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

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

complex_gen_matrix_cond_num computes an estimate of the absolute condition number of a matrix function of a complex 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 f01kb

https://www.nag.com/numeric/nl/nagdoc_28.4/flhtml/f01/f01kbf.html

Parameters
acomplex, 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 .

dataarbitrary, optional

User-communication data for callback functions.

Returns
acomplex, 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 while evaluating the matrix function . You can investigate further by calling complex_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 . complex_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).

References

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