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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_ztptri (f07uw)

## Purpose

nag_lapack_ztptri (f07uw) computes the inverse of a complex triangular matrix, using packed storage.

## Syntax

[ap, info] = f07uw(uplo, diag, n, ap)
[ap, info] = nag_lapack_ztptri(uplo, diag, n, ap)

## Description

nag_lapack_ztptri (f07uw) forms the inverse of a complex triangular matrix A$A$, using packed storage. 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:     n – int64int32nag_int scalar
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.
4:     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 n$n$ by n$n$ triangular matrix A$A$, packed by columns.
More precisely,
• if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the upper triangle of A$A$ 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 A$A$ 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$i\ge j$.
If diag = 'U'${\mathbf{diag}}=\text{'U'}$, the diagonal elements of A$A$ are assumed to be 1$1$, and are not referenced; the same storage scheme is used whether diag = 'N'${\mathbf{diag}}=\text{'N'}$ or ‘U’.

None.

None.

### Output Parameters

1:     ap( : $:$) – complex array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
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: ap, 5: info.
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_dtptri (f07uj).

## Example

```function nag_lapack_ztptri_example
uplo = 'L';
diag = 'N';
n = int64(4);
ap = [ 4.78 + 4.56i;
2 - 0.3i;
2.89 - 1.34i;
-1.89 + 1.15i;
-4.11 + 1.25i;
2.36 - 4.25i;
0.04 - 3.69i;
4.15 + 0.8i;
-0.02 + 0.46i;
0.33 - 0.26i];
[apOut, info] = nag_lapack_ztptri(uplo, diag, n, ap)
```
```

apOut =

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

info =

0

```
```function f07uw_example
uplo = 'L';
diag = 'N';
n = int64(4);
ap = [ 4.78 + 4.56i;
2 - 0.3i;
2.89 - 1.34i;
-1.89 + 1.15i;
-4.11 + 1.25i;
2.36 - 4.25i;
0.04 - 3.69i;
4.15 + 0.8i;
-0.02 + 0.46i;
0.33 - 0.26i];
[apOut, info] = f07uw(uplo, diag, n, ap)
```
```

apOut =

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

info =

0

```