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NAG Toolbox

NAG Toolbox: nag_lapack_dgetri (f07aj)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dgetri (f07aj) computes the inverse of a real matrix A, where A has been factorized by nag_lapack_dgetrf (f07ad).

Syntax

[a, info] = f07aj(a, ipiv, 'n', n)
[a, info] = nag_lapack_dgetri(a, ipiv, 'n', n)

Description

nag_lapack_dgetri (f07aj) is used to compute the inverse of a real matrix A, the function must be preceded by a call to nag_lapack_dgetrf (f07ad), 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: – double 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_dgetrf (f07ad).
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_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: n0.

Output Parameters

1:     alda: – double 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 floating-point operations is approximately 43n3.
The complex analogue of this function is nag_lapack_zgetri (f07aw).

Example

This example computes the inverse of the matrix A, where
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 .  
Here A is nonsymmetric and must first be factorized by nag_lapack_dgetrf (f07ad).
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

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