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# NAG Toolbox: nag_lapack_dtftri (f07wk)

## Purpose

nag_lapack_dtftri (f07wk) computes the inverse of a real 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] = f07wk(transr, uplo, diag, n, a)
[a, info] = nag_lapack_dtftri(transr, uplo, diag, n, a)

## Description

nag_lapack_dtftri (f07wk) forms the inverse of a real 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 RFP representation of A$A$ is normal or transposed.
transr = 'N'${\mathbf{transr}}=\text{'N'}$
The matrix A$A$ is stored in normal RFP format.
transr = 'T'${\mathbf{transr}}=\text{'T'}$
The matrix A$A$ is stored in transposed RFP format.
Constraint: transr = 'N'${\mathbf{transr}}=\text{'N'}$ or 'T'$\text{'T'}$.
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${\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$) – double 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${\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$) – double 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 floating point operations is approximately (1/3)n3$\frac{1}{3}{n}^{3}$.
The complex analogue of this function is nag_lapack_ztftri (f07wx).

## Example

```function nag_lapack_dtftri_example
a = [-8.02; 4.30; -3.96; 0.40; -0.27; -5.95; 0.12; -4.87; 0.31; 0.07];
transr = 'n';
uplo   = 'l';
diag   = 'n';
n      = int64(4);
% Compute inverse of a
[a, info] = nag_lapack_dtftri(transr, uplo, diag, n, a);

if info == 0
% Convert inverse to full array form, and print it
[f, info] = nag_matop_dtfttr(transr, uplo, n, a);
fprintf('\n');
[ifail] = nag_file_print_matrix_real_gen(uplo, 'n', f, 'Inverse');
else
fprintf('\na is singular.\n');
end
```
```

Inverse
1          2          3          4
1      0.2326
2     -0.1891    -0.2053
3      0.0043    -0.0079    -0.1247
4      0.8463    -0.2738    -6.1825     8.3333

```
```function f07wk_example
a = [-8.02; 4.30; -3.96; 0.40; -0.27; -5.95; 0.12; -4.87; 0.31; 0.07];
transr = 'n';
uplo   = 'l';
diag   = 'n';
n      = int64(4);
% Compute inverse of a
[a, info] = f07wk(transr, uplo, diag, n, a);

if info == 0
% Convert inverse to full array form, and print it
[f, info] = f01vg(transr, uplo, n, a);
fprintf('\n');
[ifail] = x04ca(uplo, 'n', f, 'Inverse');
else
fprintf('\na is singular.\n');
end
```
```

Inverse
1          2          3          4
1      0.2326
2     -0.1891    -0.2053
3      0.0043    -0.0079    -0.1247
4      0.8463    -0.2738    -6.1825     8.3333

```

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