U' or 'L'.
2:     ap: – complex array
The dimension of the array ap must be at least max1,n×n+1/2
The factorization of A stored in packed form, as returned by nag_lapack_zsptrf (f07qr).
3:     ipiv: int64int32nag_int array
The dimension of the array ipiv must be at least max1,n
Details of the interchanges and the block structure of D, as returned by nag_lapack_zsptrf (f07qr).

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the dimension of the array ipiv.
n, the order of the matrix A.
Constraint: n0.

Output Parameters

1:     ap: – complex array
The dimension of the array ap will be max1,n×n+1/2
The factorization stores the n by n matrix A-1.
More precisely,
  • if uplo='U', the upper triangle of A-1 must be stored with element Aij in api+jj-1/2 for ij;
  • if uplo='L', the lower triangle of A-1 must be stored with element Aij in api+2n-jj-1/2 for ij.
2:     info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

   info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
W  info>0
Element _ of the diagonal is exactly zero. D is singular and the inverse of A cannot be computed.

Accuracy

The computed inverse X satisfies a bound of the form cn is a modest linear function of n, and ε is the machine precision

Further Comments

The total number of real floating-point operations is approximately 83n3.
The real analogue of this function is nag_lapack_dsptri (f07pj).

Example

This example computes the inverse of the matrix A, where
A= -0.39-0.71i 5.14-0.64i -7.86-2.96i 3.80+0.92i 5.14-0.64i 8.86+1.81i -3.52+0.58i 5.32-1.59i -7.86-2.96i -3.52+0.58i -2.83-0.03i -1.54-2.86i 3.80+0.92i 5.32-1.59i -1.54-2.86i -0.56+0.12i .  
Here A is symmetric, stored in packed form, and must first be factorized by nag_lapack_zsptrf (f07qr).
function f07qw_example


fprintf('f07qw example results\n\n');

% Get Inverse of A, where A is complex symmetric matrix such that the
% lower triangular part is stored in packed format
uplo = 'L';
n = int64(4);
ap = [ -0.39 -  0.71i,   5.14 -  0.64i,  -7.86 - 2.96i,   3.80 + 0.92i, ...
        8.86 +  1.81i,  -3.52 +  0.58i,   5.32 - 1.59i,   ...
       -2.83 -  0.03i,  -1.54 -  2.86i,  ...
       -0.56 + 0.12i];

% Factorize
[apf, ipiv, info] = f07qr( ...
                           uplo, n, ap);

% Invert
[ainv, info] = f07qw(uplo, apf, ipiv);

% Display packed inverse: Integer labels, 80 colunms wide, no indent
rlabs = {'       '};
clabs = {'       '};
ncols  = int64(80);
indent = int64(0);

[ifail] = x04dd( ...
                 uplo, 'N', n, ainv, 'Brac', ' ', 'Inverse', 'Int', rlabs, ...
                 'Int', clabs, ncols, indent);


f07qw example results

 Inverse
                      1                   2                   3
 1  ( -0.1562, -0.1014)
 2  (  0.0400,  0.1527) (  0.0946, -0.1475)
 3  (  0.0550,  0.0845) ( -0.0326, -0.1370) ( -0.1320, -0.0102)
 4  (  0.2162, -0.0742) ( -0.0995, -0.0461) ( -0.1793,  0.1183)

                      4
 1
 2
 3
 4  ( -0.2269,  0.2383)

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Chapter Contents
Chapter Introduction
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