NumGfun

Mezzarobba, Marc

doi:10.1145/1837934.1837965

Cited by 30 publications

(7 citation statements)

References 24 publications

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“…These results can also be used to build sets of functions and equations where we can explicitly compute their singularities (as in Examples 5.4, 5.5 and 5.6). This is fundamental to build appropriate analytic continuation paths that are used to compute certified numerical evaluations of D-finite functions [16,17]. Future research in this direction will involve the study of the nature of these singularities.…”

Section: Discussionmentioning

confidence: 99%

Simple Differentially Definable Functions

Jiménez-Pastor

2021

Proceedings of the 2021 International Symposium on Symbolic and Algebraic Computation

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D-finite functions satisfy linear differential equations with polynomial coefficients. The solutions to this type of equations may have singularities determined by the zeros of their leading coefficient. There are algorithms to desingularize the equations, i.e., remove singularities from the equation that do not appear in its solutions. However, classical computations of closure properties (such as addition, multiplication, etc.) with D-finite functions return equations with extra zeros in the leading coefficient. In this paper we present theory and algorithms based on linear algebra to control the leading coefficients when computing these closure properties and we also extend this theory to the more general class of differentially definable functions. CCS CONCEPTS• Mathematics of computing → Mathematical software; Generating functions; Ordinary differential equations.

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Section: Discussionmentioning

confidence: 99%

Simple Differentially Definable Functions

Jiménez-Pastor

2021

Proceedings of the 2021 International Symposium on Symbolic and Algebraic Computation

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“…This would allow one to explore many variants of a numerical algorithm; this would also allow the analysis of numerical programs significantly larger than the ones we are able to deal with by pen-and-paper calculations. The pioneering work of Mezzarobba (2010Mezzarobba ( , 2020 addresses these two topics.…”

Section: Floating-point Arithmetic and Beyondmentioning

confidence: 99%

Floating-point arithmetic

Boldo¹,

Jeannerod²,

Melquiond³

et al. 2023

Acta Numerica

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Floating-point numbers have an intuitive meaning when it comes to physics-based numerical computations, and they have thus become the most common way of approximating real numbers in computers. The IEEE-754 Standard has played a large part in making floating-point arithmetic ubiquitous today, by specifying its semantics in a strict yet useful way as early as 1985. In particular, floating-point operations should be performed as if their results were first computed with an infinite precision and then rounded to the target format. A consequence is that floating-point arithmetic satisfies the ‘standard model’ that is often used for analysing the accuracy of floating-point algorithms. But that is only scraping the surface, and floating-point arithmetic offers much more.In this survey we recall the history of floating-point arithmetic as well as its specification mandated by the IEEE-754 Standard. We also recall what properties it entails and what every programmer should know when designing a floating-point algorithm. We provide various basic blocks that can be implemented with floating-point arithmetic. In particular, one can actually compute the rounding error caused by some floating-point operations, which paves the way to designing more accurate algorithms. More generally, properties of floating-point arithmetic make it possible to extend the accuracy of computations beyond working precision.

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“…For D-finite functions, there are algorithms for certified arbitrary precision numerical evaluation [11] at any regular point only using the defining differential equation, i.e., the data structure (2). These algorithms require a pre-processing step: the computation of the singularities of the differential equation (1).…”

Section: Singularities Of D-finite Functionsmentioning

confidence: 99%

Differentially definable functions: a survey

Jiménez-Pastor

Pillwein

2021

Proc Appl Math & Mech

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Most widely used special functions, such as orthogonal polynomials, Bessel functions, Airy functions, etc., are defined as solutions to differential equations with polynomial coefficients. This class of functions is referred to as D‐finite functions. There are many symbolic algorithms (and implementations thereof) to operate with these objects exactly. Recently, we have extended this notion to a more general class that also allows for good symbolic handling: differentially definable functions. In this paper, we give an overview on what is currently known about this new class.

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NumGfun

Cited by 30 publications

References 24 publications

Simple Differentially Definable Functions

Simple Differentially Definable Functions

Floating-point arithmetic

Differentially definable functions: a survey

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