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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_ztftri (f07wx)

## Purpose

nag_lapack_ztftri (f07wx) computes the inverse of a complex triangular matrix stored in Rectangular Full Packed (RFP) format. The RFP storage format is described in Section [Rectangular Full Packed (RFP) Storage] in the F07 Chapter Introduction.

## Syntax

[a, info] = f07wx(transr, uplo, diag, n, a)
[a, info] = nag_lapack_ztftri(transr, uplo, diag, n, a)

## Description

nag_lapack_ztftri (f07wx) forms the inverse of a complex triangular matrix A$A$, stored using RFP format. 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
Gustavson F G, Waśniewski J, Dongarra J J and Langou J (2010) Rectangular full packed format for Cholesky's algorithm: factorization, solution, and inversion ACM Trans. Math. Software 37, 2

## Parameters

### Compulsory Input Parameters

1:     transr – string (length ≥ 1)
Specifies whether the normal RFP representation of A$A$ or its conjugate transpose is stored.
transr = 'N'${\mathbf{transr}}=\text{'N'}$
The matrix A$A$ is stored in normal RFP format.
transr = 'C'${\mathbf{transr}}=\text{'C'}$
The conjugate transpose of the RFP representation of the matrix A$A$ is stored.
Constraint: transr = 'N'${\mathbf{transr}}=\text{'N'}$ or 'C'$\text{'C'}$.
2:     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'$\text{'L'}$.
3:     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'$\text{'U'}$.
4:     n – int64int32nag_int scalar
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.
5:     a(n × (n + 1) / 2${\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$) – complex array
The n$n$ by n$n$ triangular matrix A$A$, stored in RFP format.

None.

None.

### Output Parameters

1:     a(n × (n + 1) / 2${\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$) – complex array
A$A$ stores A1${A}^{-1}$, in the same storage format as A$A$.
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.

W INFO > 0${\mathbf{INFO}}>0$
a(_,_)$a\left(_,_\right)$ is exactly zero. A$A$ is singular 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_dtftri (f07wk).

## Example

```function nag_lapack_ztftri_example
a = [ 4.15 - 0.80i;
4.78 + 4.56i;
2.00 - 0.30i;
2.89 - 1.34i;
-1.89 + 1.15i;
-0.02 - 0.46i;
0.33 + 0.26i;
-4.11 + 1.25i;
2.36 - 4.25i;
0.04 - 3.69i];
transr = 'n';
uplo   = 'l';
diag   = 'n';
n      = int64(4);
% Compute inverse of a
[a, info] = nag_lapack_ztftri(transr, uplo, diag, n, a);

if info == 0
% Convert inverse to full array form, and print it
[f, info] = nag_matop_ztfttr(transr, uplo, n, a);
fprintf('\n');
[ifail] = ...
nag_file_print_matrix_complex_gen_comp(uplo, 'n', f, 'b', 'f7.4', 'Inverse', 'i', 'i', int64(80), int64(0));
else
fprintf('\na is singular.\n');
end
```
```

Inverse
1                 2                 3                 4
1  ( 0.1095,-0.1045)
2  ( 0.0582,-0.0411) (-0.2227,-0.0677)
3  ( 0.0032, 0.1905) ( 0.1538,-0.2192) ( 0.2323,-0.0448)
4  ( 0.7602, 0.2814) ( 1.6184,-1.4346) ( 0.1289,-0.2250) ( 1.8697, 1.4731)

```
```function f07wx_example
a = [ 4.15 - 0.80i;
4.78 + 4.56i;
2.00 - 0.30i;
2.89 - 1.34i;
-1.89 + 1.15i;
-0.02 - 0.46i;
0.33 + 0.26i;
-4.11 + 1.25i;
2.36 - 4.25i;
0.04 - 3.69i];
transr = 'n';
uplo   = 'l';
diag   = 'n';
n      = int64(4);
% Compute inverse of a
[a, info] = f07wx(transr, uplo, diag, n, a);

if info == 0
% Convert inverse to full array form, and print it
[f, info] = f01vh(transr, uplo, n, a);
fprintf('\n');
[ifail] = x04db(uplo, 'n', f, 'b', 'f7.4', 'Inverse', 'i', 'i', int64(80), int64(0));
else
fprintf('\na is singular.\n');
end
```
```

Inverse
1                 2                 3                 4
1  ( 0.1095,-0.1045)
2  ( 0.0582,-0.0411) (-0.2227,-0.0677)
3  ( 0.0032, 0.1905) ( 0.1538,-0.2192) ( 0.2323,-0.0448)
4  ( 0.7602, 0.2814) ( 1.6184,-1.4346) ( 0.1289,-0.2250) ( 1.8697, 1.4731)

```