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NAG Toolbox

NAG Toolbox: nag_lapack_dgetri (f07aj)

Purpose

nag_lapack_dgetri (f07aj) computes the inverse of a real matrix AA, where AA 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 AA, the function must be preceded by a call to nag_lapack_dgetrf (f07ad), 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, : :) – double 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_dgetrf (f07ad).
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_dgetrf (f07ad).

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, : :) – double 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 floating point operations is approximately (4/3)n343n3.
The complex analogue of this function is nag_lapack_zgetri (f07aw).

Example

function nag_lapack_dgetri_example
a = [1.8, 2.88, 2.05, -0.89;
     5.25, -2.95, -0.95, -3.8;
     1.58, -2.69, -2.9, -1.04;
     -1.11, -0.66, -0.59, 0.8];
% Factorize a
[a, ipiv, info] = nag_lapack_dgetrf(a);

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

a =

    1.7720    0.5757    0.0843    4.8155
   -0.1175   -0.4456    0.4114   -1.7126
    0.1799    0.4527   -0.6676    1.4824
    2.4944    0.7650   -0.0360    7.6119


info =

                    0


function f07aj_example
a = [1.8, 2.88, 2.05, -0.89;
     5.25, -2.95, -0.95, -3.8;
     1.58, -2.69, -2.9, -1.04;
     -1.11, -0.66, -0.59, 0.8];
% Factorize a
[a, ipiv, info] = f07ad(a);

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

a =

    1.7720    0.5757    0.0843    4.8155
   -0.1175   -0.4456    0.4114   -1.7126
    0.1799    0.4527   -0.6676    1.4824
    2.4944    0.7650   -0.0360    7.6119


info =

                    0



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