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NAG Toolbox: nag_lapack_zgetri (f07aw)

Purpose

nag_lapack_zgetri (f07aw) computes the inverse of a complex matrix AA, where AA 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 AA, the function must be preceded by a call to nag_lapack_zgetrf (f07ar), which computes the LULU factorization of AA as A = PLUA=PLU. The inverse of AA is computed by forming U1U-1 and then solving the equation XPL = U1XPL=U-1 for XX.

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:     a(lda, : :) – complex array
The first dimension of the array a must be at least max (1,n)max(1,n)
The second dimension of the array must be at least max (1,n)max(1,n)
The LULU factorization of AA, as returned by nag_lapack_zgetrf (f07ar).
2:     ipiv( : :) – int64int32nag_int array
Note: the dimension of the array ipiv must be at least max (1,n)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 The second dimension of the arrays a, ipiv.
nn, the order of the matrix AA.
Constraint: n0n0.

Input Parameters Omitted from the MATLAB Interface

lda work lwork

Output Parameters

1:     a(lda, : :) – complex array
The first dimension of the array a will be max (1,n)max(1,n)
The second dimension of the array will be max (1,n)max(1,n)
ldamax (1,n)ldamax(1,n).
The factorization stores the nn by nn matrix A1A-1.
2:     info – int64int32nag_int scalar
info = 0info=0 unless the function detects an error (see Section [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 = iinfo=-i
If info = iinfo=-i, parameter ii had an illegal value on entry. The parameters are numbered as follows:
1: n, 2: a, 3: lda, 4: ipiv, 5: work, 6: lwork, 7: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.
W INFO > 0INFO>0
If info = iinfo=i, the iith diagonal element of the factor UU is zero, UU is singular, and the inverse of AA cannot be computed.

Accuracy

The computed inverse XX satisfies a bound of the form:
|XAI|c(n)ε|X|P|L||U| ,
|XA-I|c(n)ε|X|P|L||U| ,
where c(n)c(n) is a modest linear function of nn, and εε is the machine precision.
Note that a similar bound for |AXI||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 (16/3)n3163n3.
The real analogue of this function is nag_lapack_dgetri (f07aj).

Example

function nag_lapack_zgetri_example
a = [ -1.34 + 2.55i,  0.28 + 3.17i,  -6.39 - 2.2i,  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
[a, ipiv, info] = nag_lapack_zgetrf(a);

% Compute inverse of a
[a, info] = nag_lapack_zgetri(a, ipiv)
 

a =

   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


info =

                    0


function f07aw_example
a = [ -1.34 + 2.55i,  0.28 + 3.17i,  -6.39 - 2.2i,  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
[a, ipiv, info] = f07ar(a);

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

a =

   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


info =

                    0



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