Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zpotri (f07fw)

## Purpose

nag_lapack_zpotri (f07fw) computes the inverse of a complex Hermitian positive definite matrix $A$, where $A$ has been factorized by nag_lapack_zpotrf (f07fr).

## Syntax

[a, info] = f07fw(uplo, a, 'n', n)
[a, info] = nag_lapack_zpotri(uplo, a, 'n', n)

## Description

nag_lapack_zpotri (f07fw) is used to compute the inverse of a complex Hermitian positive definite matrix $A$, the function must be preceded by a call to nag_lapack_zpotrf (f07fr), 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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{uplo}$ – 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 $\text{'L'}$.
2:     $\mathrm{a}\left(\mathit{lda},:\right)$ – 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 upper triangular matrix $U$ if ${\mathbf{uplo}}=\text{'U'}$ or the lower triangular matrix $L$ if ${\mathbf{uplo}}=\text{'L'}$, as returned by nag_lapack_zpotrf (f07fr).

### 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
$U$ stores the upper triangle of ${A}^{-1}$ if ${\mathbf{uplo}}=\text{'U'}$; $L$ stores the lower triangle of ${A}^{-1}$ if ${\mathbf{uplo}}=\text{'L'}$.
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 the Cholesky factor is zero; the Cholesky factor is singular and the inverse of $A$ cannot be computed.

## 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
 $κ2A=A2A-12 .$

## Further Comments

The total number of real floating-point operations is approximately $\frac{8}{3}{n}^{3}$.
The real analogue of this function is nag_lapack_dpotri (f07fj).

## 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 and must first be factorized by nag_lapack_zpotrf (f07fr).
```function f07fw_example

fprintf('f07fw example results\n\n');

% Lower triangular part of Hermitian matrix A
uplo = 'Lower';
a = [ 3.23 + 0i,     0    + 0i,     0    + 0i,     0    + 0i;
1.51 + 1.92i,  3.58 + 0i,     0    + 0i,     0    + 0i;
1.90 - 0.84i, -0.23 - 1.11i,  4.09 + 0i,     0    + 0i;
0.42 - 2.50i, -1.18 - 1.37i,  2.33 + 0.14i,  4.29 + 0i];

% Factorize
[L, info] = f07fr( ...
uplo, a);

% Invert
[ainv, info] = f07fw( ...
uplo, L);

[ifail] = x04da( ...
uplo, 'Non-unit', ainv, 'Inverse');

```
```f07fw example results

Inverse
1          2          3          4
1      5.4691
0.0000

2     -1.2624     1.1024
-1.5491     0.0000

3     -2.9746     0.8989     2.1589
-0.9616    -0.5672     0.0000

4      1.1962    -0.9826    -1.3756     2.2934
2.9772    -0.2566    -1.4550     0.0000
```

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2015