The 2025 ISSAC conference in Guanajuato, Mexico, marks the 50th event in this significant series, making it an ideal moment to reflect on the field’s history. This paper reviews the formative years of symbolic computation up to 1975, fifty years ago. By revisiting a period unfamiliar to most current participants, this survey aims to shed light on once-pressing issues that are now largely resolved and to highlight how some of today’s challenges were recognized earlier than expected.

A “blendstring” is a piecewise polynomial interpolant with high-degree two-point Hermite interpolational polynomials on each piece, analogous to a cubic spline. Blendstrings are smoother and can be more accurate than cubic splines, and can be used to represent smooth functions on a line segment or polygonal path in the complex plane. I sketch some properties of blendstrings, including efficient methods for evaluation, differentiation, and integration, as well as a prototype Maple implementation. Blendstrings can be differentiated and integrated exactly and can be combined algebraically. I also show applications of blendstrings to solving differential equations and computing Mathieu functions and generalized Mathieu eigenfunctions.

A Bohemian matrix family is a set of matrices all of whose entries are drawn from a fixed, usually discrete and hence bounded, subset of a field of characteristic zero. Originally these were integers---hence the name, from the acronym BOunded HEight Matrix of Integers (BOHEMI)---but other kinds of entries are also interesting. Some kinds of questions about Bohemian matrices can be answered by numerical computation, but sometimes exact computation is better. In this paper we explore some Bohemian families (symmetric, upper Hessenberg, or Toeplitz) computationally, and answer some open questions posed about the distributions of eigenvalue densities.

One useful standard method to compute eigenvalues of matrix polynomials P(z)∈ C n x n [z] of degree at most ℓ in z (denoted of grade ℓ, for short) is to first transform P(z) to an equivalent linear matrix polynomial L(z)=zB-A, called a companion pencil, where A and B are usually of larger dimension than P(z) but L(z) is now only of grade 1 in z. The eigenvalues and eigenvectors of L(z) can be computed numerically by, for instance, the QZ algorithm. The eigenvectors of P(z), including those for infinite eigenvalues, can also be recovered from eigenvectors of L(z) if L(z) is what is called a "strong linearization'' of P(z). In this paper we show how to use algorithms for computing the Hermite Normal Form of a companion matrix for a scalar polynomial to direct the discovery of unimodular matrix polynomial cofactors E(z) and F(z) which, via the equation E(z)L(z)F(z) = diag(P(z), In, …, I_n), explicitly show the equivalence of P(z) and P(z). By this method we give new explicit constructions for several linearizations using different polynomial bases. We contrast these new unimodular pairs with those constructed by strict equivalence, some of which are also new to this paper. We discuss the limitations of this experimental, computational discovery method of finding unimodular cofactors.

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.

This paper takes a classical problem in two-dimensional fluid flow-namely, flow into or out of a wedge-shaped channel with a sink or source at the vertex, which flow is known as Jeffery-Hamel flow and has "well-known" solutions containing elliptic functions-and tries to duplicate, or even extend, the classical solutions by using a CAS, in this instance Maple. The purposes of this case study include examining just how good CAS can be at elliptic functions; and, more importantly, identifying needs for improvement. Another purpose is to compare the analytical solution with modern numerical solutions. Finally, we believe that this work will motivate improvements to CAS facilities for automatic case analysis. As an aside, we present some simple methods for integration of elliptic functions that seem not to be widely known.

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.

In the present work, we extend the standard idea of numerical parameterization (i.e., parameterization by the numerical solution of initial-value problems (IVPs) for ordinary differential equations (ODEs) to affine varieties in ℂ;n for n≥2. We use these results with an efficient implementation in Maple to explore the use of numerical parameterization for the visualization of Riemann surfaces.

In this paper, we propose a new semi-numerical algorithmic method for factoring multivariate polynomials absolutely. It is based on algebraic and geometric properties after reduction to the bivariate case in a generic system of coordinates. The method combines 4 tools: zero-sum relations at triplets of points, partial information on monodromy action, Newton interpolation on a structured grid, and a homotopy method. The algorithm relies on a probabilistic approach and uses numerical computations to propose a candidate factorization (with probability almost one) which is later validated.

A new algorithm is presented for factoring bivariate approximate polynomials over C[x, y]. Given a particular polynomial, the method constructs a nearby composite polynomial, if one exists, and its irreducible factors. Subject to a conjecture, the time to produce the factors is polynomial in the degree of the problem. This method has been implemented in Maple, and has been demonstrated to be efficient and numerically robust.

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.

We describe algorithms for computing the greatest common divisor (GCD) of two univariate polynomials with inexactlyknown coefficients. Assuming that an estimate for the GCD degree is available (e.g., using an SVD-based algorithm), we formulate and solve a nonlinear optimization problem in order to determine the coefficients of the "best" GCD. We discuss various issues related to the implementation of the algorithms and present some preliminary test results.

\Ve discuss the use of a single generic linear combination of multiplication matrices, and its reordered Schur factorization, to find the roots of a system of multivariate polynomial equations.The principal contribution of this paper is to show how to reduce thr multivariate problem to a univariate problem, even in the raw of multiple roots, in a numerically stable wav,

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

This paper introduces singular value decomposition (SVD) algorithms for some standard polynomial computations, in the case where the coefficients are inexact or imperfectly known. We first give an algorithm for computing univariate GCD's which gives exact results for interesting nearby problems, and give efficient algorithms for computing precisely how nearby. We generalize this to multivariate GCD computation. Next, we adapt Lazard's u-resultant algorithm for the solution of overdetermined systems of polynomial equations to the inexact-coefficient case. We also briefly discuss an application of the modied Lazard's method to the location of singular points on approximately known projections of algebraic curves.

Many of the computations in the calculus of variations are algebraic in nature: computing the Euler-Lagrange equations and solving them, for example.However, deciding whether or not the computed extremals provide minima or maxima is an analytic problem, and one that has not been previously attempted in a computer algebra package.I describe here a Maple implementation of some techniques for making these decisions, and detail some successes and failures.Some of the failures point to areas where computer algebra systems could be improved.

Article Free Access Share on Solving linear integral equations in Maple Authors: Honglin Ye View Profile , Robert M. Corless View Profile Authors Info & Claims ISSAC '92: Papers from the international symposium on Symbolic and algebraic computationAugust 1992 Pages 95–102https://doi.org/10.1145/143242.143279Published:01 August 1992Publication History 2citation1,144DownloadsMetricsTotal Citations2Total Downloads1,144Last 12 Months59Last 6 weeks6 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 SiteeReaderPDF