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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zhetri (f07mw)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_zhetri (f07mw) computes the inverse of a complex Hermitian indefinite matrix A, where A has been factorized by nag_lapack_zhetrf (f07mr).

Syntax

[a, info] = f07mw(uplo, a, ipiv, 'n', n)
[a, info] = nag_lapack_zhetri(uplo, a, ipiv, 'n', n)

Description

nag_lapack_zhetri (f07mw) is used to compute the inverse of a complex Hermitian indefinite matrix A, the function must be preceded by a call to nag_lapack_zhetrf (f07mr), which computes the Bunch–Kaufman factorization of A.
If uplo='U', A=PUDUHPT and A-1 is computed by solving UHPTXPU=D-1 for X.
If uplo='L', A=PLDLHPT and A-1 is computed by solving LHPTXPL=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=PUDUHPT, where U is upper triangular.
uplo='L'
A=PLDLHPT, 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_zhetrf (f07mr).
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_zhetrf (f07mr).

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 Hermitian 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= -1.36+0.00i 1.58+0.90i 2.21-0.21i 3.91+1.50i 1.58-0.90i -8.87+0.00i -1.84-0.03i -1.78+1.18i 2.21+0.21i -1.84+0.03i -4.63+0.00i 0.11+0.11i 3.91-1.50i -1.78-1.18i 0.11-0.11i -1.84+0.00i .  
Here A is Hermitian indefinite and must first be factorized by nag_lapack_zhetrf (f07mr).
function f07mw_example


fprintf('f07mw example results\n\n');

% Hermitian indefinite matrix A (Lower triangular part stored)
uplo = 'L';
a = [-1.36 + 0i,     0    + 0i,     0    + 0i,      0    + 0i;
      1.58 - 0.90i, -8.87 + 0i,     0    + 0i,      0    + 0i;
      2.21 + 0.21i, -1.84 + 0.03i, -4.63 + 0i,      0    + 0i;
      3.91 - 1.50i, -1.78 - 1.18i,  0.11 - 0.11i,  -1.84 + 0i];

% Factorize
[af, ipiv, info] = f07mr( ...
                          uplo, a);

% Invert
[ainv, info] = f07mw( ...
                      uplo, af, ipiv);

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


f07mw example results

 Inverse
          1       2       3       4
 1   0.0826
     0.0000

 2  -0.0335 -0.1408
     0.0440  0.0000

 3   0.0603  0.0422 -0.2007
    -0.0105 -0.0222  0.0000

 4   0.2391  0.0304  0.0982  0.0073
    -0.0926  0.0203 -0.0635  0.0000

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