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Year 2022, Volume: 5 Issue: 2, 72 - 92, 15.06.2022
https://doi.org/10.33205/cma.1102689

Abstract

References

  • A. Abdesselam, V. Rivasseau: An explicit large versus small field multiscale cluster expansion, Rev. Math. Phys., 9 (2) (1997), 123–199.
  • A. Abdesselam, V. Rivasseau: Trees, forests and jungles: a botanical garden for cluster expansions, Constructive physics (Palaiseau, 1994), Lecture Notes in Phys., vol. 446, Springer, Berlin, 1995, pp. 7–36.
  • J. Agapito, L. Godinho: New polytope decompositions and Euler-Maclaurin formulas for simple integral polytopes, Adv. Math., 214 (1) (2007), 379–416.
  • J. Agapito, L. Godinho: Cone decompositions of non-simple polytopes, J. Symplectic Geom., 14 (3) (2016), 737–766.
  • A. I. Barvinok: A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed, Math. Oper. Res., 19 (4) (1994), 769–779.
  • N. Berline, M. Vergne: Local Euler-Maclaurin formula for polytopes, Mosc. Math. J., 7 (3) (2007), 355– 386, 573.
  • M. Brion, M. Vergne: Lattice points in simple polytopes, J. Amer. Math. Soc., 10 (2) (1997), 371–392.
  • S. E. Cappell, Julius L. Shaneson: Euler-Maclaurin expansions for lattices above dimension one, C. R. Acad. Sci. Paris Sér. I Math., 321 (7) (1995), 885–890.
  • A. Carbery, M. Christ and James Wright: Multidimensional van der Corput and sublevel set estimates, J. Amer. Math. Soc., 12 (4) (1999), 981–1015.
  • V. Guillemin, S. Sternberg: Riemann sums over polytopes, vol. 57, 2007, Festival Yves Colin de Verdière, pp. 2183–2195.
  • C. Haase: Polar decomposition and Brion’s theorem, Integer points in polyhedra—geometry, number theory, algebra, optimization, Contemp. Math., vol. 374, Amer. Math. Soc., Providence, RI, 2005, pp. 91–99.
  • L. Hörmander: Lectures on nonlinear hyperbolic differential equations, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 26, Springer-Verlag, Berlin, (1997).
  • Y. Karshon, S. Sternberg and J.Weitsman: The Euler-Maclaurin formula for simple integral polytopes, Proc. Natl. Acad. Sci. USA, 100 (2) (2003), 426–433.
  • Y. Karshon, S. Sternberg and J. Weitsman: Euler-Maclaurin with remainder for a simple integral polytope, Duke Math. J., 130 (3) (2005), 401–434.
  • N. H. Katz, E. Krop and M. Maggioni: Remarks on the box problem, Math. Res. Lett., 9 (4) (2002), 515–519.
  • K. Knopp: Theory and application of infinite series, Blackie, London, (1951).
  • J. Lawrence: Polytope volume computation, Math. Comp., 57 (195) (1991), 259–271.
  • Y. L. Floch, Á. Pelayo: Euler-MacLaurin formulas via differential operators, Adv. in Appl. Math., 73 (2016), 99–124.
  • I. Pinelis: An alternative to the Euler–Maclaurin summation formula: approximating sums by integrals only, Numerische Mathematik, 140 (3) (2018), 755–790.
  • A. V. Pukhlikov, A. G. Khovanski: The Riemann-Roch theorem for integrals and sums of quasipolynomials on virtual polytopes, Algebra i Analiz, 4 (4) (1992), 188–216.
  • A. Szenes, M. Vergne: Residue formulae for vector partitions and Euler-MacLaurin sums, Advances in Applied Mathematics, 30 (1–2) (2003), 295–342.
  • T. Tate: Asymptotic Euler-Maclaurin formula over lattice polytopes, J. Funct. Anal., 260 (2) (2011), 501–540.
  • G. N. Watson: A note on Gamma functions, Edinburgh Math. Notes, 42 (1959), 7–9.
  • H. Whitney: Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., 36 (1) (1934), 63–89.

