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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zsytri (f07nw)

## Purpose

nag_lapack_zsytri (f07nw) computes the inverse of a complex symmetric matrix A$A$, where A$A$ has been factorized by nag_lapack_zsytrf (f07nr).

## Syntax

[a, info] = f07nw(uplo, a, ipiv, 'n', n)
[a, info] = nag_lapack_zsytri(uplo, a, ipiv, 'n', n)

## Description

nag_lapack_zsytri (f07nw) is used to compute the inverse of a complex symmetric matrix A$A$, the function must be preceded by a call to nag_lapack_zsytrf (f07nr), which computes the Bunch–Kaufman factorization of A$A$.
If uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, A = PUDUTPT$A=PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$ and A1${A}^{-1}$ is computed by solving UTPTXPU = D1${U}^{\mathrm{T}}{P}^{\mathrm{T}}XPU={D}^{-1}$ for X$X$.
If uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, A = PLDLTPT$A=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$ and A1${A}^{-1}$ is computed by solving LTPTXPL = D1${L}^{\mathrm{T}}{P}^{\mathrm{T}}XPL={D}^{-1}$ 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 = PUDUTPT$A=PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$, where U$U$ is upper triangular.
uplo = 'L'${\mathbf{uplo}}=\text{'L'}$
A = PLDLTPT$A=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$, 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_zsytrf (f07nr).
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_zsytrf (f07nr).

### 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$n$ symmetric 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'${\mathbf{uplo}}=\text{'U'}$, |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_dsytri (f07mj).

## Example

```function nag_lapack_zsytri_example
uplo = 'L';
a = [ -0.39 - 0.71i,  0 + 0i,  0 + 0i,  0 + 0i;
5.14 - 0.64i,  8.86 + 1.81i,  0 + 0i,  0 + 0i;
-7.86 - 2.96i,  -3.52 + 0.58i,  -2.83 - 0.03i,  0 + 0i;
3.8 + 0.92i,  5.32 - 1.59i,  -1.54 - 2.86i,  -0.56 + 0.12i];
[a, ipiv, info] = nag_lapack_zsytrf(uplo, a);
[aOut, info] = nag_lapack_zsytri(uplo, a, ipiv)
```
```

aOut =

-0.1562 - 0.1014i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
0.0400 + 0.1527i   0.0946 - 0.1475i   0.0000 + 0.0000i   0.0000 + 0.0000i
0.0550 + 0.0845i  -0.0326 - 0.1370i  -0.1320 - 0.0102i   0.0000 + 0.0000i
0.2162 - 0.0742i  -0.0995 - 0.0461i  -0.1793 + 0.1183i  -0.2269 + 0.2383i

info =

0

```
```function f07nw_example
uplo = 'L';
a = [ -0.39 - 0.71i,  0 + 0i,  0 + 0i,  0 + 0i;
5.14 - 0.64i,  8.86 + 1.81i,  0 + 0i,  0 + 0i;
-7.86 - 2.96i,  -3.52 + 0.58i,  -2.83 - 0.03i,  0 + 0i;
3.8 + 0.92i,  5.32 - 1.59i,  -1.54 - 2.86i,  -0.56 + 0.12i];
[a, ipiv, info] = f07nr(uplo, a);
[aOut, info] = f07nw(uplo, a, ipiv)
```
```

aOut =

-0.1562 - 0.1014i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
0.0400 + 0.1527i   0.0946 - 0.1475i   0.0000 + 0.0000i   0.0000 + 0.0000i
0.0550 + 0.0845i  -0.0326 - 0.1370i  -0.1320 - 0.0102i   0.0000 + 0.0000i
0.2162 - 0.0742i  -0.0995 - 0.0461i  -0.1793 + 0.1183i  -0.2269 + 0.2383i

info =

0

```