Title: Improving the accuracy of the trapezoidal rule
The trapezoidal rule uses function values at equispaced nodes. It is very accurate for integrals over periodic intervals, but is usually quite inaccurate in non-periodic cases. Commonly used improvements, such as Simpson’s rule and the Newton-Cotes formulas, are not much (if at all) better than the even more classical quadrature method by James Gregory (1638-1675). For increasing orders of accuracy, these methods all suffer from the Runge phenomenon (the fact that polynomial interpolants on equispaced grids become violently oscillatory as their degree increases), causing quadrature weights to become of oscillating signs and large magnitudes.
When looking further into a recently developed (radial basis function-based) method for numerical quadrature over curved bounded surfaces, it was noted that this approach somehow managed avoid adverse Runge phenomenon-type effects even when their orders of accuracy was increased. This inspired the method that will be focused on here. It transpires that there in fact does exist weight sets that can be of very high order of accuracy without any weights becoming either negative or large in magnitude (based on equispaced nodes over the interval). In particular, we have found a 10th order accurate set of weights that are all relatively simple rational numbers, and a very brief algorithm (10 lines in Matlab) produces Runge-phenomenon-free weight sets of 20th (and still higher) orders.
Contact Name: Pavel Lushnikov