naginterfaces.library.matop.real_​gen_​matrix_​cond_​std

naginterfaces.library.matop.real_gen_matrix_cond_std(fun, a)[source]

real_gen_matrix_cond_std computes an estimate of the absolute condition number of a matrix function at a real matrix in the -norm, where is either the exponential, logarithm, sine, cosine, hyperbolic sine (sinh) or hyperbolic cosine (cosh). The evaluation of the matrix function, , is also returned.

For full information please refer to the NAG Library document for f01ja

https://www.nag.com/numeric/nl/nagdoc_28.4/flhtml/f01/f01jaf.html

Parameters
funstr

Indicates which matrix function will be used.

The matrix exponential, , will be used.

The matrix sine, , will be used.

The matrix cosine, , will be used.

The hyperbolic matrix sine, , will be used.

The hyperbolic matrix cosine, , will be used.

The matrix logarithm, , will be used.

afloat, array-like, shape

The matrix .

Returns
afloat, ndarray, shape

The matrix, .

condafloat

An estimate of the absolute condition number of at .

normafloat

The -norm of .

normfafloat

The -norm of .

Raises
NagValueError
(errno )

On entry, .

Constraint: .

(errno )

Input argument number is invalid.

(errno )

An internal error occurred when evaluating the matrix function . Please contact NAG.

(errno )

An internal error occurred when estimating the norm of the Fréchet derivative of at . Please contact NAG.

Notes

The absolute condition number of at , is given by the norm of the Fréchet derivative of , , which is defined by

where is the Fréchet derivative in the direction . is linear in and can, therefore, be written as

where the operator stacks the columns of a matrix into one vector, so that is . real_gen_matrix_cond_std computes an estimate such that , where . The relative condition number can then be computed via

The algorithm used to find is detailed in Section 3.4 of Higham (2008).

References

Higham, N J, 2008, Functions of Matrices: Theory and Computation, SIAM, Philadelphia, PA, USA