freealg.AlgebraicForm.atoms

AlgebraicForm.atoms(eta=1e-06, tol=1e-12, real_tol=None, w_tol=1e-10, merge_tol=1e-08)

Detect atom locations and weights of distribution

This routine uses the necessary condition for a finite pole: a_s(z0)=0, where a_s(z) is the leading coefficient of P in powers of m. Candidate atom locations are the (nearly) real roots of a_s(z). The atom weight is estimated numerically from the Stieltjes transform as

w ~= eta * Im(m(z0 + i*eta)),

which follows from m(z) ~ -w/(z - z0) near an atom at z0.

Parameters:

etafloat, default=1e-6

Small imaginary part used to probe the pole strength.

tolfloat, default=1e-12

Tolerance for trimming polynomial coefficients.

real_tolfloat or None, default=None

Tolerance for treating a complex root as real. If None, uses 1e3*tol.

w_tolfloat, default=1e-10

Minimum atom weight to report.

merge_tolfloat, default=1e-8

Merge roots whose real parts differ by at most this tolerance.

Returns:

atomslist of (float, float)

List of (atom_loc, atom_w). Locations are real numbers and weights are nonnegative.

See also

branch_points