Creative Telescoping for Holonomic Functions

Koutschan, Christoph

doi:10.1007/978-3-7091-1616-6_7

Cited by 47 publications

(41 citation statements)

References 79 publications

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“…In other words, a linear recursive sequence is a special case of a holonomic (or P-recursive) sequence whose recurrence relations only have constant coefficients (see [11]). …”

Section: Definition and Characterization Of Linear Recursive Sequencesmentioning

confidence: 99%

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Linear Algebra for Computing Gröbner Bases of Linear Recursive Multidimensional Sequences

Berthomieu

Boyer

Faugère

2015

Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

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Sakata generalized the Berlekamp -Massey algorithm to n dimensions in 1988. The Berlekamp -Massey -Sakata (BMS) algorithm can be used for finding a Gröbner basis of a 0-dimensional ideal of relations verified by a table. We investigate this problem using linear algebra techniques, with motivations such as accelerating change of basis algorithms (FGLM) or improving their complexity.We first define and characterize multidimensional linear recursive sequences for 0-dimensional ideals. Under genericity assumptions, we propose a randomized preprocessing of the table that corresponds to performing a linear change of coordinates on the polynomials associated with the linear recurrences. This technique then essentially reduces our problem to using the efficient 1-dimensional Berlekamp -Massey (BM) algorithm. However, the number of probes to the table in this scheme may be elevated. We thus consider the table in the black-box model: we assume probing the table is expensive and we minimize the number of probes to the table in our complexity model. We produce an FGLM-like algorithm for finding the relations in the table, which lets us use linear algebra techniques. Under some additional assumptions, we make this algorithm adaptive and reduce further the number of table probes. This number can be estimated by counting the number of distinct elements in a multi-Hankel matrix (a multivariate generalization of Hankel matrices); we can relate this quantity with the geometry of the final staircase. Hence, in favorable cases such as convex ones, the complexity is essentially linear in the size of the output. Finally, when using the LEX ordering, we can make use of fast structured linear algebra similarly to the Hankel interpretation of Berlekamp -Massey.

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“…In other words, a linear recursive sequence is a special case of a holonomic (or P-recursive) sequence whose recurrence relations only have constant coefficients (see [11]). …”

Section: Definition and Characterization Of Linear Recursive Sequencesmentioning

confidence: 99%

“…In other words, if u is linear recursive, then K[x]/I is a finite dimensional K-vector space. This is related to the definition of holonomic function in an Ore algebra A = K(z) ∂ z , see [11,Def. 1].…”

Section: Gröbner Bases and Characterization Of Linear Recursive Sequementioning

confidence: 99%

Linear Algebra for Computing Gröbner Bases of Linear Recursive Multidimensional Sequences

Berthomieu

Boyer

Faugère

2015

Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

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“…We describe here the general context and a few of the recent developments. More information can be found in recent surveys [53,91].…”

Section: Example 12mentioning

confidence: 99%

Linear Differential Equations as a Data Structure

Salvy

2019

Found Comput Math

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A. A lot of information concerning solutions of linear differential equations can be computed directly from the equation. It is therefore natural to consider these equations as a data-structure, from which mathematical properties can be computed. 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.1991 Mathematics Subject Classification. 68W30 and 33F10.

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“…This successful story has been pushed forward for single nested sums and related problems, see, e.g., [25,26,10] . Further generalizations opened up substantially the class of applications, like the holonomic approach [48,11,21] dealing with objects that can be desribed by recurrence systems or the multi-summation approach of (q-)hypergeometric products [47,46,7].…”

Section: Introductionmentioning

confidence: 99%

Minimal Representations and Algebraic Relations for Single Nested Products

Schneider

2020

Program Comput Soft

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Recently, it has been shown constructively how a finite set of hypergeometric products, multibasic hypergeometric products or their mixed versions can be modeled properly in the setting of formal difference rings. Here special emphasis is put on robust constructions: whenever further products have to be considered, one can reuse -up to some mild modifications-the already existing difference ring. In this article we relax this robustness criteria and seek for another form of optimality. We will elaborate a general framework to represent a finite set of products in a formal difference ring where the number of transcendental product generators is minimal. As a bonus we are able to describe explicitly all relations among the given input products.This work was supported by the Austrian Science Fund (FWF) grant SFB F50 (F5009-N15).3 In this example the evaluation of an element from É(ι)(x) is carried out by replacing x with concrete values n ∈ AE. Later we will generalize this simplest case to formal difference rings equipped with an evaluation function acting on the ring elements.4 More generally, if R is a commutative ring with 1, we define the ideal I generated by a 1 , . . . , ar ∈ R with I = a 1 , . . . , ar R = {f 1 a 1 + · · · + fr ar | f 1 , . . . , fr ∈ R}.

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Creative Telescoping for Holonomic Functions

Cited by 47 publications

References 79 publications

Linear Algebra for Computing Gröbner Bases of Linear Recursive Multidimensional Sequences

Linear Algebra for Computing Gröbner Bases of Linear Recursive Multidimensional Sequences

Linear Differential Equations as a Data Structure

Minimal Representations and Algebraic Relations for Single Nested Products

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