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

naginterfaces.library.matop.real_gen_matrix_cond_num(a, f, data=None)[source]

real_gen_matrix_cond_num computes an estimate of the absolute condition number of a matrix function at a real matrix in the -norm. Numerical differentiation is used to evaluate the derivatives of when they are required.

For full information please refer to the NAG Library document for f01jb

https://www.nag.com/numeric/nl/nagdoc_28.4/flhtml/f01/f01jbf.html

Parameters
afloat, array-like, shape

The matrix .

fcallable fz = f(z, data=None)

The function evaluates at a number of points .

Parameters
zcomplex, ndarray, shape

The points at which the function is to be evaluated.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns
fzcomplex, array-like, shape

The function values. should return the value , for . If lies on the real line, then so must .

dataarbitrary, optional

User-communication data for callback functions.

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 )

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

(errno )

An internal error occurred when evaluating the matrix function . You can investigate further by calling real_gen_matrix_fun_num() with the matrix and the function .

Warns
NagCallbackTerminateWarning
(errno )

Termination requested in .

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_num 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).

The function is supplied via function which evaluates at a number of points .

References

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