NAG Toolbox: nag_lapack_zhetri (f07mw)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{uplo}$ – string (length ≥ 1)

Specifies how $A$ has been factorized.

$\mathbf{uplo}=\text{'U'}$

$A=PUD{U}^{\mathrm{H}}{P}^{\mathrm{T}}$, where $U$ is upper triangular.

$\mathbf{uplo}=\text{'L'}$

$A=PLD{L}^{\mathrm{H}}{P}^{\mathrm{T}}$, where $L$ is lower triangular.

Constraint: $\mathbf{uplo}=\text{'U'}$ or $\text{'L'}$.

2:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array

The first dimension of the array a must be at least $\max \left(1,\mathbf{n}\right)$.

The second dimension of the array a must be at least $\max \left(1,\mathbf{n}\right)$.

Details of the factorization of $A$, as returned by nag_lapack_zhetrf (f07mr).

3:     $\mathrm{ipiv}\left(:\right)$ – int64int32nag_int array

The dimension of the array ipiv must be at least $\max \left(1,\mathbf{n}\right)$

Details of the interchanges and the block structure of $D$, as returned by nag_lapack_zhetrf (f07mr).

Optional Input Parameters

1:     $\mathrm{n}$ – 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: $\mathbf{n}\ge 0$.

Output Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array

The first dimension of the array a will be $\max \left(1,\mathbf{n}\right)$.

The second dimension of the array a will be $\max \left(1,\mathbf{n}\right)$.

The factorization stores the $n$ by $n$ Hermitian matrix $A^{-1}$.

If $\mathbf{uplo}=\text{'U'}$, the upper triangle of $A^{-1}$ is stored in the upper triangular part of the array.

If $\mathbf{uplo}=\text{'L'}$, the lower triangle of $A^{-1}$ is stored in the lower triangular part of the array.

2:     $\mathrm{info}$ – int64int32nag_int scalar

$\mathbf{info}=0$ 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.

   $\mathbf{info}<0$

If $\mathbf{info}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

W  $\mathbf{info}>0$

Element $\_$ of the diagonal is exactly zero. $D$ is singular and the inverse of $A$ cannot be computed.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f07mw_example

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