seemps.analysis.expansion.LegendreExpansion#

class seemps.analysis.expansion.LegendreExpansion#

Bases: PolynomialExpansion

Expansion in the Legendre basis.

The Legendre polynomials \(P_k(x)\) are orthogonal on the interval \([−1, 1]\) with respect to the uniform weight \(w(x)=1\). They are widely used in approximation theory since truncated Legendre series minimize the error in the \(L^2([-1,1])\) norm.

See https://en.wikipedia.org/wiki/Legendre_polynomials for more information.

classmethod project(func: Callable[[ndarray[tuple[Any, ...], dtype[_ScalarT]]], float], order: int, approximation_domain: tuple[float, float] = (-1.0, 1.0)) → LegendreExpansion[source]#

Project a scalar function onto the Legendre basis on the given approximation domain.

The approximation domain must contain the full range of arguments on which the expansion will be evaluated; otherwise, rescaling maps the argument outside the orthogonality domain where the basis is not defined, leading to large errors.

recurrence_coefficients(k: int) → tuple[float, float, float][source]#: Returns the three-term coefficients of the Legendre recursion:

\[(k+1) P_{k+1}(x) = (2k+1) x P_k(x) - k P_{k-1}(x)\]

rescale_mpo(mpo: MPO) → MPO[source]#: Rescale the argument MPO from the approximation domain to the orthogonality domain of the basis, if applicable. This is delegated to the polynomial basis subclass to correctly account for their orthogonality domain.

rescale_mps(mps: MPS) → MPS[source]#: Rescale the argument MPS from the approximation domain to the orthogonality domain of the basis, if applicable. This is delegated to the polynomial basis subclass to correctly account for their orthogonality domain.