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

2017/05/24 by Iosif Pinelis, Pinelis, Iosif
Mathematics · #Advanced Mathematical Identities #Mathematical functions and polynomials #Functional Equations Stability Results

paper · pdf · doi:10.48550/arxiv.1705.09159

Abstract

The Euler--Maclaurin (EM) summation formula is used in many theoretical\nstudies and numerical calculations. It approximates the sum \∑_k=0^n-1\nf(k) of values of a function f by a linear combination of a corresponding\nintegral of f and values of its higher-order derivatives f^(j). An\nalternative (Alt) summation formula was recently presented by the author, which\napproximates the sum by a linear combination of integrals only, without using\nhigh-order derivatives of f. It was shown that the Alt formula will in most\ncases outperform, or greatly outperform, the EM formula in terms of the\nexecution time and memory use. In the present paper, a\nmultiple-sum/multi-index-sum extension of the Alt formula is given, with\napplications to summing possibly divergent multi-index series and to sums over\nthe integral points of integral lattice polytopes.\n

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