research-article

Generalized Hermite Reduction, Creative Telescoping and Definite Integration of D-Finite Functions

Authors:

Frédéric Chyzak,

Bruno SalvyAuthors Info & Claims

ISSAC '18: Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation

Pages 95 - 102

https://doi.org/10.1145/3208976.3208992

Published: 11 July 2018 Publication History

Abstract

Hermite reduction is a classical algorithmic tool in symbolic integration. It is used to decompose a given rational function as a sum of a function with simple poles and the derivative of another rational function. We extend Hermite reduction to arbitrary linear differential operators instead of the pure derivative, and develop efficient algorithms for this reduction. We then apply the generalized Hermite reduction to the computation of linear operators satisfied by single definite integrals of D-finite functions of several continuous or discrete parameters. The resulting algorithm is a generalization of reduction-based methods for creative telescoping.

References

[1]

S. A. Abramov. EG-eliminations. J. Differ. Equations Appl., 5(4--5):393--433, 1999.

[2]

S. A. Abramov and K. Y. Kvashenko. Fast algorithms for the search of the rational solutions of linear differential equations with polynomial coefficients. In ISSAC'91, pages 267--270, 1991.

Digital Library

[3]

S. A. Abramov and M. Petkovšek. Minimal decomposition of indefinite hypergeometric sums. In ISSAC'01, pages 7--14. ACM, 2001.

Digital Library

[4]

S. A. Abramov and M. van Hoeij. Integration of solutions of linear functional equations. Integral Transform. Spec. Funct., 8(1--2):3--12, 1999.

[5]

K. Adjamagbo. Sur l'effectivité du lemme du vecteur cyclique. C. R. Acad. Sci. Paris Sér. I Math., 306(13):543--546, 1988.

[6]

A. Adolphson. An index theorem for p-adic differential operators. Trans. Amer. Math. Soc., 216:279--293, 1976.

[7]

G. Almkvist and D. Zeilberger. The method of differentiating under the integral sign. J. Symbolic Comput., 10(6):571--591, 1990.

Digital Library

[8]

M. A. Barkatou. On rational solutions of systems of linear differential equations. J. Symbolic Comput., 28(4--5):547--567, 1999.

[9]

A. Bostan, S. Chen, F. Chyzak, and Z. Li. Complexity of creative telescoping for bivariate rational functions. In ISSAC'10, pages 203--210. ACM, 2010.

Digital Library

[10]

A. Bostan, S. Chen, F. Chyzak, Z. Li, and G. Xin. Hermite reduction and creative telescoping for hyperexponential functions. In ISSAC'13, pages 77--84. ACM, 2013.

Digital Library

[11]

A. Bostan, L. Dumont, and B. Salvy. Efficient algorithms for mixed creative telescoping. In ISSAC'16, pages 127--134. ACM, 2016.

Digital Library

[12]

A. Bostan, P. Lairez, and B. Salvy. Creative telescoping for rational functions using the Griffiths-Dwork method. In ISSAC'13, pages 93--100. ACM, 2013.

Digital Library

[13]

S. Chen, H. Huang, M. Kauers, and Z. Li. A modified Abramov-Petkovšek reduction and creative telescoping for hypergeometric terms. In ISSAC'15, pages 117--124. ACM, 2015.

Digital Library

[14]

S. Chen, M. Kauers, and C. Koutschan. Reduction-based creative telescoping for algebraic functions. In ISSAC'16, pages 175--182. ACM, 2016.

Digital Library

[15]

S. Chen, M. Kauers, and M. F. Singer. Telescopers for rational and algebraic functions via residues. In ISSAC'12, pages 130--137. ACM, 2012.

Digital Library

[16]

S. Chen, M. van Hoeij, M. Kauers, and C. Koutschan. Reduction-based creative telescoping for fuchsian D-finite functions. J. Symbolic Comput., 85:108--127, 2018.

[17]

R. C. Churchill and J. J. Kovacic. Cyclic vectors. In Differential Algebra and Related Topics, pages 191--218. World Scientific, 2002.

[18]

F. Chyzak. Fonctions holonomes en calcul formel. PhD Thesis, École polytechnique, 1998.

[19]

F. Chyzak. An extension of Zeilberger's fast algorithm to general holonomic functions. Discrete Math., 217(1--3):115--134, 2000.

Digital Library

[20]

F. Chyzak. The ABC of Creative Telescoping - Algorithms, Bounds, Complexity. Accreditation to supervise research (HDR), École polytechnique, Apr. 2014.

