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# NAG Toolbox: nag_lapack_ztrtri (f07tw)

## Purpose

nag_lapack_ztrtri (f07tw) computes the inverse of a complex triangular matrix.

## Syntax

[a, info] = f07tw(uplo, diag, a, 'n', n)
[a, info] = nag_lapack_ztrtri(uplo, diag, a, 'n', n)

## Description

nag_lapack_ztrtri (f07tw) forms the inverse of a complex triangular matrix $A$. Note that the inverse of an upper (lower) triangular matrix is also upper (lower) triangular.

## 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 whether $A$ is upper or lower triangular.
${\mathbf{uplo}}=\text{'U'}$
$A$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
$A$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2:     $\mathrm{diag}$ – string (length ≥ 1)
Indicates whether $A$ is a nonunit or unit triangular matrix.
${\mathbf{diag}}=\text{'N'}$
$A$ is a nonunit triangular matrix.
${\mathbf{diag}}=\text{'U'}$
$A$ is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be $1$.
Constraint: ${\mathbf{diag}}=\text{'N'}$ or $\text{'U'}$.
3:     $\mathrm{a}\left(\mathit{lda},:\right)$ – 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 $n$ by $n$ triangular matrix $A$.
• If ${\mathbf{uplo}}=\text{'U'}$, $a$ is upper triangular and the elements of the array below the diagonal are not referenced.
• If ${\mathbf{uplo}}=\text{'L'}$, $a$ is lower triangular and the elements of the array above the diagonal are not referenced.
• If ${\mathbf{diag}}=\text{'U'}$, the diagonal elements of $a$ are assumed to be $1$, and are not referenced.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the first dimension of the array a and the second dimension of the array a.
$n$, 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
$A$ stores ${A}^{-1}$, using the same storage format as described above.
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. $A$ is singular its inverse cannot be computed.

## Accuracy

The computed inverse $X$ satisfies
 $XA-I≤cnεXA ,$
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.
The computed inverse satisfies the forward error bound
 $X-A-1≤cnεA-1AX .$
See Du Croz and Higham (1992).

## Further Comments

The total number of real floating-point operations is approximately $\frac{4}{3}{n}^{3}$.
The real analogue of this function is nag_lapack_dtrtri (f07tj).

## Example

This example computes the inverse of the matrix $A$, where
 $A= 4.78+4.56i 0.00+0.00i 0.00+0.00i 0.00+0.00i 2.00-0.30i -4.11+1.25i 0.00+0.00i 0.00+0.00i 2.89-1.34i 2.36-4.25i 4.15+0.80i 0.00+0.00i -1.89+1.15i 0.04-3.69i -0.02+0.46i 0.33-0.26i .$
```function f07tw_example

fprintf('f07tw example results\n\n');

% Invert A, where A is Lower triangular
a = [  4.78 +  4.56i,   0    +  0i,     0    + 0i,    0    + 0i;
2.00 -  0.30i,  -4.11 +  1.25i,  0    + 0i,    0    + 0i;
2.89 -  1.34i,   2.36 -  4.25i,  4.15 + 0.8i,  0    + 0i;
-1.89 +  1.15i,   0.04 -  3.69i, -0.02 + 0.46i, 0.33 - 0.26i];

% Invert
uplo = 'L';
diag = 'N';
[ainv, info] = f07tw(uplo, diag, a);

% Display inverse
disp('Inverse');
disp(ainv);

```
```f07tw example results

Inverse
0.1095 - 0.1045i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
0.0582 - 0.0411i  -0.2227 - 0.0677i   0.0000 + 0.0000i   0.0000 + 0.0000i
0.0032 + 0.1905i   0.1538 - 0.2192i   0.2323 - 0.0448i   0.0000 + 0.0000i
0.7602 + 0.2814i   1.6184 - 1.4346i   0.1289 - 0.2250i   1.8697 + 1.4731i

```

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