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

Alin Bostan, Frédéric Chyzak, Pierre Lairez, Bruno Salvy
2018 Proceedings of the 2018 ACM on International Symposium on Symbolic and Algebraic Computation - ISSAC '18  
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.  ...  D-finiteness of the Telescoping Ideal In the general case, the telescoping ideal T f of a D-finite function f need not be D-finite.  ... 
doi:10.1145/3208976.3208992 dblp:conf/issac/BostanCLS18 fatcat:wqy77bl5xfdp5ey2ogjhvh5tvi

Reduction-Based Creative Telescoping for Fuchsian D-finite Functions [article]

Shaoshi Chen, Mark van Hoeij, Manuel Kauers, Christoph Koutschan
2016 arXiv   pre-print
Continuing a series of articles in the past few years on creative telescoping using reductions, we adapt Trager's Hermite reduction for algebraic functions to fuchsian D-finite functions and develop a  ...  reduction-based creative telescoping algorithm for this class of functions, thereby generalizing our recent reduction-based algorithm for algebraic functions, presented at ISSAC 2016.  ...  Acknowledgements We would like to thank Ruyong Feng and Michael F. Singer for helpful discussions.  ... 
arXiv:1611.07421v1 fatcat:aoglxboaqzbivhbamiz7ebshce

Reduction-Based Creative Telescoping for Algebraic Functions

Shaoshi Chen, Manuel Kauers, Christoph Koutschan
2016 Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation - ISSAC '16  
Continuing a series of articles in the past few years on creative telescoping using reductions, we develop a new algorithm to construct minimal telescopers for algebraic functions.  ...  This algorithm is based on Trager's Hermite reduction and on polynomial reduction, which was originally designed for hyperexponential functions and extended to the algebraic case in this paper.  ...  Acknowledgements We would like to thank Ruyong Feng and Michael F. Singer for helpful discussions, and the anonymous referees for their constructive and helpful comments.  ... 
doi:10.1145/2930889.2930901 dblp:conf/issac/ChenKK16 fatcat:mvfcg5dghnbgrfu2pnggevqj6a

Reduction-Based Creative Telescoping for Algebraic Functions [article]

Shaoshi Chen, Manuel Kauers, Christoph Koutschan
2016 arXiv   pre-print
Continuing a series of articles in the past few years on creative telescoping using reductions, we develop a new algorithm to construct minimal telescopers for algebraic functions.  ...  This algorithm is based on Trager's Hermite reduction and on polynomial reduction, which was originally designed for hyperexponential functions and extended to the algebraic case in this paper.  ...  Acknowledgements We would like to thank Ruyong Feng and Michael F. Singer for helpful discussions.  ... 
arXiv:1602.00424v1 fatcat:lynwzloosrbzrpwtkhard6dyca

Lazy Hermite Reduction and Creative Telescoping for Algebraic Functions [article]

Shaoshi Chen, Lixin Du, Manuel Kauers
2021 arXiv   pre-print
Bronstein's lazy Hermite reduction is a symbolic integration technique that reduces algebraic functions to integrands with only simple poles without the prior computation of an integral basis.  ...  We sharpen the lazy Hermite reduction by combining it with the polynomial reduction to solve the decomposition problem of algebraic functions.  ...  Lazy Hermite Reduction Trager's generalization of Hermite reduction to algebraic functions works as follows [36, 21, 12, 15] .  ... 
arXiv:2102.06538v2 fatcat:3wjupkdkuzgjxgcmjtsefkhebi

Complexity of creative telescoping for bivariate rational functions

Alin Bostan, Shaoshi Chen, Frédéric Chyzak, Ziming Li
2010 Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation - ISSAC '10  
By considering this constrained class of inputs, we are able to blend the general method of creative telescoping with the well-known Hermite reduction.  ...  The long-term goal initiated in this work is to obtain fast algorithms and implementations for definite integration in Almkvist and Zeilberger's framework of (differential) creative telescoping.  ...  Given a Dfinite function f of the variables x and y, the definite integral F (x) = β α f (x, y) dy is D-finite, and a linear differential equation satisfied by F can be constructed [20] .  ... 
doi:10.1145/1837934.1837975 dblp:conf/issac/BostanCCL10 fatcat:fzcc5bejtrfmtk2fv7eckv2jwi

Complexity of Creative Telescoping for Bivariate Rational Functions [article]

Alin Bostan, Shaoshi Chen, Frédéric Chyzak, Ziming Li
2013 arXiv   pre-print
By considering this constrained class of inputs, we are able to blend the general method of creative telescoping with the well-known Hermite reduction.  ...  The long-term goal initiated in this work is to obtain fast algorithms and implementations for definite integration in Almkvist and Zeilberger's framework of (differential) creative telescoping.  ...  , we are indeed able to blend the general method of creative telescoping with the well-known Hermite reduction [10] .  ... 
arXiv:1301.5045v1 fatcat:nbuw5h4k6zhldj4uuf2elg76ay

Some Open Problems related to Creative Telescoping [article]

