We study discrete dynamical systems, or recurrence relations, of the general form

This article describes an efficient and robust algorithm and implementation for the evaluation of the Wright ω function in IEEE double precision arithmetic over the complex plane.

Given a real parametric polynomial p(x) and an interval (a,b) ⊂ R, the Complete Root Classification (CRC) of p(x) on (a,b) is a collection of all possible cases of its root classification on (a,b), together with the conditions its coefficients must satisfy for each case. In this paper, a new algorithm is proposed for the automatic computation of the complete root classification of a parametric polynomial on an interval. As a direct application, the new algorithm is applied to some real quantifier elimination problems.

We study the solutions of the matrix equation S exp(S) = A. Our motivation comes from the study of systems of delay differential equations y' (t) = Ay(t - 1), which occur in some models of practical interest, especially in mathematical biology. This paper concentrates on the distinction between evaluating a matrix function and solving a matrix equation. In particular,it shows that the matrix Lambert W function evaluated at the matrix A does not represent all possible solutions of S exp(S) = A. These results can easily be extended to more general matrix equations.

We consider algebraic numbers defined by univariate polynomials over the rationals. In the syntax of Maple, such numbers are expressed using the RootOf function. This paper defines a canonical form for RootOf with respect to affine transformations. The affine shifts of monic irreducible polynomials form a group, and the orbits of the polynomials can be used to define a canonical form. The canonical form of the polynomials then defines a canonical form for the corresponding algebraic numbers. Reducing any RootOf to its canonical form has the advantage that affine relations between algebraic numbers are readily identified. More generally, the reduction minimizes the number of algebraic numbers appearing in a computation, and also allows the Maple indexed RootOf to be used more easily.

The (4 0,) (1:) or fadoretl fornl.For exau~ple~ a. demise polynomial of degree IL woultl take approsimately 2r1 operat.ionst.o cva1uat.e in eitlwr espandcd or factin form.-4 presentatiori itS two cornlx~sit.ionfac:tor.r;,however: ~voldtl t,akr l>etw:rn 4&i and II aritlm&ic operations.main results of this pilpcr illC ail iterative nictliod t,o conlput~e il.decomposit.ion of a giveu itpprosillla~.e pOl~IlOIlliill, giveIl a starting point.

An algorithm is given for the integration of a class of piecewise-continuous functions. The integration is with respect to a real variable, because the functions considered do not in general allow integration in the complex plane to be defined. The class of integrands includes commonly occurring waveforms, such as square waves, triangular waves, and the floor function; it also includes the signum function. The algorithm can be implemented recursively, and it has the property of ensuring that integrals are continuous on domains of maximum extent. 1 Introduction The integration of a function expressed using the Maple function piecewise or the signum function was considered in [3], where the fundamental definitions and theorems on integrating discontinuous functions were presented. We recall that a function F (x) is said to have breakpoints at those values of x where the function is discontinuous. It was also pointed out in [3] that the problem of integrating piecewisecontinuous functio...

Article A sequence of series for the Lambert W function Share on Authors: Robert M. Corless Department of Applied Mathematics, University of Western Ontario, London, Canada N6A 5B7 Department of Applied Mathematics, University of Western Ontario, London, Canada N6A 5B7View Profile , David J. Jeffrey Department of Applied Mathematics, University of Western Ontario, London, Canada N6A 5B7 Department of Applied Mathematics, University of Western Ontario, London, Canada N6A 5B7View Profile , Donald E. Knuth Computer Science Department, Gates 4B, Stanford University, Stanford, CA Computer Science Department, Gates 4B, Stanford University, Stanford, CAView Profile Authors Info & Claims ISSAC '97: Proceedings of the 1997 international symposium on Symbolic and algebraic computationJuly 1997 Pages 197–204https://doi.org/10.1145/258726.258783Online:01 July 1997Publication History 87citation1,653DownloadsMetricsTotal Citations87Total Downloads1,653Last 12 Months71Last 6 weeks9 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

When a computer algebra system has an assumption facility, it is possible to distinguish between integration problems with respect to a real variable, and those with respect to a complex variable. Here, a class of integration problems is defined in which the integrand consists of compositions of continuous functions and signum functions, and integration is with respect to a real variable. Algorithms are given for evaluating such integrals. 1 Introduction In recent years, `assume' or `declare' facilities have been implemented in most of the available computer algebra systems (CAS). As well, such facilities have been gaining wider acceptance within the user community. The presence of these facilities has altered the way CAS behave, and many established areas of symbolic computation need to be reconsidered. The topic of this paper is an example of the impact on one traditional field of computer algebra, namely, symbolic integration. Because the early versions of many present-day CAS cou...

The tan(x/2) substitution, also called the Weierstrass substitution, is one method currently used by computer-algebra systems for the evaluation of trigonometric integrals. The method needs to be improved, because the expressions obtained using it sometimes contain discontinuities, which unnecessarily limit the domains over which the expressions are correct. We show that the discontinuities are spurious in the following sense: Given an integrand and an expression for its antiderivative that was obtained by the Weierstrass substition, a better expression can be found that is continuous on wider intervals than the first expression and yet is still an antiderivative of the integrand. The origin of the discontinuities is identified, and an algorithm is presented for automatically finding the improved type of antiderivative. The new algorithm also enlarges the set of functions that can be used in the substitution. The algorithm works by first evaluating the given integral using the Weierstrass substitution in the usual way and then removing any spurious discontinuities present in the antiderivative.

In certain circumstances, the integration routines used by computer algebra systems return expressions whose domains of validity are unnecessarily restricted by the presence of discontinuities. It is argued that this is undesirable and that integration routines should meet an additional requirement: they should return expressions that are valid on domains of maximum extent. The contention is supported by general mathematical arguments, by an examination of existing practises and by a demonstration that two standard algorithms can be modified to meet the requirement.