An alternative to the Euler--Maclaurin formula: Approximating sums by integrals only

TL;DR

This paper introduces an alternative summation formula that approximates sums using only integrals, avoiding derivatives, and demonstrates its efficiency and parallelizability compared to the traditional Euler--Maclaurin formula.

Contribution

A new summation method is proposed that replaces derivatives with integrals, improving computational speed and memory efficiency, especially for parallel processing.

Findings

Alt formula outperforms EM in speed and memory usage

Both formulas are faster than Mathematica's HurwitzZeta for high-accuracy calculations

Alt formula is highly parallelizable

Abstract

The Euler--Maclaurin (EM) summation formula is used in many theoretical studies and numerical calculations. It approximates the sum $\sum_{k=0}^{n-1} f(k)$ of values of a function $f$ by a linear combination of a corresponding integral of $f$ and values of its higher-order derivatives $f^{(j)}$. An alternative (Alt) summation formula is proposed, which approximates the sum by a linear combination of integrals only, without using high-order derivatives of $f$. Explicit and rather easy to use bounds on the remainder are given. Possible extensions to multi-index summation are suggested. Applications to summing possibly divergent series are presented. It is shown that the Alt formula will in most cases outperform, or greatly outperform, the EM formula in terms of the execution time and memory use. One of the advantages of the Alt calculations is that, in contrast with the EM ones, they can be almost completely parallelized. Illustrative examples are given. In one of the examples, where an array of values of the Hurwitz generalized zeta function is computed with high accuracy, it is shown that both our implementation of the EM formula and, especially, the Alt formula perform much faster than the built-in Mathematica command HurwitzZeta[].