[21]

F. Chyzak and B. Salvy. Non-commutative elimination in Ore algebras proves multivariate identities. J. Symbolic Comput., 26(2):187--227, 1998.

Digital Library

[22]

J. H. Davenport. The Risch differential equation problem. SIAM J. Comput., 15(4):903--918, 1986.

Digital Library

[23]

L. E. Dickson. Finiteness of the Odd Perfect and Primitive Abundant Numbers with n Distinct Prime Factors. Amer. J. Math., 35(4):413--422, 1913.

[24]

J. C. Faugère, P. Gianni, D. Lazard, and T. Mora. Efficient computation of zero-dimensional Gröbner bases by change of ordering. J. Symbolic Comput., 16(4):329--344, 1993.

Digital Library

[25]

L. Fuchs. Die Periodicitätsmoduln der hyperelliptischen Integrale als Functionen eines Parameters aufgefasst. J. Reine Angew. Math., 71:91--127, 1870.

[26]

K. Geddes, H. Le, and Z. Li. Differential rational normal forms and a reduction algorithm for hyperexponential functions. In ISSAC'04, pages 183--190, 2004.

Digital Library

[27]

C. Hermite. Sur l'intégration des fractions rationnelles. Ann. Sci. École Norm. Sup. (2), 1:215--218, 1872.

[28]

H. Huang. New bounds for hypergeometric creative telescoping. In ISSAC'16, pages 279--286. ACM, 2016.

Digital Library

[29]

E. L. Ince. Ordinary Differential Equations. Dover Publications, New York, 1944.

[30]

C. Koutschan. Examplesv11.nb. On the Holonomic Functions web page.

[31]

C. Koutschan. Advanced Applications of the Holonomic Systems Approach. PhD thesis, RISC-Linz, 2009.

[32]

P. Lairez. Computing periods of rational integrals. Math. Comp., 85(300):1719--1752, 2016.

[33]

J. Liouville. Second mémoire sur la détermination des intégrales dont la valeur est algébrique. Journal de l'École polytechnique, 14:149--193, 1833.

[34]

L. Lipshitz. The diagonal of a D-finite power series is D-finite. J. Algebra, 113(2):373--378, 1988.

[35]

B. Malgrange. Sur les points singuliers des équations différentielles. Enseignement Math. (2), 20:147--176, 1974.

[36]

P. Monsky. Finiteness of de Rham cohomology. Amer. J. Math., 94:237--245, 1972.

[37]

M. Ostrogradsky. De l'intégration des fractions rationnelles. Bull. classe phys.-math. Acad. Impériale des Sciences Saint-Pétersbourg, 4:145--167, 286--300, 1845.

[38]

É. Picard. Sur les intégrales doubles de fonctions rationnelles dont tous les résidus sont nuls. Bull. Sci. Math. (2), 26:143--152, 1902.

[39]

É. Picard and G. Simart. Théorie des fonctions algébriques de deux variables indépendantes, volume I (1897) and II (1906). Gauthier-Villars et fils, 1897.

[40]

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev. Integrals and series. Vol. 2. Gordon & Breach Science Publishers, NY, second edition, 1988. Special functions.

[41]

M. S. Rezaoui. Indice polynomial d'une matrice d'opérateurs différentiels. C. R. Acad. Sci. Paris Sér. I Math., 332(6):505--508, 2001.

[42]

N. Takayama. An approach to the zero recognition problem by Buchberger algorithm. Journal of Symbolic Computation, 14:265--282, 1992.

Digital Library

[43]

B. M. Trager. Integration of Algebraic Functions. PhD Thesis, MIT, 1984.

[44]

J. van der Hoeven. Constructing reductions for creative telescoping, 2017. Technical Report,

[45]

M. van der Put and M. Reversat. A local-global problem for linear differential equations. Pacific J. Math., 238(1):171--199, 2008.

[46]

M. van der Put and M. F. Singer. Galois theory of linear differential equations, volume 328 of Grundlehren der Mathematischen Wissenschaften. Springer, 2003.

[47]

D. Y. Y. Yun. On square-free decompositions algorithms. In Proc. 1976 ACM Symposium on Symbolic and Algebraic Computation, pages 26--35. ACM, 1976.

Digital Library

[48]

D. Y. Y. Yun. Fast algorithm for rational function integration. In Proc. IFIP'77 Congr., Toronto, Ont., pages 493--498. North-Holland, 1977.

