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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dtptri (f07uj)

Purpose

nag_lapack_dtptri (f07uj) computes the inverse of a real triangular matrix, using packed storage.

Syntax

[ap, info] = f07uj(uplo, diag, n, ap)
[ap, info] = nag_lapack_dtptri(uplo, diag, n, ap)

Description

nag_lapack_dtptri (f07uj) forms the inverse of a real triangular matrix $A$, 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:     $\mathrm{uplo}$ – string (length ≥ 1)
Specifies whether $A$ is upper or lower triangular.
${\mathbf{uplo}}=\text{'U'}$
$A$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
$A$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2:     $\mathrm{diag}$ – string (length ≥ 1)
Indicates whether $A$ is a nonunit or unit triangular matrix.
${\mathbf{diag}}=\text{'N'}$
$A$ is a nonunit triangular matrix.
${\mathbf{diag}}=\text{'U'}$
$A$ is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be $1$.
Constraint: ${\mathbf{diag}}=\text{'N'}$ or $\text{'U'}$.
3:     $\mathrm{n}$int64int32nag_int scalar
$n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:     $\mathrm{ap}\left(:\right)$ – double array
The dimension of the array ap must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$
The $n$ by $n$ triangular matrix $A$, packed by columns.
More precisely,
• if ${\mathbf{uplo}}=\text{'U'}$, the upper triangle of $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{ap}}\left(i+j\left(j-1\right)/2\right)$ for $i\le j$;
• if ${\mathbf{uplo}}=\text{'L'}$, the lower triangle of $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{ap}}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for $i\ge j$.
If ${\mathbf{diag}}=\text{'U'}$, the diagonal elements of $A$ are assumed to be $1$, and are not referenced; the same storage scheme is used whether ${\mathbf{diag}}=\text{'N'}$ or ‘U’.

None.

Output Parameters

1:     $\mathrm{ap}\left(:\right)$ – double array
The dimension of the array ap will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$
$A$ stores ${A}^{-1}$, using the same storage format as described above.
2:     $\mathrm{info}$int64int32nag_int scalar
${\mathbf{info}}=0$ unless the function detects an error (see 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.

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
W  ${\mathbf{info}}>0$
Element $_$ of the diagonal is exactly zero. $A$ is singular its inverse cannot be computed.

Accuracy

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

The total number of floating-point operations is approximately $\frac{1}{3}{n}^{3}$.
The complex analogue of this function is nag_lapack_ztptri (f07uw).

Example

This example computes the inverse of the matrix $A$, where
 $A= 4.30 0.00 0.00 0.00 -3.96 -4.87 0.00 0.00 0.40 0.31 -8.02 0.00 -0.27 0.07 -5.95 0.12 ,$
using packed storage.
function f07uj_example

fprintf('f07uj example results\n\n');

% Invert A, where A is Lower triangular and packed
n = int64(4);
ap = [ 4.30; -3.96;  0.40; -0.27;
-4.87;  0.31;  0.07;
-8.02; -5.95;
0.12];
uplo = 'L';
diag = 'N';

% Invert
[ainv, info] = f07uj(uplo, diag, n, ap);

[ifail] = x04cc( ...
uplo, 'Non-unit', n, ainv, 'Inverse');

f07uj example results

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