quadax.GenzMalikRule

class quadax.GenzMalikRule(ndim: int, degree: int = 9, norm: Callable | float | int = inf, batch_size: int | None = None)Source

Integrate a function over a box using a fixed degree Genz-Malik rule.

Integration is performed using a fully symmetric rule of the given polynomial degree, with error estimated from an embedded rule two degrees lower sharing its nodes [1] and from null rules on those nodes and one extra orbit [2].

Parameters:

• ndim (int) – Dimension of the box to integrate over. One dimension is allowed, where the rule reduces to a degree point rule on the interval, but a 1D rule such as GaussKronrodRule reaches a far higher degree for the same nodes and is the better choice there.

• degree (int) – Polynomial degree of the rule, one of 7, 9, 11 or 13. Higher degrees are worth their extra nodes only on integrands smooth enough to be resolved by them.

• norm (int, callable) – Norm to use for measuring error for vector valued integrands. No effect if the integrand is scalar valued. If an int, uses p-norm of the given order, otherwise should be callable.

• batch_size (int, optional) – Number of points to evaluate the integrand at simultaneously. The default evaluates all the rule’s nodes at once.

Notes

The number of integrand evaluations per application is 2**ndim from the corner orbit, plus a polynomial in ndim of degree (degree - 3) // 2 from the rest, plus 2*ndim for the orbit the error estimate adds:

ndim	degree 7	degree 9	degree 11	degree 13
1	9	11	13	15
2	21	33	49	69
3	39	77	143	245
4	65	153	345	705
5	103	273	723	1725
6	161	453	1365	3729
8	417	1105	3921	13345
10	1265	2605	9405	37409

The extra orbit is what lets the error be estimated over a space of null rules rather than from a single one, which is worth its 2*ndim nodes several times over; the same counts appear in [2]. It carries no integration weight, so the value the rule returns is the classical one.

Raising the degree therefore costs more the higher the dimension, and no degree escapes the exponential corner term, which overtakes the polynomial part around ndim of 7 at degree 7 and progressively later at higher degrees. Both growths are in the rule itself, before an adaptive routine multiplies them by a number of regions that grows with ndim in its own right.

The rule has negative weights, increasingly so as ndim grows, so it is not guaranteed to return a positive value for a positive integrand and can lose precision to cancellation in high dimensions. Degrees 9 and 11 carry rather less negative weight than degree 7 does, so their extra nodes buy conditioning as well as degree; degree 13 carries considerably more, and in low precision can give back what its degree gains.

References

[1]

A. C. Genz, A. A. Malik. “An imbedded family of fully symmetric numerical integration rules”. SIAM Journal on Numerical Analysis, vol. 20, no. 3, 1983, pp. 580-588.

[2] (1,2)

J. Berntsen, T. O. Espelid, A. Genz. “An adaptive algorithm for the approximate calculation of multiple integrals”. ACM Transactions on Mathematical Software, vol. 17, no. 4, 1991, pp. 437-451.

Methods

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

Attributes

ndim	Dimension of the domain this rule integrates over.
nodes_per_call	How many evaluations of the integrand one application of the rule costs.