quadax.NestedRule

class quadax.NestedRule(_xh: Array, _wh: Array, _wl: Array, _norm: float | int | Callable, _batch_size: int | None)Source

Base class for nested quadrature rules.

Nested rules consist of a set of nodes (xh) and weights (wh) for a high order rule, along with an additional set of weights (wl) for a lower order rule that shares nodes with the high order rule [1].

Notes

The error estimate is derived from the difference between the two rules, so it is only meaningful while the nodes resolve the integrand. For oscillatory integrands, below roughly three points per oscillation both rules alias and agree spuriously, and the estimate can fall well below the true error; above roughly eight it is reliably conservative. This is a property of nested rules in general rather than of any particular order, since raising the order only raises the frequency at which it sets in. Strongly oscillatory integrands are better served by a specialized method.

References

[1]

R. Piessens, E. de Doncker-Kapenga, C. W. Überhuber, D. K. Kahaner. “QUADPACK: A Subroutine Package for Automatic Integration”. Springer Series in Computational Mathematics, vol. 1. Springer-Verlag, Berlin, 1983. doi:10.1007/978-3-642-61786-7

Methods

integrate(fun, a, b, args)	Integrate a function from a to b using a nested rule.
norm(x)	Norm to use for measuring error for vector valued integrands.

Attributes

nodes_per_call

How many evaluations of the integrand one application of the rule costs.