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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zpftri (f07ww)

## Purpose

nag_lapack_zpftri (f07ww) computes the inverse of a complex Hermitian positive definite matrix using the Cholesky factorization computed by nag_lapack_zpftrf (f07wr) stored in Rectangular Full Packed (RFP) format.

## Syntax

[ar, info] = f07ww(transr, uplo, n, ar)
[ar, info] = nag_lapack_zpftri(transr, uplo, n, ar)

## Description

nag_lapack_zpftri (f07ww) is used to compute the inverse of a complex Hermitian positive definite matrix $A$, stored in RFP format. The RFP storage format is described in Rectangular Full Packed (RFP) Storage in the F07 Chapter Introduction. The function must be preceded by a call to nag_lapack_zpftrf (f07wr), which computes the Cholesky factorization of $A$.
If ${\mathbf{uplo}}=\text{'U'}$, $A={U}^{\mathrm{H}}U$ and ${A}^{-1}$ is computed by first inverting $U$ and then forming $\left({U}^{-1}\right){U}^{-\mathrm{H}}$.
If ${\mathbf{uplo}}=\text{'L'}$, $A=L{L}^{\mathrm{H}}$ and ${A}^{-1}$ is computed by first inverting $L$ and then forming ${L}^{-\mathrm{H}}\left({L}^{-1}\right)$.

## 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 how $A$ has been factorized.
${\mathbf{uplo}}=\text{'U'}$
$A={U}^{\mathrm{H}}U$, where $U$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
$A=L{L}^{\mathrm{H}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
3:     $\mathrm{n}$int64int32nag_int scalar
$n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:     $\mathrm{ar}\left({\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$ – complex array
The Cholesky factorization of $A$ stored in RFP format, as returned by nag_lapack_zpftrf (f07wr).

None.

### Output Parameters

1:     $\mathrm{ar}\left({\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$ – complex array
The factorization stores the $n$ by $n$ matrix ${A}^{-1}$ stored in RFP format.
2:     $\mathrm{info}$int64int32nag_int scalar
${\mathbf{info}}=0$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and 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.
${\mathbf{info}}>0$
The leading minor of order $_$ is not positive definite and the factorization could not be completed. Hence $A$ itself is not positive definite. This may indicate an error in forming the matrix $A$. There is no function specifically designed to invert a Hermitian matrix stored in RFP format which is not positive definite; the matrix must be treated as a full Hermitian matrix, by calling nag_lapack_zhetri (f07mw).

## Accuracy

The computed inverse $X$ satisfies
 $XA-I2≤cnεκ2A and AX-I2≤cnεκ2A ,$
where $c\left(n\right)$ is a modest function of $n$, $\epsilon$ is the machine precision and ${\kappa }_{2}\left(A\right)$ is the condition number of $A$ defined by
 $κ2A=A2A-12 .$

The total number of real floating-point operations is approximately $\frac{8}{3}{n}^{3}$.
The real analogue of this function is nag_lapack_dpftri (f07wj).

## Example

This example computes the inverse of the matrix $A$, where
 $A= 3.23+0.00i 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58+0.00i -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09+0.00i 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29+0.00i .$
Here $A$ is Hermitian positive definite, stored in RFP format, and must first be factorized by nag_lapack_zpftrf (f07wr).
```function f07ww_example

fprintf('f07ww example results\n\n');

% Symmetric matrix in RFP format
transr = 'n';
uplo   = 'l';
ar = [ 4.09 + 0.00i  2.33 - 0.14i;
3.23 + 0.00i  4.29 + 0.00i;
1.51 + 1.92i  3.58 + 0.00i;
1.90 - 0.84i -0.23 - 1.11i;
0.42 - 2.50i -1.18 - 1.37i];
n  = int64(4);
n2 = (n*(n+1))/2;
ar  = reshape(ar,[n2,1]);

% Factorize a
[ar, info] = f07wr(transr, uplo, n, ar);

if info == 0
% Compute inverse of a
[ar, info] = f07ww( ...
transr, uplo, n, ar);
% 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 not positive definite.\n');
end

```
```f07ww example results

Inverse, lower triangle:
1                 2                 3                 4
1  ( 5.4691, 0.0000)
2  (-1.2624,-1.5491) ( 1.1024, 0.0000)
3  (-2.9746,-0.9616) ( 0.8989,-0.5672) ( 2.1589,-0.0000)
4  ( 1.1962, 2.9772) (-0.9826,-0.2566) (-1.3756,-1.4550) ( 2.2934,-0.0000)
```