orthax.recurrence.generate_recurrence

orthax.recurrence.generate_recurrence(weight: Callable, domain: tuple, n: int, scale: str = 'monic', quadrule=None, quadopts: dict | None = None) → TabulatedRecurrenceRelationSource

Generate recurrence relation coefficients for orthogonal polynomial family.

Finds coefficients \(a_i, b_i, g_i\) such that

\[\begin{split}\begin{align} p_{-1}(x) &= 0 \\ p_0(x) &= 1 \\ p_{i+1}(x) &= (x-a_i) p_i(x) - b_i p_{i-1}(x) \\ \int_D p_i(x) p_j(x) w(x) dx &= g^2_i \delta_{i,j} \end{align}\end{split}\]
Parameters:

  • weight (callable) – Weight function.

  • domain (tuple of float) – Lower and upper bounds for the domain of the polynomials.

  • n (int) – Number of terms to generate, ie, highest order of polynomial desired.

  • scale ({"monic", "normalized"}) – How to scale the resulting polynomials.

  • quadrule (quadax.AbstractQuadratureRule, optional) – Quadrature rule to use for computing integrals in generating recurrence coefficients. Defaults to quadax.TanhSinhRule(order=129).

  • quadopts (dict, optional) – Additional options passed to quadax.adaptive_quadrature. Default options are epsabs=1e-15, epsrel=1e-15, max_ninter=500.

Returns:

rec (TabulatedRecurrenceRelation) – Recurrence relation coefficients and polynomial norms.

Notes

Requires the quadax package to be installed.