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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zsytri (f07nw)

## 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 ${\mathbf{uplo}}=\text{'U'}$, $A=PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$ and ${A}^{-1}$ is computed by solving ${U}^{\mathrm{T}}{P}^{\mathrm{T}}XPU={D}^{-1}$ for $X$.
If ${\mathbf{uplo}}=\text{'L'}$, $A=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$ and ${A}^{-1}$ is computed by solving ${L}^{\mathrm{T}}{P}^{\mathrm{T}}XPL={D}^{-1}$ 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:     $\mathrm{uplo}$ – string (length ≥ 1)
Specifies how $A$ has been factorized.
${\mathbf{uplo}}=\text{'U'}$
$A=PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$, where $U$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
$A=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$, 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
Details of the factorization of $A$, as returned by nag_lapack_zsytrf (f07nr).
3:     $\mathrm{ipiv}\left(:\right)$int64int32nag_int array
The dimension of the array ipiv must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
Details of the interchanges and the block structure of $D$, as returned by nag_lapack_zsytrf (f07nr).

### 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The factorization stores the $n$ by $n$ symmetric 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

The computed inverse $X$ satisfies a bound of the form
• if ${\mathbf{uplo}}=\text{'U'}$, $\left|D{U}^{\mathrm{T}}{P}^{\mathrm{T}}XPU-I\right|\le c\left(n\right)\epsilon \left(\left|D\right|\left|{U}^{\mathrm{T}}\right|{P}^{\mathrm{T}}\left|X\right|P\left|U\right|+\left|D\right|\left|{D}^{-1}\right|\right)$;
• if ${\mathbf{uplo}}=\text{'L'}$, $\left|D{L}^{\mathrm{T}}{P}^{\mathrm{T}}XPL-I\right|\le c\left(n\right)\epsilon \left(\left|D\right|\left|{L}^{\mathrm{T}}\right|{P}^{\mathrm{T}}\left|X\right|P\left|L\right|+\left|D\right|\left|{D}^{-1}\right|\right)$,
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision

The total number of real floating-point operations is approximately $\frac{8}{3}{n}^{3}$.
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
```