Real root isolation for exp-log-arctan functions
type: Item
instance of: scholarly article
title: Real root isolation for exp-log-arctan functions
zbMATH Open document ID: 1244.65070
published in: Journal of Symbolic Computation
publication date: 2012-03-22T00:00:00Z
review text: Using the set of exp-log-arctan functions -- defined as the smallest set of partial functions \(\mathbb{R}\longrightarrow \mathbb{R}\) containing exp, log, arctan, identity and constant functions that are closed under addition, multiplication and composition of functions -- the author proposes a real root isolation procedure which is an extension of an algorithm defined by \textit{A. Strzeboński} [Real root isolation for exp-log functions, Proceedings of the twenty-first international symposium on symbolic and algebraic computation 2008, Linz/Hagenberg, Austria, July 20--23, 2008. New York, NY: Association for Computing Machinery (ACM). 303--314 (2008; Zbl 1236.65057)] for exp-log functions. Basically, the algorithm RootInInterval (defined in Section 5 together with several examples and a detailed study of complexity) offers a new proof of the claim: \textit{The domain of an exp-log-arctan function consists of a finite number of open intervals and an exp-log-arctan function has a finite number of real roots}. The algorithm uses the notion of semi-Fourier sequence and its applicability in root isolation proposed in an unpublished paper by \textit{D. Richardson} [Recognising zero among implicitly defined elementary numbers. (submitted for publication)]. The theoretical construction offered by the paper is finally verified by an implementation using a C source code and Mathematica system. All experimental results confirm the validity of the RootInInterval algorithm proposed.
reviewed by: Adrian Atanasiu
Mathematics Subject Classification ID: 65H05
Mathematics Subject Classification ID: 65H04
zbMATH DE Number: 6017591
zbMATH Keywords: exp-log-arctan functions
zbMATH Keywords: elementary functions
zbMATH Keywords: real root isolation
zbMATH Keywords: solving equations
zbMATH Keywords: numerical examples
zbMATH Keywords: algorithm
author: Adam Strzeboński
describes a project that uses: Mathematica
MaRDI profile type: Publication
full work available at URL: j.jsc.2011.11.004
OpenAlex ID: W2102283069
cites work: Q3686991
cites work: Q5520004
cites work: Q3963114
cites work: How to recognize zero?
cites work: Polynomial Minimum Root Separation
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cites work: The elementary theory of restricted analytic fields with exponentiation
cites work: Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function
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DOI: 10.1016/J.JSC.2011.11.004
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description: scientific article; zbMATH DE number 6017591