Cited By

van Hoeij M(2024)Submodule approach to creative telescoping In memory of Marko PetkovšekJournal of Symbolic Computation10.1016/j.jsc.2024.102342(102342)Online publication date: Jun-2024
https://doi.org/10.1016/j.jsc.2024.102342
Brochet HSalvy B(2024)Reduction-Based Creative Telescoping for Definite Summation of D-finite FunctionsJournal of Symbolic Computation10.1016/j.jsc.2024.102329(102329)Online publication date: Apr-2024
https://doi.org/10.1016/j.jsc.2024.102329
Chen SFeng RLi ZSinger MWatt S(2024)Telescopers for differential forms with one parameterSelecta Mathematica10.1007/s00029-024-00926-630:3Online publication date: 9-Mar-2024
https://doi.org/10.1007/s00029-024-00926-6
Show More Cited By

Index Terms

Generalized Hermite Reduction, Creative Telescoping and Definite Integration of D-Finite Functions
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms
      1. Algebraic algorithms

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cover image ACM Other conferences

ISSAC '18: Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation

July 2018

418 pages

ISBN:9781450355506

DOI:10.1145/3208976

Editor:
Carlos Arreche
University of Texas at Dallas, USA
,
General Chairs:
Manuel Kauers
Johannes Kepler University, Austria
,
Alexey Ovchinnikov
City University of New York, USA
,
Program Chair:
Éric Schost
University of Waterloo, Canada

Copyright © 2018 ACM.

© 2018 Association for Computing Machinery. ACM acknowledges that this contribution was authored or co-authored by an employee, contractor or affiliate of a national government. As such, the Government retains a nonexclusive, royalty-free right to publish or reproduce this article, or to allow others to do so, for Government purposes only.

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SIGSAM: ACM Special Interest Group on Symbolic and Algebraic Manipulation

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New York, NY, United States

Publication History

Published: 11 July 2018

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ISSAC '18

ISSAC '18: International Symposium on Symbolic and Algebraic Computation

July 16 - 19, 2018

NY, New York, USA

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Cited By

van Hoeij M(2024)Submodule approach to creative telescoping In memory of Marko PetkovšekJournal of Symbolic Computation10.1016/j.jsc.2024.102342(102342)Online publication date: Jun-2024
https://doi.org/10.1016/j.jsc.2024.102342
Brochet HSalvy B(2024)Reduction-Based Creative Telescoping for Definite Summation of D-finite FunctionsJournal of Symbolic Computation10.1016/j.jsc.2024.102329(102329)Online publication date: Apr-2024
https://doi.org/10.1016/j.jsc.2024.102329
Chen SFeng RLi ZSinger MWatt S(2024)Telescopers for differential forms with one parameterSelecta Mathematica10.1007/s00029-024-00926-630:3Online publication date: 9-Mar-2024
https://doi.org/10.1007/s00029-024-00926-6
Bostan ANeiger VYurkevich S(2023)Beating binary powering for polynomial matricesProceedings of the 2023 International Symposium on Symbolic and Algebraic Computation10.1145/3597066.3597118(70-79)Online publication date: 24-Jul-2023
https://dl.acm.org/doi/10.1145/3597066.3597118
Chen SDu LKauers M(2023)Hermite Reduction for D-finite Functions via Integral BasesProceedings of the 2023 International Symposium on Symbolic and Algebraic Computation10.1145/3597066.3597092(155-163)Online publication date: 24-Jul-2023
https://dl.acm.org/doi/10.1145/3597066.3597092
Kauers MKauers M(2023)Summation and IntegrationD-Finite Functions10.1007/978-3-031-34652-1_5(393-509)Online publication date: 22-May-2023
https://doi.org/10.1007/978-3-031-34652-1_5
Kauers MKauers M(2023)OperatorsD-Finite Functions10.1007/978-3-031-34652-1_4(287-391)Online publication date: 22-May-2023
https://doi.org/10.1007/978-3-031-34652-1_4
Kauers MKauers M(2023)The Differential Case in One VariableD-Finite Functions10.1007/978-3-031-34652-1_3(185-286)Online publication date: 22-May-2023
https://doi.org/10.1007/978-3-031-34652-1_3
Kauers MKauers M(2023)The Recurrence Case in One VariableD-Finite Functions10.1007/978-3-031-34652-1_2(81-183)Online publication date: 22-May-2023
https://doi.org/10.1007/978-3-031-34652-1_2
Kauers MKauers M(2023)Background and Fundamental ConceptsD-Finite Functions10.1007/978-3-031-34652-1_1(1-80)Online publication date: 22-May-2023
https://doi.org/10.1007/978-3-031-34652-1_1
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