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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zhptri (f07pw)

## Purpose

nag_lapack_zhptri (f07pw) computes the inverse of a complex Hermitian indefinite matrix A$A$, where A$A$ has been factorized by nag_lapack_zhptrf (f07pr), using packed storage.

## Syntax

[ap, info] = f07pw(uplo, ap, ipiv, 'n', n)
[ap, info] = nag_lapack_zhptri(uplo, ap, ipiv, 'n', n)

## Description

nag_lapack_zhptri (f07pw) is used to compute the inverse of a complex Hermitian indefinite matrix A$A$, the function must be preceded by a call to nag_lapack_zhptrf (f07pr), which computes the Bunch–Kaufman factorization of A$A$, using packed storage.
If uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, A = PUDUHPT$A=PUD{U}^{\mathrm{H}}{P}^{\mathrm{T}}$ and A1${A}^{-1}$ is computed by solving UHPTXPU = D1${U}^{\mathrm{H}}{P}^{\mathrm{T}}XPU={D}^{-1}$ for X$X$.
If uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, A = PLDLHPT$A=PLD{L}^{\mathrm{H}}{P}^{\mathrm{T}}$ and A1${A}^{-1}$ is computed by solving LHPTXPL = D1${L}^{\mathrm{H}}{P}^{\mathrm{T}}XPL={D}^{-1}$ for X$X$.

## 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:     uplo – string (length ≥ 1)
Specifies how A$A$ has been factorized.
uplo = 'U'${\mathbf{uplo}}=\text{'U'}$
A = PUDUHPT$A=PUD{U}^{\mathrm{H}}{P}^{\mathrm{T}}$, where U$U$ is upper triangular.
uplo = 'L'${\mathbf{uplo}}=\text{'L'}$
A = PLDLHPT$A=PLD{L}^{\mathrm{H}}{P}^{\mathrm{T}}$, where L$L$ is lower triangular.
Constraint: uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or 'L'$\text{'L'}$.
2:     ap( : $:$) – complex array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
The factorization of A$A$ stored in packed form, as returned by nag_lapack_zhptrf (f07pr).
3:     ipiv( : $:$) – int64int32nag_int array
Note: the dimension of the array ipiv must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
Details of the interchanges and the block structure of D$D$, as returned by nag_lapack_zhptrf (f07pr).

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The dimension of the array ipiv.
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.

work

### Output Parameters

1:     ap( : $:$) – complex array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
The factorization stores the n$n$ by n$n$ matrix A1${A}^{-1}$.
More precisely,
• if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the upper triangle of A1${A}^{-1}$ must be stored with element Aij${A}_{ij}$ in ap(i + j(j1) / 2)${\mathbf{ap}}\left(i+j\left(j-1\right)/2\right)$ for ij$i\le j$;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the lower triangle of A1${A}^{-1}$ must be stored with element Aij${A}_{ij}$ in ap(i + (2nj)(j1) / 2)${\mathbf{ap}}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for ij$i\ge j$.
2:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [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.

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ had an illegal value on entry. The parameters are numbered as follows:
1: uplo, 2: n, 3: ap, 4: ipiv, 5: work, 6: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.
W INFO > 0${\mathbf{INFO}}>0$
If info = i${\mathbf{info}}=i$, d(i,i)$d\left(i,i\right)$ is exactly zero; D$D$ is singular and the inverse of A$A$ cannot be computed.

## Accuracy

The computed inverse X$X$ satisfies a bound of the form
• if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, |DUTPTXPUI|c(n)ε(|D||UT|PT|X|P|U| + |D||D1|)$|D{U}^{\mathrm{T}}{P}^{\mathrm{T}}XPU-I|\le c\left(n\right)\epsilon \left(|D||{U}^{\mathrm{T}}|{P}^{\mathrm{T}}|X|P|U|+|D||{D}^{-1}|\right)$;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, |DLTPTXPLI|c(n)ε(|D||LT|PT|X|P|L| + |D||D1|)$|D{L}^{\mathrm{T}}{P}^{\mathrm{T}}XPL-I|\le c\left(n\right)\epsilon \left(|D||{L}^{\mathrm{T}}|{P}^{\mathrm{T}}|X|P|L|+|D||{D}^{-1}|\right)$,
c(n)$c\left(n\right)$ is a modest linear function of n$n$, and ε$\epsilon$ is the machine precision

The total number of real floating point operations is approximately (8/3)n3$\frac{8}{3}{n}^{3}$.
The real analogue of this function is nag_lapack_dsptri (f07pj).

## Example

```function nag_lapack_zhptri_example
uplo = 'L';
ap = [-1.36;
3.91 - 1.5i;
0.3100287981271241 + 0.04333020743962702i;
-0.1518120207240102 + 0.3742958425613706i;
-1.84 + 0i;
0.5637050486508776 + 0.2850349501519716i;
0.339658279960361 + 0.03031451811355637i;
-5.417624387291579 + 0i;
0.2997244646075835 + 0.1578268372785777i;
-7.102809895801842 + 0i];
ipiv = [int64(-4);-4;3;4];
[apOut, info] = nag_lapack_zhptri(uplo, ap, ipiv)
```
```

apOut =

0.0826 + 0.0000i
-0.0335 + 0.0440i
0.0603 - 0.0105i
0.2391 - 0.0926i
-0.1408 + 0.0000i
0.0422 - 0.0222i
0.0304 + 0.0203i
-0.2007 + 0.0000i
0.0982 - 0.0635i
0.0073 + 0.0000i

info =

0

```
```function f07pw_example
uplo = 'L';
ap = [-1.36;
3.91 - 1.5i;
0.3100287981271241 + 0.04333020743962702i;
-0.1518120207240102 + 0.3742958425613706i;
-1.84 + 0i;
0.5637050486508776 + 0.2850349501519716i;
0.339658279960361 + 0.03031451811355637i;
-5.417624387291579 + 0i;
0.2997244646075835 + 0.1578268372785777i;
-7.102809895801842 + 0i];
ipiv = [int64(-4);-4;3;4];
[apOut, info] = f07pw(uplo, ap, ipiv)
```
```

apOut =

0.0826 + 0.0000i
-0.0335 + 0.0440i
0.0603 - 0.0105i
0.2391 - 0.0926i
-0.1408 + 0.0000i
0.0422 - 0.0222i
0.0304 + 0.0203i
-0.2007 + 0.0000i
0.0982 - 0.0635i
0.0073 + 0.0000i

info =

0

```