naginterfaces.library.matop.real_​gen_​pseudinv

naginterfaces.library.matop.real_gen_pseudinv(t, a)[source]

real_gen_pseudinv calculates the rank and pseudo-inverse of an real matrix, , using a factorization with column interchanges.

For full information please refer to the NAG Library document for f01bl

https://www.nag.com/numeric/nl/nagdoc_28.4/flhtml/f01/f01blf.html

Parameters
tfloat

The tolerance used to decide when elements can be regarded as zero (see Further Comments).

afloat, array-like, shape

The rectangular matrix .

Returns
afloat, ndarray, shape

The transpose of the pseudo-inverse of .

aijmaxfloat, ndarray, shape

contains the element of largest modulus in the reduced matrix at the th stage. If , then only the first elements of have values assigned to them; the remaining elements are unused. The ratio usually gives an indication of the condition number of the original matrix (see Further Comments).

irankint

, the rank of as determined using the tolerance .

incint, ndarray, shape

The record of the column interchanges in the Householder factorization.

Raises
NagValueError
(errno )

Inverse not found. Incorrect .

(errno )

Invalid tolerance, too large: .

(errno )

Invalid tolerance, : .

(errno )

On entry, and .

Constraint: .

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

Householder’s factorization with column interchanges is used in the decomposition , where is with its columns permuted, is the first columns of an orthogonal matrix and is an upper-trapezoidal matrix of rank . The pseudo-inverse of is given by where

If the matrix is found to be of maximum rank, , is a nonsingular upper-triangular matrix and the pseudo-inverse of simplifies to . The transpose of the pseudo-inverse of is overwritten on .

References

Peters, G and Wilkinson, J H, 1970, The least squares problem and pseudo-inverses, Comput. J. (13), 309–316

Wilkinson, J H and Reinsch, C, 1971, Handbook for Automatic Computation II, Linear Algebra, Springer–Verlag