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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_ztftri (f07wx)

## Purpose

nag_lapack_ztftri (f07wx) computes the inverse of a complex triangular matrix stored in Rectangular Full Packed (RFP) format.

## Syntax

[ar, info] = f07wx(transr, uplo, diag, n, ar)
[ar, info] = nag_lapack_ztftri(transr, uplo, diag, n, ar)

## Description

nag_lapack_ztftri (f07wx) forms the inverse of a complex triangular matrix $A$, stored using RFP format. The RFP storage format is described in Rectangular Full Packed (RFP) Storage in the F07 Chapter Introduction. 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
Gustavson F G, Waśniewski J, Dongarra J J and Langou J (2010) Rectangular full packed format for Cholesky's algorithm: factorization, solution, and inversion ACM Trans. Math. Software 37, 2

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{transr}$ – string (length ≥ 1)
Specifies whether the normal RFP representation of $A$ or its conjugate transpose is stored.
${\mathbf{transr}}=\text{'N'}$
The matrix $A$ is stored in normal RFP format.
${\mathbf{transr}}=\text{'C'}$
The conjugate transpose of the RFP representation of the matrix $A$ is stored.
Constraint: ${\mathbf{transr}}=\text{'N'}$ or $\text{'C'}$.
2:     $\mathrm{uplo}$ – 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 $\text{'L'}$.
3:     $\mathrm{diag}$ – 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 $\text{'U'}$.
4:     $\mathrm{n}$int64int32nag_int scalar
$n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
5:     $\mathrm{ar}\left({\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$ – complex array
The upper or lower triangular part (as specified by uplo) of the $n$ by $n$ Hermitian matrix $A$, in either normal or transposed RFP format (as specified by transr). The storage format is described in detail in Rectangular Full Packed (RFP) Storage in the F07 Chapter Introduction.

None.

### Output Parameters

1:     $\mathrm{ar}\left({\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$ – complex array
$A$ stores ${A}^{-1}$, in the same storage format as $A$.
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  ${\mathbf{info}}>0$
Diagonal element $_$ of $A$ 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_dtftri (f07wk).

## Example

This example computes the inverse of the matrix $A$, where
 $A= 4.78+4.56i 0.00+0.00i 0.00+0.00i 0.00+0.00i 2.00-0.30i -4.11+1.25i 0.00+0.00i 0.00+0.00i 2.89-1.34i 2.36-4.25i 4.15+0.80i 0.00+0.00i -1.89+1.15i 0.04-3.69i -0.02+0.46i 0.33-0.26i$
and is stored using RFP format.
```function f07wx_example

fprintf('f07wx example results\n\n');

% Symmetric matrix in RFP format
transr = 'n';
uplo   = 'l';
diag   = 'n';
ar = [ 4.15 - 0.80i  -0.02 - 0.46i;
4.78 + 4.56i   0.33 + 0.26i;
2.00 - 0.30i  -4.11 + 1.25i;
2.89 - 1.34i   2.36 - 4.25i;
-1.89 + 1.15i   0.04 - 3.69i];
n  = int64(4);
n2 = (n*(n+1))/2;
ar  = reshape(ar,[n2,1]);

% Compute inverse of a
[ar, info] = f07wx( ...
transr, uplo, diag, n, ar);

if info == 0
% Convert inverse to full array form for display
[a, info] = f01vh( ...
transr, uplo, n, ar);
fprintf('\n');
ncols  = int64(80);
indent = int64(0);
form   = 'f7.4';
title  = 'Inverse, lower triangle:';
diag   = 'n';
[ifail] = x04db( ...
uplo, diag, a, 'brackets', form, title, ...
'int', 'int', ncols, indent);
else
fprintf('\na is singular.\n');
end

```
```f07wx example results

Inverse, lower triangle:
1                 2                 3                 4
1  ( 0.1095,-0.1045)
2  ( 0.0582,-0.0411) (-0.2227,-0.0677)
3  ( 0.0032, 0.1905) ( 0.1538,-0.2192) ( 0.2323,-0.0448)
4  ( 0.7602, 0.2814) ( 1.6184,-1.4346) ( 0.1289,-0.2250) ( 1.8697, 1.4731)
```