freealg.distributions.FreeLevy.density

FreeLevy.density(x=None, eta=0.0002, max_iter=100, tol=1e-12, return_atoms=False, plot=False, latex=False, save=False, eig=None)

Density of distribution.

Parameters:

xnumpy.array, default=None

The locations where density is evaluated at. If None, an interval slightly larger than the support interval of the spectral density is used.

etafloat, default=2e-4

The offset \(\eta\) from the real axis where the density is evaluated using Plemelj formula at \(z = x + i \eta\).

max_iterint, default=100

Maximum number of Newton iterations to solve for the Stieltjes root.

tolfloat, default=1e-12

Tolerance for Newton iterations to solve for the Stieltjes root.

return_atomsbool, default=False

If True, the atoms (if any) of the distribution will also be returned.

plotbool, default=False

If True, density is plotted.

latexbool, default=False

If True, the plot is rendered using LaTeX. This option is relevant only if plot=True.

savebool, default=False

If not False, the plot is saved. If a string is given, it is assumed to the save filename (with the file extension). This option is relevant only if plot=True.

eignumpy.array, default=None

A collection of eigenvalues to compare to via histogram. This option is relevant only if plot=True.

Returns:

rhonumpy.array

Absolutely-continuous part of the spectral density.

if return_atoms is True:

atomslist

A list of tuples (loc, wight) containing the location and the weight of the atom.

See also

stieltjes

sample

Examples

>>> import numpy
>>> from freealg.distributions import FreeLevy

>>> # Create an object of the class
>>> fl = FreeLevy(t=[2.0, 5.5], w=[0.75, 1-0.75], lam=0.1, a=0,
...               sigma=0.9)

>>> # Plot density
>>> x = numpy.linspace(0, 2, 100)
>>> rho, atoms = fl.density(x, return_atoms=True, plot=True)