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

Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zhetri (f07mw)

## Purpose

nag_lapack_zhetri (f07mw) computes the inverse of a complex Hermitian indefinite matrix A$A$, where A$A$ has been factorized by nag_lapack_zhetrf (f07mr).

## Syntax

[a, info] = f07mw(uplo, a, ipiv, 'n', n)
[a, info] = nag_lapack_zhetri(uplo, a, ipiv, 'n', n)

## Description

nag_lapack_zhetri (f07mw) 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_zhetrf (f07mr), which computes the Bunch–Kaufman factorization of A$A$.
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:     a(lda, : $:$) – complex array
The first dimension of the array a must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
Details of the factorization of A$A$, as returned by nag_lapack_zhetrf (f07mr).
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_zhetrf (f07mr).

### Optional Input Parameters

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

lda work

### Output Parameters

1:     a(lda, : $:$) – complex array
The first dimension of the array a will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
ldamax (1,n)$\mathit{lda}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The factorization stores the n$n$ by n$n$ Hermitian matrix A1${A}^{-1}$.
If uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the upper triangle of A1${A}^{-1}$ is stored in the upper triangular part of the array.
If uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the lower triangle of A1${A}^{-1}$ is stored in the lower triangular part of the array.
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: a, 4: lda, 5: ipiv, 6: work, 7: 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'${\mathbf{uplo}}=\text{'U'}$, |DUHPTXPUI|c(n)ε(|D||UH|PT|X|P|U| + |D||D1|)$|D{U}^{\mathrm{H}}{P}^{\mathrm{T}}XPU-I|\le c\left(n\right)\epsilon \left(|D||{U}^{\mathrm{H}}|{P}^{\mathrm{T}}|X|P|U|+|D||{D}^{-1}|\right)$;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, |DLHPTXPLI|c(n)ε(|D||LH|PT|X|P|L| + |D||D1|)$|D{L}^{\mathrm{H}}{P}^{\mathrm{T}}XPL-I|\le c\left(n\right)\epsilon \left(|D||{L}^{\mathrm{H}}|{P}^{\mathrm{T}}|X|P|L|+|D||{D}^{-1}|\right)$,
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_dsytri (f07mj).

## Example

```function nag_lapack_zhetri_example
uplo = 'L';
a = [complex(-1.36),  0 + 0i,  0 + 0i,  0 + 0i;
1.58 - 0.9i,  -8.87 + 0i,  0 + 0i,  0 + 0i;
2.21 + 0.21i,  -1.84 + 0.03i,  -4.63 + 0i,  0 + 0i;
3.91 - 1.5i,  -1.78 - 1.18i,  0.11 - 0.11i,  -1.84 + 0i];
[a, ipiv, info] = nag_lapack_zhetrf(uplo, a);
[aOut, info] = nag_lapack_zhetri(uplo, a, ipiv)
```
```

aOut =

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

info =

0

```
```function f07mw_example
uplo = 'L';
a = [complex(-1.36),  0 + 0i,  0 + 0i,  0 + 0i;
1.58 - 0.9i,  -8.87 + 0i,  0 + 0i,  0 + 0i;
2.21 + 0.21i,  -1.84 + 0.03i,  -4.63 + 0i,  0 + 0i;
3.91 - 1.5i,  -1.78 - 1.18i,  0.11 - 0.11i,  -1.84 + 0i];
[a, ipiv, info] = f07mr(uplo, a);
[aOut, info] = f07mw(uplo, a, ipiv)
```
```

aOut =

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

info =

0

```