1: $a (lda :)$ – complex 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_zgetrf (f07ar).
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_zgetrf (f07ar).

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: $a (lda :)$ – complex 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 real floating-point operations is approximately

\frac{16}{3} n^{3}

The real analogue of this function is nag_lapack_dgetri (f07aj).

Example

This example computes the inverse of the matrix

A

, where

A = (\begin{array}{r} - 1.34 + 2.55 i & 0.28 + 3.17 i & - 6.39 - 2.20 i & 0.72 - 0.92 i \\ - 0.17 - 1.41 i & 3.31 - 0.15 i & - 0.15 + 1.34 i & 1.29 + 1.38 i \\ - 3.29 - 2.39 i & - 1.91 + 4.42 i & - 0.14 - 1.35 i & 1.72 + 1.35 i \\ 2.41 + 0.39 i & - 0.56 + 1.47 i & - 0.83 - 0.69 i & - 1.96 + 0.67 i \end{array}) .

Here

A

is nonsymmetric and must first be factorized by nag_lapack_zgetrf (f07ar).

Open in the MATLAB editor: f07aw_example

function f07aw_example


fprintf('f07aw example results\n\n');

a = [-1.34 + 2.55i,  0.28 + 3.17i, -6.39 - 2.20i,  0.72 - 0.92i;
     -0.17 - 1.41i,  3.31 - 0.15i, -0.15 + 1.34i,  1.29 + 1.38i;
     -3.29 - 2.39i, -1.91 + 4.42i, -0.14 - 1.35i,  1.72 + 1.35i;
      2.41 + 0.39i, -0.56 + 1.47i, -0.83 - 0.69i, -1.96 + 0.67i];

% Factorize a
[LU, ipiv, info] = f07ar(a);

% Compute inverse of a
[ainv, info] = f07aw(LU, ipiv);

disp('Inverse');
disp(ainv);

f07aw example results

Inverse
   0.0757 - 0.4324i   1.6512 - 3.1342i   1.2663 + 0.0418i   3.8181 + 1.1195i
  -0.1942 + 0.0798i  -1.1900 - 0.1426i  -0.2401 - 0.5889i  -0.0101 - 1.4969i
  -0.0957 - 0.0491i   0.7371 - 0.4290i   0.3224 + 0.0776i   0.6887 + 0.7891i
   0.3702 - 0.5040i   3.7253 - 3.1813i   1.7014 + 0.7267i   3.9367 + 3.3255i

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

Chapter Contents

Chapter Introduction

NAG Toolbox