Choice of Newton-Cotes formulae for regularly gridded multi-dimensional data

Question

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!

Answer 1

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.