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

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

References

  • A. Abdesselam, V. Rivasseau: An explicit large versus small field multiscale cluster expansion, Rev. Math. Phys., 9 (2) (1997), 123–199.
  • A. Abdesselam, V. Rivasseau: Trees, forests and jungles: a botanical garden for cluster expansions, Constructive physics (Palaiseau, 1994), Lecture Notes in Phys., vol. 446, Springer, Berlin, 1995, pp. 7–36.
  • J. Agapito, L. Godinho: New polytope decompositions and Euler-Maclaurin formulas for simple integral polytopes, Adv. Math., 214 (1) (2007), 379–416.
  • J. Agapito, L. Godinho: Cone decompositions of non-simple polytopes, J. Symplectic Geom., 14 (3) (2016), 737–766.
  • A. I. Barvinok: A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed, Math. Oper. Res., 19 (4) (1994), 769–779.
  • N. Berline, M. Vergne: Local Euler-Maclaurin formula for polytopes, Mosc. Math. J., 7 (3) (2007), 355– 386, 573.
  • M. Brion, M. Vergne: Lattice points in simple polytopes, J. Amer. Math. Soc., 10 (2) (1997), 371–392.
  • S. E. Cappell, Julius L. Shaneson: Euler-Maclaurin expansions for lattices above dimension one, C. R. Acad. Sci. Paris Sér. I Math., 321 (7) (1995), 885–890.
  • A. Carbery, M. Christ and James Wright: Multidimensional van der Corput and sublevel set estimates, J. Amer. Math. Soc., 12 (4) (1999), 981–1015.
  • V. Guillemin, S. Sternberg: Riemann sums over polytopes, vol. 57, 2007, Festival Yves Colin de Verdière, pp. 2183–2195.
  • C. Haase: Polar decomposition and Brion’s theorem, Integer points in polyhedra—geometry, number theory, algebra, optimization, Contemp. Math., vol. 374, Amer. Math. Soc., Providence, RI, 2005, pp. 91–99.
  • L. Hörmander: Lectures on nonlinear hyperbolic differential equations, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 26, Springer-Verlag, Berlin, (1997).
  • Y. Karshon, S. Sternberg and J.Weitsman: The Euler-Maclaurin formula for simple integral polytopes, Proc. Natl. Acad. Sci. USA, 100 (2) (2003), 426–433.
  • Y. Karshon, S. Sternberg and J. Weitsman: Euler-Maclaurin with remainder for a simple integral polytope, Duke Math. J., 130 (3) (2005), 401–434.
  • N. H. Katz, E. Krop and M. Maggioni: Remarks on the box problem, Math. Res. Lett., 9 (4) (2002), 515–519.
  • K. Knopp: Theory and application of infinite series, Blackie, London, (1951).
  • J. Lawrence: Polytope volume computation, Math. Comp., 57 (195) (1991), 259–271.
  • Y. L. Floch, Á. Pelayo: Euler-MacLaurin formulas via differential operators, Adv. in Appl. Math., 73 (2016), 99–124.
  • I. Pinelis: An alternative to the Euler–Maclaurin summation formula: approximating sums by integrals only, Numerische Mathematik, 140 (3) (2018), 755–790.
  • A. V. Pukhlikov, A. G. Khovanski: The Riemann-Roch theorem for integrals and sums of quasipolynomials on virtual polytopes, Algebra i Analiz, 4 (4) (1992), 188–216.
  • A. Szenes, M. Vergne: Residue formulae for vector partitions and Euler-MacLaurin sums, Advances in Applied Mathematics, 30 (1–2) (2003), 295–342.
  • T. Tate: Asymptotic Euler-Maclaurin formula over lattice polytopes, J. Funct. Anal., 260 (2) (2011), 501–540.
  • G. N. Watson: A note on Gamma functions, Edinburgh Math. Notes, 42 (1959), 7–9.
  • H. Whitney: Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., 36 (1) (1934), 63–89.
There are 24 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Iosif Pinelis 0000-0003-4742-5789

Publication Date June 15, 2022
Published in Issue Year 2022 Volume: 5 Issue: 2

Cite

APA Pinelis, I. (2022). Approximating sums by integrals only: multiple sums and sums over lattice polytopes. Constructive Mathematical Analysis, 5(2), 72-92. https://doi.org/10.33205/cma.1102689
AMA Pinelis I. Approximating sums by integrals only: multiple sums and sums over lattice polytopes. CMA. June 2022;5(2):72-92. doi:10.33205/cma.1102689
Chicago Pinelis, Iosif. “Approximating Sums by Integrals Only: Multiple Sums and Sums over Lattice Polytopes”. Constructive Mathematical Analysis 5, no. 2 (June 2022): 72-92. https://doi.org/10.33205/cma.1102689.
EndNote Pinelis I (June 1, 2022) Approximating sums by integrals only: multiple sums and sums over lattice polytopes. Constructive Mathematical Analysis 5 2 72–92.
IEEE I. Pinelis, “Approximating sums by integrals only: multiple sums and sums over lattice polytopes”, CMA, vol. 5, no. 2, pp. 72–92, 2022, doi: 10.33205/cma.1102689.
ISNAD Pinelis, Iosif. “Approximating Sums by Integrals Only: Multiple Sums and Sums over Lattice Polytopes”. Constructive Mathematical Analysis 5/2 (June 2022), 72-92. https://doi.org/10.33205/cma.1102689.
JAMA Pinelis I. Approximating sums by integrals only: multiple sums and sums over lattice polytopes. CMA. 2022;5:72–92.
MLA Pinelis, Iosif. “Approximating Sums by Integrals Only: Multiple Sums and Sums over Lattice Polytopes”. Constructive Mathematical Analysis, vol. 5, no. 2, 2022, pp. 72-92, doi:10.33205/cma.1102689.
Vancouver Pinelis I. Approximating sums by integrals only: multiple sums and sums over lattice polytopes. CMA. 2022;5(2):72-9.