Algebraic Algorithms for D-Modules and Numerical Analysis

Oaku, Toshinori; Shiraki, Yoshinao; Takayama, Nobuki

doi:10.1142/9789812704436_0003

Cited by 17 publications

(33 citation statements)

References 17 publications

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“…Various researches have revisited the theory of D-modules with a view to determining computational methods. One of the remarkable discoveries is the restriction and integration algorithm found by Oaku [25], and we recommend his introductory book on the subject [26]. In the theory of D-modules, the notions of restriction and the integration of modules play a central role, and they lead to algorithms for several objects.…”

Section: Integration Algorithmmentioning

confidence: 98%

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Gröbner Basis for Rings of Differential Operators and Applications

Takayama

2013

Gröbner Bases

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We introduce the theory and present some applications of Gröbner bases for the rings of differential operators with rational function coefficients R and for those with polynomial coefficients D.The discussion with R, in the first half, is elementary. In the ring of polynomials, zero-dimensional ideals form the biggest class, and this is also true in R. However, in D, there is no zero-dimensional ideal, and holonomic ideals form the biggest class. Most algorithms for D use holonomic ideals.As an application, we present an algorithm for finding local minimums of holonomic functions; it can be applied to the maximum-likelihood estimate. The last part of this chapter considers A-hypergeometric systems; topics covered in other chapters will reappear in the study of A-hypergeometric systems. We have provided many of the proofs, but some technical proofs in the second half of this chapter have been omitted; these may be found in the references at the end of this chapter.

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Section: Integration Algorithmmentioning

confidence: 98%

“…1 also hold here. It is left as an exercise to prove them rigorously in the case of D. If you need guidance, we refer you to [26,29] We have called this a "polynomial", but, strictly speaking, it is a differential operator.…”

Section: Gröbner Bases In the Ring Of Differential Operators With Polmentioning

confidence: 99%