Cone decompositions of non-simple polytopes

Agapito, José; Godinho, Leonor

doi:10.4310/jsg.2016.v14.n3.a4

Cited by 2 publications

(3 citation statements)

References 12 publications

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“…Note also that [14, Theorem 2] is obtained for simple polytopes. In [3], this result was extended to allow more general weights, and then further generalized to non-simple polytopes in [4].…”

Section: Dxf (X)mentioning

confidence: 99%

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

Pinelis

2022

Constructive Mathematical Analysis

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The Euler-Maclaurin (EM) summation formula is used in many theoretical studies and numerical calculations. It approximates the sum n−1 k=0 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/multiindex-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.

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Section: Dxf (X)mentioning

confidence: 99%

“…Extensions of the EM formula to the multiple sums, including sums over the integral points of integral lattice polytopes, have been of significant interest; see e.g. [20,8,7,13,21,14,6,3,10,22,18,4]. In the present paper, a multiple-sum/multi-index-sum extension of the Alt formula will be given.…”

Section: Introductionmentioning

confidence: 99%

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

Pinelis

2022

Constructive Mathematical Analysis

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“…Note also that [7, Theorem 2] is obtained for simple polytopes. In [1], this result was extended to allow more general weights, and then further generalized to non-simple polytopes in [2].…”

Section: Application To Sums Over the Integral Points Of Integral Lat...mentioning

confidence: 99%

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

Pinelis¹

2017

Preprint

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The Euler-Maclaurin (EM) summation formula is used in many theoretical studies and numerical calculations. It approximates the sum ∑ n−1 k=0 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 recently presented by the author, which approximates the sum by a linear combination of integrals only, without using high-order derivatives of f . 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.

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Cone decompositions of non-simple polytopes

Abstract: A polytope cone decomposition expresses the characteristic function of a polytope as a sum of characteristic functions of convex cones associated to its faces. In this work we generalize the decompositions for simple polytopes introduced in [3] to any convex polytope.

Cited by 2 publications

References 12 publications

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

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

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

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