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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zgetri (f07aw)

## Purpose

nag_lapack_zgetri (f07aw) computes the inverse of a complex matrix $A$, where $A$ has been factorized by nag_lapack_zgetrf (f07ar).

## Syntax

[a, info] = f07aw(a, ipiv, 'n', n)
[a, info] = nag_lapack_zgetri(a, ipiv, 'n', n)

## Description

nag_lapack_zgetri (f07aw) is used to compute the inverse of a complex matrix $A$, the function must be preceded by a call to nag_lapack_zgetrf (f07ar), which computes the $LU$ factorization of $A$ as $A=PLU$. The inverse of $A$ is computed by forming ${U}^{-1}$ and then solving the equation $XPL={U}^{-1}$ 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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $LU$ factorization of $A$, as returned by nag_lapack_zgetrf (f07ar).
2:     $\mathrm{ipiv}\left(:\right)$int64int32nag_int array
The dimension of the array ipiv must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The pivot indices, as returned by nag_lapack_zgetrf (f07ar).

### 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 $n$ matrix ${A}^{-1}$.
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 zero. $U$ is singular, and the inverse of $A$ cannot be computed.

## Accuracy

The computed inverse $X$ satisfies a bound of the form:
 $XA-I≤cnεXPLU ,$
where $c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision.
Note that a similar bound for $\left|AX-I\right|$ cannot be guaranteed, although it is almost always satisfied. See Du Croz and Higham (1992).

The total number of real floating-point operations is approximately $\frac{16}{3}{n}^{3}$.
The real analogue of this function is nag_lapack_dgetri (f07aj).

## Example

This example computes the inverse of the matrix $A$, where
 $A= -1.34+2.55i 0.28+3.17i -6.39-2.20i 0.72-0.92i -0.17-1.41i 3.31-0.15i -0.15+1.34i 1.29+1.38i -3.29-2.39i -1.91+4.42i -0.14-1.35i 1.72+1.35i 2.41+0.39i -0.56+1.47i -0.83-0.69i -1.96+0.67i .$
Here $A$ is nonsymmetric and must first be factorized by nag_lapack_zgetrf (f07ar).
```function f07aw_example

fprintf('f07aw example results\n\n');

a = [-1.34 + 2.55i,  0.28 + 3.17i, -6.39 - 2.20i,  0.72 - 0.92i;
-0.17 - 1.41i,  3.31 - 0.15i, -0.15 + 1.34i,  1.29 + 1.38i;
-3.29 - 2.39i, -1.91 + 4.42i, -0.14 - 1.35i,  1.72 + 1.35i;
2.41 + 0.39i, -0.56 + 1.47i, -0.83 - 0.69i, -1.96 + 0.67i];

% Factorize a
[LU, ipiv, info] = f07ar(a);

% Compute inverse of a
[ainv, info] = f07aw(LU, ipiv);

disp('Inverse');
disp(ainv);

```
```f07aw example results

Inverse
0.0757 - 0.4324i   1.6512 - 3.1342i   1.2663 + 0.0418i   3.8181 + 1.1195i
-0.1942 + 0.0798i  -1.1900 - 0.1426i  -0.2401 - 0.5889i  -0.0101 - 1.4969i
-0.0957 - 0.0491i   0.7371 - 0.4290i   0.3224 + 0.0776i   0.6887 + 0.7891i
0.3702 - 0.5040i   3.7253 - 3.1813i   1.7014 + 0.7267i   3.9367 + 3.3255i

```