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# NAG Toolbox: nag_lapack_dpptri (f07gj)

## Purpose

nag_lapack_dpptri (f07gj) computes the inverse of a real symmetric positive definite matrix $A$, where $A$ has been factorized by nag_lapack_dpptrf (f07gd), using packed storage.

## Syntax

[ap, info] = f07gj(uplo, n, ap)
[ap, info] = nag_lapack_dpptri(uplo, n, ap)

## Description

nag_lapack_dpptri (f07gj) is used to compute the inverse of a real symmetric positive definite matrix $A$, the function must be preceded by a call to nag_lapack_dpptrf (f07gd), which computes the Cholesky factorization of $A$, using packed storage.
If ${\mathbf{uplo}}=\text{'U'}$, $A={U}^{\mathrm{T}}U$ and ${A}^{-1}$ is computed by first inverting $U$ and then forming $\left({U}^{-1}\right){U}^{-\mathrm{T}}$.
If ${\mathbf{uplo}}=\text{'L'}$, $A=L{L}^{\mathrm{T}}$ and ${A}^{-1}$ is computed by first inverting $L$ and then forming ${L}^{-\mathrm{T}}\left({L}^{-1}\right)$.

## 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.
${\mathbf{uplo}}=\text{'U'}$
$A={U}^{\mathrm{T}}U$, where $U$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
$A=L{L}^{\mathrm{T}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2:     $\mathrm{n}$int64int32nag_int scalar
$n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathrm{ap}\left(:\right)$ – double 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 Cholesky factor of $A$ stored in packed form, as returned by nag_lapack_dpptrf (f07gd).

None.

### Output Parameters

1:     $\mathrm{ap}\left(:\right)$ – double array
The dimension of the array ap will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$
The factorization stores the $n$ by $n$ matrix ${A}^{-1}$.
More precisely,
• if ${\mathbf{uplo}}=\text{'U'}$, the upper triangle of ${A}^{-1}$ 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 ${A}^{-1}$ must be stored with element ${A}_{ij}$ in ${\mathbf{ap}}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for $i\ge j$.
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
 $κ2A=A2A-12 .$

The total number of floating-point operations is approximately $\frac{2}{3}{n}^{3}$.
The complex analogue of this function is nag_lapack_zpptri (f07gw).

## Example

This example computes the inverse of the matrix $A$, where
 $A= 4.16 -3.12 0.56 -0.10 -3.12 5.03 -0.83 1.18 0.56 -0.83 0.76 0.34 -0.10 1.18 0.34 1.18 .$
Here $A$ is symmetric positive definite, stored in packed form, and must first be factorized by nag_lapack_dpptrf (f07gd).
```function f07gj_example

fprintf('f07gj example results\n\n');

% Symmetric matrix A, lower triangular part packed in ap
uplo = 'L';
n = int64(4);
ap = [4.16 -3.12  0.56 -0.10 ...
5.03 -0.83  1.18 ...
0.76  0.34 ...
1.18];

[L, info] = f07gd( ...
uplo, n, ap);

[ainv, info] = f07gj( ...
uplo, n, L);

[ifail] = x04cc( ...
uplo, 'N', n, ainv, 'Inverse');

```
```f07gj example results

Inverse
1          2          3          4
1      0.6995
2      0.7769     1.4239
3      0.7508     1.8255     4.0688
4     -0.9340    -1.8841    -2.9342     3.4978
```

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