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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… (More)

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 high-precision numerical evaluation algorithms of Chudnovsky &… (More)

We describe the main features of the Dynamic Dictionary of Mathematical Functions (version 1.5). It is a website consisting of interactive tables of mathematical formulas on elementary and special functions. The formulas are automatically generated by computer algebra routines. The user can ask for more terms of the expansions, more digits of the numerical… (More)

—We introduce an algorithm that allows one to quickly compare a binary64 floating-point (FP) number and a decimal64 FP number, assuming the " binary encoding " of the decimal formats specified by the IEEE 754-2008 standard for FP arithmetic is used. It is a two-step algorithm: a first pass, based on the exponents only, makes it possible to quickly eliminate… (More)

- Marc Mezzarobba
- 2011

—The series expansion at the origin of the Airy function Ai(x) is alternating and hence problematic to evaluate for x > 0 due to cancellation. Based on a method recently proposed by Gawronski, Müller, and Reinhard, we exhibit two functions F and G, both with nonnegative Taylor expansions at the origin, such that Ai(x) = G(x)/F(x). The sums are now… (More)

We present a new open source implementation in the Sage-Math computer algebra system of algorithms for the numerical solution of linear ODEs with polynomial coefficients. Our code supports regular singular connection problems and provides rigorous error bounds. Extended abstract for a talk given at the 5th International Congress on Mathematical Software… (More)

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 D-finite functions (i.e., functions described as solutions of linear differential equations with polynomial coefficients) may be computed… (More)

—We introduce an algorithm to compare a binary floating-point (FP) number and a decimal FP number, assuming the " binary encoding " of the decimal formats is used, and with a special emphasis on the basic interchange formats specified by the IEEE 754-2008 standard for FP arithmetic. It is a two-step algorithm: a first pass, based on the exponents only,… (More)

—This work introduces an approach to the computer-assisted implementation of mathematical functions geared toward special functions such as those occurring in mathematical physics. The general idea is to start with an exact symbolic representation of a function and automate as much as possible of the process of implementing it. In order to deal with a large… (More)