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# NAG Toolbox: nag_lapack_ztrtri (f07tw)

## Purpose

nag_lapack_ztrtri (f07tw) computes the inverse of a complex triangular matrix.

## Syntax

[a, info] = f07tw(uplo, diag, a, 'n', n)
[a, info] = nag_lapack_ztrtri(uplo, diag, a, 'n', n)

## Description

nag_lapack_ztrtri (f07tw) forms the inverse of a complex triangular matrix A$A$. Note that the inverse of an upper (lower) triangular matrix is also upper (lower) triangular.

## 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 whether A$A$ is upper or lower triangular.
uplo = 'U'${\mathbf{uplo}}=\text{'U'}$
A$A$ is upper triangular.
uplo = 'L'${\mathbf{uplo}}=\text{'L'}$
A$A$ is lower triangular.
Constraint: uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or 'L'$\text{'L'}$.
2:     diag – string (length ≥ 1)
Indicates whether A$A$ is a nonunit or unit triangular matrix.
diag = 'N'${\mathbf{diag}}=\text{'N'}$
A$A$ is a nonunit triangular matrix.
diag = 'U'${\mathbf{diag}}=\text{'U'}$
A$A$ is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be 1$1$.
Constraint: diag = 'N'${\mathbf{diag}}=\text{'N'}$ or 'U'$\text{'U'}$.
3:     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)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The n$n$ by n$n$ triangular matrix A$A$.
• If uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, a$a$ is upper triangular and the elements of the array below the diagonal are not referenced.
• If uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, a$a$ is lower triangular and the elements of the array above the diagonal are not referenced.
• If diag = 'U'${\mathbf{diag}}=\text{'U'}$, the diagonal elements of a$a$ are assumed to be 1$1$, and are not referenced.

### Optional Input Parameters

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

lda

### 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)$\mathit{lda}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
A$A$ stores A1${A}^{-1}$, using the same storage format as described above.
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: diag, 3: n, 4: a, 5: lda, 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${\mathbf{info}}=i$, a(i,i)$a\left(i,i\right)$ is exactly zero; A$A$ is singular and its inverse cannot be computed.

## Accuracy

The computed inverse X$X$ satisfies
 |XA − I| ≤ c(n)ε|X||A| , $|XA-I|≤c(n)ε|X||A| ,$
where c(n)$c\left(n\right)$ is a modest linear function of n$n$, and ε$\epsilon$ is the machine precision.
Note that a similar bound for |AXI|$|AX-I|$ cannot be guaranteed, although it is almost always satisfied.
The computed inverse satisfies the forward error bound
 |X − A − 1| ≤ c(n)ε|A − 1||A||X| . $|X-A-1|≤c(n)ε|A-1||A||X| .$
See Du Croz and Higham (1992).

The total number of real floating point operations is approximately (4/3)n3$\frac{4}{3}{n}^{3}$.
The real analogue of this function is nag_lapack_dtrtri (f07tj).

## Example

```function nag_lapack_ztrtri_example
uplo = 'L';
diag = 'N';
a = [ 4.78 + 4.56i,  0 + 0i,  0 + 0i,  0 + 0i;
2 - 0.3i,  -4.11 + 1.25i,  0 + 0i,  0 + 0i;
2.89 - 1.34i,  2.36 - 4.25i,  4.15 + 0.8i,  0 + 0i;
-1.89 + 1.15i,  0.04 - 3.69i,  -0.02 + 0.46i,  0.33 - 0.26i];
[aOut, info] = nag_lapack_ztrtri(uplo, diag, a)
```
```

aOut =

0.1095 - 0.1045i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
0.0582 - 0.0411i  -0.2227 - 0.0677i   0.0000 + 0.0000i   0.0000 + 0.0000i
0.0032 + 0.1905i   0.1538 - 0.2192i   0.2323 - 0.0448i   0.0000 + 0.0000i
0.7602 + 0.2814i   1.6184 - 1.4346i   0.1289 - 0.2250i   1.8697 + 1.4731i

info =

0

```
```function f07tw_example
uplo = 'L';
diag = 'N';
a = [ 4.78 + 4.56i,  0 + 0i,  0 + 0i,  0 + 0i;
2 - 0.3i,  -4.11 + 1.25i,  0 + 0i,  0 + 0i;
2.89 - 1.34i,  2.36 - 4.25i,  4.15 + 0.8i,  0 + 0i;
-1.89 + 1.15i,  0.04 - 3.69i,  -0.02 + 0.46i,  0.33 - 0.26i];
[aOut, info] = f07tw(uplo, diag, a)
```
```

aOut =

0.1095 - 0.1045i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
0.0582 - 0.0411i  -0.2227 - 0.0677i   0.0000 + 0.0000i   0.0000 + 0.0000i
0.0032 + 0.1905i   0.1538 - 0.2192i   0.2323 - 0.0448i   0.0000 + 0.0000i
0.7602 + 0.2814i   1.6184 - 1.4346i   0.1289 - 0.2250i   1.8697 + 1.4731i

info =

0

```

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