Shaoshi Chen, Manuel Kauers
2016 arXiv   pre-print
Creative telescoping is the method of choice for obtaining information about definite sums or integrals.  ...  It has been intensively studied since the early 1990s, and can now be considered as a classical technique in computer algebra. At the same time, it is still subject of ongoing research.  ...  Since there is no analogous theorem for general D-finite functions, not even for solutions of Fuchsian equations, it is not clear how to generalize the reduction based algorithm from algebraic functions  ... 
arXiv:1609.03768v1 fatcat:4d256sszknhazct3bazjg5l5ea

Creative telescoping for rational functions using the griffiths

Alin Bostan, Pierre Lairez, Bruno Salvy
2013 Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation - ISSAC '13  
We describe a precise and elementary algorithmic version of the Griffiths-Dwork method for the creative telescoping of rational functions.  ...  Creative telescoping algorithms compute linear differential equations satisfied by multiple integrals with parameters.  ...  Christol for many rewarding discussions, and we thank G. Villard and W. Zhou for communicating their complexity results in linear algebra.  ... 
doi:10.1145/2465506.2465935 dblp:conf/issac/BostanLS13 fatcat:i3zxw62hdbbppnuvjpbx4nsc4i

Efficient Algorithms for Mixed Creative Telscoping

Alin Bostan, Louis Dumont, Bruno Salvy
2016 Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation - ISSAC '16  
We design a new creative telescoping algorithm operating on this class of inputs, based on a Hermite-like reduction procedure.  ...  Creative telescoping is a powerful computer algebra paradigm -initiated by Doron Zeilberger in the 90's- for dealing with definite integrals and sums with parameters.  ...  We are grateful to the referees for their thorough work and helpful comments. This work has been supported in part by FastRelax ANR-14-CE25-0018-01.  ... 
doi:10.1145/2930889.2930907 dblp:conf/issac/BostanDS16 fatcat:kzmyji4yz5f5pfnvqtxfhllv7q

Additive Decompositions in Primitive Extensions [article]

Shaoshi Chen and Hao Du and Ziming Li
2018 arXiv   pre-print
Additive decompositions may lead to reduction-based creative-telescoping methods for nested logarithmic functions, which are not necessarily D-finite.  ...  This paper extends the classical Ostrogradsky-Hermite reduction for rational functions to more general functions in primitive extensions of certain types.  ...  Blending reductions with creative telescoping [2, 31] leads to the fourth and most recent generation of creative telescoping algorithms, which are called reduction-based algorithms [3, 4, 5, 9, 10]  ... 
arXiv:1802.02329v1 fatcat:gvd5uaoh2zgxfhiuraweoga25i

Creative Telescoping for Holonomic Functions [chapter]

Christoph Koutschan
2013 Texts & Monographs in Symbolic Computation  
the impact of creative telescoping in numerous contexts.  ...  The aim of this article is twofold: on the one hand it is intended to serve as a gentle introduction to the topic of creative telescoping, from a practical point of view; for this purpose its application  ...  Further innovations include an algorithm for hyperexponential functions based on Hermite reduction [19] and new algorithm for rational functions [22] using the Griffiths-Dwork method.  ... 
doi:10.1007/978-3-7091-1616-6_7 fatcat:7kmlabkcejfsvpwcjfi6vwrth4

Linear Differential Equations as a Data-Structure [article]

Bruno Salvy
2018 arXiv   pre-print
A lot of information concerning solutions of linear differential equations can be computed directly from the equation.  ...  A variety of algorithms has thus been designed in recent years that do not aim at "solving", but at computing with this representation. Many of these results are surveyed here.  ...  ∂-finite ideals. The notions of D-finiteness or P-recursiveness generalize as follows. Definition 14.1.  ... 
arXiv:1811.08616v1 fatcat:uvtvvzspxzgl5bgjsrgqfi5ygm

Definite Sums of Hypergeometric Terms and Limits of P-Recursive Sequences [article]

Hui Huang
2017 arXiv   pre-print
In the first part, we generalize the reduction-based creative telescoping algorithms to the hypergeometric setting, which allows to deal with definite sums of hypergeometric terms more quickly.  ...  The ubiquity of the class of D-finite functions and P-recursive sequences in symbolic computation is widely recognized. In this thesis, the presented work consists of two parts related to this class.  ...  In general, D-finite power series are called D-finite functions instead. A formal power series is D-finite if and only if its coefficient sequence is P-recursive.  ... 
arXiv:1710.08566v1 fatcat:he6mnewnd5cfdojbymo5byxpfa

Creative Telescoping for Parametrised Integration and Summation

Frédéric Chyzak
2011 Les cours du CIRM  
II-12 Course n o II-Creative Telescoping for Parametrised Integration and Summation The case studied by Lipshitz is that of a differentiably finite series, in short D-finite series, that is, of a series  ...  The algorithm I formulated in (2000) applies to general operators in place of just the derivations D x and D y , as long as the same kind of finiteness as with D-finite functions is preserved.  ... 
doi:10.5802/ccirm.14 fatcat:5uu7v6zlljda7jlbbqmfxostte
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