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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_ztptri (f07uw)

## Purpose

nag_lapack_ztptri (f07uw) computes the inverse of a complex triangular matrix, using packed storage.

## Syntax

[ap, info] = f07uw(uplo, diag, n, ap)
[ap, info] = nag_lapack_ztptri(uplo, diag, n, ap)

## Description

nag_lapack_ztptri (f07uw) forms the inverse of a complex triangular matrix $A$, using packed storage. 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:     $\mathrm{n}$int64int32nag_int scalar
$n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:     $\mathrm{ap}\left(:\right)$ – complex array
The dimension of the array ap must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$
The $n$ by $n$ triangular matrix $A$, packed by columns.
More precisely,
• if ${\mathbf{uplo}}=\text{'U'}$, the upper triangle of $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{ap}}\left(i+j\left(j-1\right)/2\right)$ for $i\le j$;
• if ${\mathbf{uplo}}=\text{'L'}$, the lower triangle of $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{ap}}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for $i\ge j$.
If ${\mathbf{diag}}=\text{'U'}$, the diagonal elements of $A$ are assumed to be $1$, and are not referenced; the same storage scheme is used whether ${\mathbf{diag}}=\text{'N'}$ or ‘U’.

None.

### Output Parameters

1:     $\mathrm{ap}\left(:\right)$ – complex array
The dimension of the array ap will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$
$A$ 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).

The total number of real floating-point operations is approximately $\frac{4}{3}{n}^{3}$.
The real analogue of this function is nag_lapack_dtptri (f07uj).

## Example

This example computes the inverse of the matrix $A$, where
 $A= 4.78+4.56i 0.00+0.00i 0.00+0.00i 0.00+0.00i 2.00-0.30i -4.11+1.25i 0.00+0.00i 0.00+0.00i 2.89-1.34i 2.36-4.25i 4.15+0.80i 0.00+0.00i -1.89+1.15i 0.04-3.69i -0.02+0.46i 0.33-0.26i ,$
using packed storage.
```function f07uw_example

fprintf('f07uw example results\n\n');

% Invert A, where A is Complex Lower triangular and packed.
n = int64(4);
ap = [ 4.78 + 4.56i;  2.00 - 0.30i;  2.89 - 1.34i; -1.89 + 1.15i;
-4.11 + 1.25i;  2.36 - 4.25i;  0.04 - 3.69i;
4.15 + 0.80i; -0.02 + 0.46i;
0.33 - 0.26i];

% Invert
uplo = 'L';
diag = 'N';
[ainv, info] = f07uw(uplo, diag, n, ap);

[ifail] = x04dc( ...
uplo, 'Non-unit', n, ainv, 'Inverse');

```
```f07uw example results

Inverse
1          2          3          4
1      0.1095
-0.1045

2      0.0582    -0.2227
-0.0411    -0.0677

3      0.0032     0.1538     0.2323
0.1905    -0.2192    -0.0448

4      0.7602     1.6184     0.1289     1.8697
0.2814    -1.4346    -0.2250     1.4731
```