Approximating sums by integrals only: multiple sums and sums over lattice polytopes

Year 2022, Volume: 5 Issue: 2, 72 - 92, 15.06.2022

Iosif Pinelis

https://doi.org/10.33205/cma.1102689

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 was presented by the author, which approximates the sum by a linear combination of integrals only, without using derivatives of $f$. It was shown that the Alt formula will in most cases outperform the EM formula. In the present paper, a multiple-sum/multi-index-sum extension of the Alt formula is given, with applications to summing possibly divergent multi-index series and to sums over the integral points of integral lattice polytopes.

Keywords

Euler-Maclaurin summation formula, alternative summation formula, multiple sums, multi-index series, approximation, lattice polytopes

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