Approximating sums by integrals only: multiple sums and sums over lattice polytopes
Abstract
The Euler--Maclaurin (EM) summation formula is used in many theoretical studies and numerical calculations. It approximates the sum of values of a function by a linear combination of a corresponding integral of and values of its higher-order derivatives . An alternative (Alt) summation formula was recently presented by the author, which approximates the sum by a linear combination of integrals only, without using high-order derivatives of . It was 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. 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.
Cite
@article{arxiv.1705.09159,
title = {Approximating sums by integrals only: multiple sums and sums over lattice polytopes},
author = {Iosif Pinelis},
journal= {arXiv preprint arXiv:1705.09159},
year = {2017}
}
Comments
Version 4: a minor mistake in line 3 of Step 3 in the proof of Lemma 5.3 is fixed. Version 5: restriction in Th. 2.1 is removed. Version 6: restriction that the polytope be simple is removed. Version 7: mistake in definition of \mathbb{R}^+_J is fixed, w/ corresponding small changes in proof of Proposition 4.1