An Invitation to Hermite's Integration and Summation: A Comparison between Hermite's and Simpson's Rules

Abstract

Hermite's rule for numerical integration is presented and compared with the more familiar Simpson's rule. Through several examples, the power of Hermite's rule for numerical summation is then demonstrated. It is established that Hermite's rule surpasses Simpson's rule due to better error estimates and due to the fact that Simpson's rule is not easily applicable to numerical summation. For these reasons, it is argued that Hermite's rule should replace Simpson's rule in the undergraduate curriculum. The derivation of Hermite's rule is accessible to undergraduates, and the topic should be included in the undergraduate curriculum.