Searcharxiv⌕ Search

arXiv subjects

Robert M. Corless

Publications and source records attributed to Robert M. Corless.

At least 19 recordsLinked to original sources
arXiv2025

Minimizing Residuals in ODE Integration Using Optimal Control

Given the set of discrete solution points or nodes, called the skeleton, generated by an ODE solver, we study the problem of fitting a curve passing through the nodes in the skeleton minimizing a norm of the residual vector of the ODE. We reformulate this interpolation problem as a multi-stage optimal control problem and, for the minimization of two different norms, we apply the associated maximum principle to obtain the necessary conditions of optimality. We solve the problem analytically for the Dahlquist test problem and a variant of the leaky bucket problem, in terms of the given skeleton. We also consider the Van der Pol equation, for which we obtain interpolating curves with minimal residual norms by numerically solving a direct discretization of the problem through optimization software. With the skeletons obtained by various ODE solvers of MATLAB, we make comparisons between the residuals obtained by our approach and those obtained by the MATLAB function deval.

math.NA↗
arXiv2025

On the maximal spread of symmetric Bohemian matrices

Let A be a square matrix with real entries. The spread of A is defined as the maximum of the distances among the eigenvalues of A. Let $S_m[a,b]$ denote the set of all $m\times m$ symmetric matrices with entries in the real interval $[a,b]$ and let $S_m\{a,b\}$ be the subset of $S_m[a,b]$ of Bohemian matrices with population from only the extremal elements $\{a,b\}$. S. M. Fallat and J. J. Xing in 2012 proposed the following conjecture: the maximum spread in $S_m[a,b]$ is attained by a rank $2$ matrix in $S_m\{a,b\}$. X. Zhan had proved previously that the conjecture was true for $S_m[-a,a]$ with $a>0$. We will show how to interpret this problem geometrically, via polynomial resultants, in order to be able to treat this conjecture from a computational point of view. This will allow us to prove that this conjecture is true for several formerly open cases.

math.SP↗
arXiv2025

Well-Conditioned Polynomial Representations for Mathematical Handwriting Recognition

Previous work has made use of a parameterized plane curve polynomial representation for mathematical handwriting, with the polynomials represented in a Legendre or Legendre-Sobolev graded basis. This provides a compact geometric representation for the digital ink. Preliminary results have also been shown for Chebyshev and Chebyshev-Sobolev bases. This article explores the trade-offs between basis choice and polynomial degree to achieve accurate modeling with a low computational cost. To do this, we consider the condition number for polynomial evaluation in these bases and bound how the various inner products give norms for the variations between symbols.

cs.CV↗
arXiv2025

On Parametric Linear System Solving

Parametric linear systems are linear systems of equations in which some symbolic parameters, that is, symbols that are not considered to be candidates for elimination or solution in the course of analyzing the problem, appear in the coefficients of the system. In this paper we assume that the symbolic parameters appear polynomially in the coefficients and that the only variables to be solved for are those of the linear system. The consistency of the system and expression of the solutions may vary depending on the values of the parameters. It is well-known that it is possible to specify a covering set of regimes, each of which is a Zariski-constructible condition on the parameters together with a solution description valid under that condition. We provide a method of solution that requires time polynomial in the matrix dimension and the degrees of the polynomials when there are up to three parameters. We also discuss examples suggesting how the method may be useful beyond the formal three-parameter setting. In previous methods the number of regimes needed is exponential in the system dimension and polynomial degree of the parameters. Our approach exploits the Hermite and Smith normal forms that may be computed when the system coefficient domain is mapped to the univariate polynomial domain over suitably constructed fields. Our method identifies {intrinsic singularities} and {ramification points} where the algebraic and geometric structure of the matrix changes. Parametric eigenvalue problems are addressed as well.

math.RA↗
arXiv2025

Symbolic Mathematical Computation 1965--1975: The emergence of a discipline

Today, symbolic mathematical computation is taken for granted as part of the scientific infrastructure, but it has not always been so. This paper provides a historical survey of the discipline's formative decade, 1965-1975, viewed from a 50 year perspective. This span of years saw the evolution from a few specialized programs with naive algorithms to integrated systems with substantial capabilities. We highlight some of the important early figures in the field and the innovations upon which the current generation of systems and algorithms are built. By revisiting a period unfamiliar to most current readers, 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.

