naginterfaces.library.matop.real_gen_matrix_cond_usd¶
- naginterfaces.library.matop.real_gen_matrix_cond_usd(a, f, data=None)[source]¶
real_gen_matrix_cond_usd
computes an estimate of the absolute condition number of a matrix function at a real matrix in the -norm, using analytical derivatives of you have supplied.For full information please refer to the NAG Library document for f01jc
https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f01/f01jcf.html
- Parameters
- afloat, array-like, shape
The matrix .
- fcallable fz = f(m, z, data=None)
Given an integer , the function evaluates at a number of points .
- Parameters
- mint
The order, , of the derivative required.
If , should be returned.
For , should be returned.
- 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 or derivative 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_usd()
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_usd
computes an estimate such that , where . The relative condition number can then be computed viaThe algorithm used to find is detailed in Section 3.4 of Higham (2008).
The function , and the derivatives of , are returned by function which, given an integer , evaluates at a number of (generally complex) points , for . For any on the real line, must also be real.
real_gen_matrix_cond_usd
is, therefore, appropriate for functions that can be evaluated on the complex plane and whose derivatives, of arbitrary order, can also be evaluated on the complex plane.
- References
Higham, N J, 2008, Functions of Matrices: Theory and Computation, SIAM, Philadelphia, PA, USA