L�sungstypen von Differenzengleichungen und Summengleichungen in normierten abelschen Gruppen

Schäfke, Friedrich Wilhelm

doi:10.1007/bf01112693

Cited by 13 publications

(4 citation statements)

References 15 publications

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“…Various further extensions and renements of these results are available, see, e.g., Schäfke (1965), Kooman and Tijdeman (1990), Pituk (1997), Buslaev and Buslaeva (2005), and the references therein.…”

Section: The Perron-kreuser Theorem a Linear Recurrence Relationmentioning

confidence: 90%

Effective bounds for P-recursive sequences

Mezzarobba¹,

Salvy²

2010

Journal of Symbolic Computation

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We describe an algorithm that takes as input a complex sequence (un) given by a linear recurrence relation with polynomial coecients along with initial values, and outputs a simple explicit upper bound (vn) such that |un| ≤ vn for all n. Generically, the bound is tight, in the sense that its asymptotic behaviour matches that of un. We discuss applications to the evaluation of power series with guaranteed precision.

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Section: The Perron-kreuser Theorem a Linear Recurrence Relationmentioning

confidence: 90%

Effective bounds for P-recursive sequences

Mezzarobba¹,

Salvy²

2010

Journal of Symbolic Computation

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“…Besides, under mild additional assumptions on the Chebyshev recurrence, all the information needed in our analysis is already provided by the more elementary Perron-Kreuser theorem 3 (cf. [26,41,43]) or its extensions by Schäfke [55]. See also Immink [28] and the references therein for an alternative approach in the case of recurrences with polynomial coefficients, originating in unpublished work by Ramis.…”

Section: Elements Of Birkhoff-trjitzinsky Theorymentioning

confidence: 99%

Rigorous uniform approximation of D-finite functions using Chebyshev expansions

Benoit¹,

Joldeş²,

Mezzarobba³

2016

Math. Comp.

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Abstract. A wide range of numerical methods exists for computing polynomial approximations of solutions of ordinary differential equations based on Chebyshev series expansions or Chebyshev interpolation polynomials. We consider the application of such methods in the context of rigorous computing (where we need guarantees on the accuracy of the result), and from the complexity point of view.It is well-known that the order-truncation of the Chebyshev expansion of a function over a given interval is a near-best uniform polynomial approximation of the function on that interval. In the case of solutions of linear differential equations with polynomial coefficients, the coefficients of the expansions obey linear recurrence relations with polynomial coefficients. Unfortunately, these recurrences do not lend themselves to a direct recursive computation of the coefficients, owing among other things to a lack of initial conditions.We show how they can nevertheless be used, as part of a validated process, to compute good uniform approximations of D-finite functions together with rigorous error bounds, and we study the complexity of the resulting algorithms. Our approach is based on a new view of a classical numerical method going back to Clenshaw, combined with a functional enclosure method.

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“…Bei den klassischen Arbeiten von Poincar'e, Perron, Birkhoff und Trjitzinsky handelt es sich allerdings um reine Existenzaussagen, die keinen Hinweis auf die T. I-IANscn~ algebraische Charakterisierung und numerische Konstruktion der gesuchten L6sungen geben. Dieses Problem wurde erst viele Jahre sparer yon SCH~,FKE [24,25] und HOFFMANN [13] untersucht. Zun~ichst ordnet man die speziellen LSsungen (1.2) der GrSBe ihres Wachstums nach, so dab xl zum betraglich kleinsten und xr zum betraglich gr6Bten Eigenwert geh6rt.…”

Section: Einfiihrungunclassified

Ein verallgemeinerter Jacobi-Perron-Algorithmus zur Reduktion linearer Differenzengleichungssysteme

Hanschke

1998

Monatshefte f�r Mathematik

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L�sungstypen von Differenzengleichungen und Summengleichungen in normierten abelschen Gruppen

Cited by 13 publications

References 15 publications

Effective bounds for P-recursive sequences

Effective bounds for P-recursive sequences

Rigorous uniform approximation of D-finite functions using Chebyshev expansions

Ein verallgemeinerter Jacobi-Perron-Algorithmus zur Reduktion linearer Differenzengleichungssysteme

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