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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zsytri (f07nw)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_zsytri (f07nw) computes the inverse of a complex symmetric matrix A, where A has been factorized by nag_lapack_zsytrf (f07nr).

Syntax

[a, info] = f07nw(uplo, a, ipiv, 'n', n)
[a, info] = nag_lapack_zsytri(uplo, a, ipiv, 'n', n)

Description

nag_lapack_zsytri (f07nw) is used to compute the inverse of a complex symmetric matrix A, the function must be preceded by a call to nag_lapack_zsytrf (f07nr), which computes the Bunch–Kaufman factorization of A.
If uplo='U', A=PUDUTPT and A-1 is computed by solving UTPTXPU=D-1 for X.
If uplo='L', A=PLDLTPT and A-1 is computed by solving LTPTXPL=D-1 for X.

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:     alda: – complex array
The first dimension of the array a must be at least max1,n.
The second dimension of the array a must be at least max1,n.
Details of the factorization of A, as returned by nag_lapack_zsytrf (f07nr).
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_zsytrf (f07nr).

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the array a and the second dimension of the arrays a, ipiv.
n, the order of the matrix A.
Constraint: n0.

Output Parameters

1:     alda: – complex array
The first dimension of the array a will be max1,n.
The second dimension of the array a will be max1,n.
The factorization stores the n by n symmetric matrix A-1.
If uplo='U', the upper triangle of A-1 is stored in the upper triangular part of the array.
If uplo='L', the lower triangle of A-1 is stored in the lower triangular part of the array.
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_dsytri (f07mj).

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 and must first be factorized by nag_lapack_zsytrf (f07nr).
function f07nw_example


fprintf('f07nw example results\n\n');

% Symmetric indefinite matrix A (Upper triangular part stored)
uplo = 'L';
a = [-0.39 - 0.71i,  0    + 0i,     0    + 0i,     0    + 0i;
      5.14 - 0.64i,  8.86 + 1.81i,  0    + 0i,     0    + 0i;
     -7.86 - 2.96i, -3.52 + 0.58i, -2.83 - 0.03i,  0    + 0i;
      3.80 + 0.92i,  5.32 - 1.59i, -1.54 - 2.86i, -0.56 + 0.12i];

% Factorize
[af, ipiv, info] = f07nr( ...
                          uplo, a);

% Invert
[ainv, info] = f07nw( ...
                      uplo, af, ipiv);

[ifail] = x04da( ...
                 uplo, 'Non-unit', ainv, 'Inverse');


f07nw example results

 Inverse
          1       2       3       4
 1  -0.1562
    -0.1014

 2   0.0400  0.0946
     0.1527 -0.1475

 3   0.0550 -0.0326 -0.1320
     0.0845 -0.1370 -0.0102

 4   0.2162 -0.0995 -0.1793 -0.2269
    -0.0742 -0.0461  0.1183  0.2383

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Chapter Contents
Chapter Introduction
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