seemps.analysis.expansion.PowerExpansion

class seemps.analysis.expansion.PowerExpansion

Bases: PolynomialExpansion

Polynomial expansion in the power basis {1, x, x^2, …}.

The canonical domain is [-1, 1]. The recurrence relation is trivial:

P_{k+1}(x) = x · P_k(x).

This is equivalent to a standard Taylor/power series expansion. When combined with Clenshaw evaluations, this enables the evaluation following Horner’s method, which is more efficient and stable than evaluating naive monomials x^i directly.

get_recurrence(k: int) → tuple[float, float, float]

Return the three-term recurrence coefficients (α_k, β_k, γ_k) for P_{k+1}(x) = (α_k x + β_k) P_k(x) - γ_k P_{k-1}(x).

property p1_factor

Return the scalar κ such that the first-degree basis polynomial satisfies P_1(x) = κ·x. Used to correctly seed the three-term recurrence relation.