0
$\begingroup$

I have a function evaluated on a regular 5D grid with 21 points per dimension (so $21^{5}$ points total).

I need to evaluate the integral of the function over all 5 dimensions, so I was planning on using one of the composite Newton-Cotes formulae (i.e. trapezium rule, Simpson's rule, Boole's rule etc.)

I'm wondering:

  • Assuming we can't re-evaluate the function on a different grid, is there a way of approaching this that is significantly better than Newton-Cotes?

  • If we are using Newton-Cotes, is higher-order always better? More specifically is it always better to divide the data into a smaller number of intervals evaluated with a higher-order scheme rather than a larger number of intervals evaluated with a lower-order scheme?

Thanks!

$\endgroup$
4
$\begingroup$

First, it can be found that Newton-Cotes can suffer from Runge's Phenomena when you use higher order version of it (because of the underlying Lagrange interpolants used in the formulation). So I suspect there is some optimal Newton-Cotes approach which does a balance of using a refined number of intervals with a lower order approach.

Additionally, a quadrature that might be better is Gauss Quadrature, which uses $n$ points to integrate a $2n-1$ order polynomial (approximation) exactly.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.