freealg.AlgebraicForm.cusp

AlgebraicForm.cusp(t_grid, kind='free', supp=None, max_iter=50, tol=1e-12, dedup_t_tol=1e-06, dedup_x_tol=1e-06, return_info=False)

Find cusp (merge/split) point of evolving spectral edges.

Parameters:

t_gridarray_like

A time grid to search for cusp points.

kind{'free', 'deformed'}, default= 'free'

The type of operation:

• 'free': evolve the spectral curve using free decompression

• 'deformed': evolve the spectral curve using deformed deformation.

supplist, default=None

Estimated support of density as a list of tuples. If not given, support is estimated from the fitted polynomial.

max_iterint, default=50

Maximum number of Newton iterations

tolfloat, default=1e-12

Tolerance in Newton root finding method

dedup_t_tolfloat, default=1e-6

Tolerance along t axis to identify duplicity (de-duplication.)

dedup_x_tolfloat, default=1e-6

Tolerance along x axis to identify duplicity (de-duplication.)

return_infobool, default=False

If True, a list of debugging information per each cusp point is returned.

Returns:

cuspslist

A list of tuples (x, t) of the location x and time t of cusp points.

infolist

A list of dictionaries, each contain the debugging information for a cusp point. This is returned if return_info is set to True.

See also

edge

Examples

>>> import numpy
>>> from freealg import AlgebraicForm
>>> from freealg.distributions import CompoundFreePoisson
>>> from freealg import submatrix

>>> # Create a distribution with two bulks
>>> cfp = CompoundFreePoisson(t=[2.0,  5.5], w=[0.75, 0.25],
...                           lam=0.1)

>>> # Get a matrix realization of the distribution
>>> A = cfp.matrix(size=6000, seed=0)

>>> # Compress the matrix to smaller size
>>> As = submatrix(A, size=1000)

>>> # Create AlgebraicForm and fit the smaller matrix
>>> af = AlgebraicForm(As)
>>> af.fit(deg_m=3, deg_z=1)

>>> # Find cusp points
>>> t_final = numpy.log(A.shape[0] / As.shape[0])
>>> t = numpy.linspace(0, t_final)
>>> cusps = af.cusp(t)