1: $a (lda :)$ – double array: The first dimension of the array a must be at least $\max (1, n)$ .
The second dimension of the array a must be at least $\max (1, n)$ .

The $L U$ factorization of $A$ , as returned by nag_lapack_dgetrf (f07ad).
2: $ipiv (:)$ – int64int32nag_int array: The dimension of the array ipiv must be at least $\max (1, n)$

The pivot indices, as returned by nag_lapack_dgetrf (f07ad).

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the first dimension of the array a and the second dimension of the arrays a, ipiv.
$n$ , the order of the matrix $A$ .

Constraint: $n \geq 0$ .

Output Parameters

1: $a (lda :)$ – double array: The first dimension of the array a will be $\max (1, n)$ .
The second dimension of the array a will be $\max (1, n)$ .

The factorization stores the $n$ by $n$ matrix $A^{- 1}$ .
2: $info$ – int64int32nag_int scalar: $info = 0$ 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.

$info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

W $info > 0$: Element $_$ of the diagonal is zero. $U$ is singular, and the inverse of $A$ cannot be computed.

Accuracy

The computed inverse

X

satisfies a bound of the form:

|X A - I| \leq c (n) ε |X| P |L| |U|,

where

c (n)

is a modest linear function of

n

, and

ε

is the machine precision.

Note that a similar bound for

|A X - I|

cannot be guaranteed, although it is almost always satisfied. See Du Croz and Higham (1992).

Further Comments

The total number of floating-point operations is approximately

\frac{4}{3} n^{3}

The complex analogue of this function is nag_lapack_zgetri (f07aw).

Example

This example computes the inverse of the matrix

A

, where

A = (\begin{array}{r} 1.80 & 2.88 & 2.05 & - 0.89 \\ 5.25 & - 2.95 & - 0.95 & - 3.80 \\ 1.58 & - 2.69 & - 2.90 & - 1.04 \\ - 1.11 & - 0.66 & - 0.59 & 0.80 \end{array}) .

Here

A

is nonsymmetric and must first be factorized by nag_lapack_dgetrf (f07ad).

Open in the MATLAB editor: f07aj_example

function f07aj_example


fprintf('f07aj example results\n\n');

a = [ 1.80,  2.88,  2.05, -0.89;
      5.25, -2.95, -0.95, -3.80;
      1.58, -2.69, -2.90, -1.04;
     -1.11, -0.66, -0.59,  0.80];

% Factorize a
[af, ipiv, info] = f07ad(a);

% Compute inverse of a
[ainv, info] = f07aj(af, ipiv);

[ifail] = x04ca( ...
                 'General', ' ', ainv, 'Inverse');

f07aj example results

 Inverse
             1          2          3          4
 1      1.7720     0.5757     0.0843     4.8155
 2     -0.1175    -0.4456     0.4114    -1.7126
 3      0.1799     0.4527    -0.6676     1.4824
 4      2.4944     0.7650    -0.0360     7.6119

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox