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 nonperiodic 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 formulas described by James Gregory in 1670. 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). In the context of quadrature methods on equispaced nodes, and for orders of accuracy around 10 or higher, this leads to weights of oscillating signs and large magnitudes. This article develops further a recently discovered approach for avoiding these adverse effects.

Automated summary

The trapezoidal rule is accurate for integrals over periodic intervals but inaccurate in nonperiodic cases. Common improvements, such as Simpson's rule and the Newton-Cotes formulas, may not be better than classical quadrature formulas. These methods suffer from the Runge phenomenon for increasing orders of accuracy.