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NumGfun: a Package for Numerical and Analytic Computation with Dfinite Functions
"... This article describes the implementation in the software package NumGfun of classical algorithms that operate on solutions of linear differential equations or recurrence relations with polynomial coefficients, including what seems to be the first general implementation of the fast highprecision nu ..."
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This article describes the implementation in the software package NumGfun of classical algorithms that operate on solutions of linear differential equations or recurrence relations with polynomial coefficients, including what seems to be the first general implementation of the fast highprecision numerical evaluation algorithms of Chudnovsky & Chudnovsky. In some cases, our descriptions contain improvements over existing algorithms. We also provide references to relevant ideas not currently used in NumGfun.
The Holonomic Toolkit
"... This is an overview over standard techniques for holonomic functions, written for readers who are new to the subject. We state the definition for holonomy in a couple of different ways, including some concrete special cases as well as a more abstract and more general version. We give a collection ..."
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This is an overview over standard techniques for holonomic functions, written for readers who are new to the subject. We state the definition for holonomy in a couple of different ways, including some concrete special cases as well as a more abstract and more general version. We give a collection of standard examples and state several fundamental properties of holonomic objects. Two techniques which are most useful in applications are explained in some more detail: closure properties, which can be used to prove identities among holonomic functions, and guessing, which can be used to generate plausible conjectures for equations satisfied by a given function.
A Note on the Space Complexity of Fast DFinite Function Evaluation
"... Abstract. We state and analyze a generalization of the “truncation trick ” suggested by Gourdon and Sebah to improve the performance of power series evaluation by binary splitting. It follows from our analysis that the values of Dfinite functions (i.e., functions described as solutions of linear di ..."
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Abstract. We state and analyze a generalization of the “truncation trick ” suggested by Gourdon and Sebah to improve the performance of power series evaluation by binary splitting. It follows from our analysis that the values of Dfinite functions (i.e., functions described as solutions of linear differential equations with polynomial coefficients) may be computed with error bounded by 2 −p in timeO(p(lgp) 3+o(1) ) and spaceO(p). The standard fast algorithm for this task, due to Chudnovsky and Chudnovsky, achieves the same time complexity bound but requires Θ(p lgp) bits of memory. 1.
Termination Conditions for Positivity Proving Procedures
"... Proving positivity of a sequence given by a linear recurrence with polynomial coefficients (Pfinite recurrence) is a nontrivial task for both humans and computers. Algorithms dealing with this task are rare or nonexistent. One method that was introduced in the last decade by Gerhold and Kauers su ..."
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Proving positivity of a sequence given by a linear recurrence with polynomial coefficients (Pfinite recurrence) is a nontrivial task for both humans and computers. Algorithms dealing with this task are rare or nonexistent. One method that was introduced in the last decade by Gerhold and Kauers succeeds on many examples, but termination of this procedure has been proven so far only up to order three for special cases. Here we present an analysis that extends the previously known termination results on recurrences of order three, and also provides termination conditions for recurrences of higher order.
9. Bibliography............................................................................111. Team Research Scientist
"... c t i v it y e p o r t 2009 Table of contents 1. Team.................................................................................... 1 ..."
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c t i v it y e p o r t 2009 Table of contents 1. Team.................................................................................... 1
Automatic Proofs of Identities
, 2011
"... We present the ideas behind algorithmic proofs of identities involving sums and integrals of large classes of special functions. Recent results allowed a new extension of the class of holonomic functions. 1 ..."
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We present the ideas behind algorithmic proofs of identities involving sums and integrals of large classes of special functions. Recent results allowed a new extension of the class of holonomic functions. 1
bInstitute for Computing and Information Sciences
"... Contrary to linear difference equations, there is no general theory of difference equations of the form G(P (x − τ1),..., P (x − τs)) + G0(x)=0, with τi ∈ K, G(x1,..., xs) ∈ K[x1,..., xs] of total degree D ≥ 2 and G0(x) ∈ K[x], where K is a field of characteristic zero. This article is concerned w ..."
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Contrary to linear difference equations, there is no general theory of difference equations of the form G(P (x − τ1),..., P (x − τs)) + G0(x)=0, with τi ∈ K, G(x1,..., xs) ∈ K[x1,..., xs] of total degree D ≥ 2 and G0(x) ∈ K[x], where K is a field of characteristic zero. This article is concerned with the following problem: given τi, G and G0, find an upper bound on the degree d of a polynomial solution P (x), if it exists. In the presented approach the problem is reduced to constructing a univariate polynomial for which d is a root. The authors formulate a sufficient condition under which such a polynomial exists. Using this condition, they give an effective bound on d, for instance, for all difference equations of the form G P (x − a), P (x − a − 1), P (x − a − 2)) + G0(x) = 0 with quadratic G, and all difference equations of the form
approximation errors
, 2010
"... For purposes of actual evaluation, mathematical functions f are commonly replaced by approximation polynomials p. Examples include floatingpoint implementations of elementary functions, quadrature or more theoretical proof work involving transcendental functions. Replacing f by p induces a relative ..."
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For purposes of actual evaluation, mathematical functions f are commonly replaced by approximation polynomials p. Examples include floatingpoint implementations of elementary functions, quadrature or more theoretical proof work involving transcendental functions. Replacing f by p induces a relative error ε = p/f −1. In order to ensure the validity of the use of p instead of f, the maximum error, i.e. the supremum norm ‖ε‖ ∞ must be safely bounded above. Numerical algorithms for supremum norms are efficient but cannot offer the required safety. Previous validated approaches often require tedious manual intervention. If they are automated, they have several drawbacks, such as the lack of quality guarantees. In this article a novel, automated supremum norm algorithm with a priori quality is proposed. It focuses on the validation step and paves the way for formally certified supremum norms.