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NAG Toolbox

NAG Toolbox: nag_lapack_zgetri (f07aw)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_zgetri (f07aw) computes the inverse of a complex matrix A, where A has been factorized by nag_lapack_zgetrf (f07ar).

Syntax

[a, info] = f07aw(a, ipiv, 'n', n)
[a, info] = nag_lapack_zgetri(a, ipiv, 'n', n)

Description

nag_lapack_zgetri (f07aw) is used to compute the inverse of a complex matrix A, the function must be preceded by a call to nag_lapack_zgetrf (f07ar), which computes the LU factorization of A as A=PLU. The inverse of A is computed by forming U-1 and then solving the equation XPL=U-1 for X.

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:     alda: – complex array
The first dimension of the array a must be at least max1,n.
The second dimension of the array a must be at least max1,n.
The LU 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 max1,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: n0.

Output Parameters

1:     alda: – complex array
The first dimension of the array a will be max1,n.
The second dimension of the array a will be max1,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:
XA-IcnεXPLU ,  
where cn is a modest linear function of n, and ε is the machine precision.
Note that a similar bound for AX-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 163n3.
The real analogue of this function is nag_lapack_dgetri (f07aj).

Example

This example computes the inverse of the matrix A, where
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 .  
Here A is nonsymmetric and must first be factorized by nag_lapack_zgetrf (f07ar).
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


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