cs.GL↗
arXiv2025

Numerical methods for Chaotic ODE

This paper explores backward error analysis for numerical solutions of ordinary differential equations, particularly focusing on chaotic systems. Three approaches are examined: residual assessment, the method of modified equations, and shadowing. We investigate how these methods explain the success of numerical simulations in capturing the behavior of chaotic systems, even when facing issues like spurious chaos introduced by numerical methods or suppression of chaos by numerical methods. Finally, we point out an open problem, namely to explain why the statistics of long orbits are usually correct, even though we do not have a theoretical guarantee why this should be so.

math.NA↗
arXiv2024

Structured Backward Error for the WKB method

The classical WKB method (also known as the WKBJ method, the LG method, or the phase integral method) for solving singularly perturbed linear differential equations has never, as far as we know, been looked at from the structured backward error (BEA) point of view. This is somewhat surprising, because a simple computation shows that for some important problems, the WKB method gives the exact solution of a problem of the same structure that can be expressed in finitely many terms. This kind of analysis can be extremely useful in assessing the validity of a solution provided by the WKB method. In this paper we show how to do this and explore some of the consequences, which include a new iterative algorithm to improve the quality of the WKB solution. We also explore a new hybrid method where the potential is approximated by Chebyshev polynomials, which can be implemented in a few lines of Chebfun.

math.NA↗
arXiv2023

Teaching Linear Algebra in a Mechanized Mathematical Environment

This paper outlines our ideas on how to teach linear algebra in a mechanized mathematical environment, and discusses some of our reasons for thinking that this is a better way to teach linear algebra than the ``old fashioned way''. We discuss some technological tools such as Maple, Matlab, Python, and Jupyter Notebooks, and some choices of topics that are especially suited to teaching with these tools. The discussion is informed by our experience over the past thirty or more years teaching at various levels, especially at the University of Western Ontario.

math.HO↗
arXiv2023

Blendstrings: an environment for computing with smooth functions

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.

math.NA↗
arXiv2023

Elliptic cross sections in blood flow regulation

Arterial deformations arise in blood flow when surrounding tissue invades the space available for a blood vessel to maintain its circular cross section, the most immediate effects being a reduction in blood flow and redistribution of shear stress. Here we consider deformations from circular to elliptic cross sections. Solution of this problem in steady flow is fairly straightforward. The focus in the present paper is on pulsatile flow where the change from circular to elliptic cross sections is associated with a transition in the character of the equations governing the flow from Bessel to Mathieu equations. The study of this problem has been hampered in the past because of difficulties involved in the solution of the governing equations. In the present study we describe methods we have used to overcome some of these difficulties and present a comprehensive set of results based on these methods. In particular, vessel deformation is examined under two different conditions relevant to blood flow regulation: (i) keeping cross sectional area constant and (ii) keeping cross sectional circumference constant. The results provide an important context for the mechanism of neurovascular control of blood flow under the pathological conditions of vessel deformation.

q-bio.TO↗
arXiv2022

Perturbation Theory

This article aims to explain essential elements of perturbation theory and their conceptual underpinnings. It is not meant as a summary of popular perturbation methods, though some illustrative examples are given to underline the main methodological insights and concerns. We also give brief explications of the mathematical notions of limit, continuity, differentiability, convergence, and divergence, which provide the necessary foundation.

math.HO↗
arXiv2022

Bohemian Matrix Geometry

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.

cs.SC↗
arXiv2021

Computational Discovery with Newton Fractals, Bohemian Matrices, & Mandelbrot Polynomials

The authors have been using a largely algebraic form of ``computational discovery'' in various undergraduate classes at their respective institutions for some decades now to teach pure mathematics, applied mathematics, and computational mathematics. This paper describes what we mean by ``computational discovery,'' what good it does for the students, and some specific techniques that we used.

math.HO↗
arXiv2021

Computation and applications of Mathieu functions: A historical perspective

Mathieu functions of period $π$ or $2π$, also called elliptic cylinder functions, were introduced in 1868 by Émile Mathieu together with so-called modified Mathieu functions, in order to help understand the vibrations of an elastic membrane set in a fixed elliptical hoop. These functions still occur frequently in applications today: our interest, for instance, was stimulated by a problem of pulsatile blood flow in a blood vessel compressed into an elliptical cross-section. This paper surveys and recapitulates the historical development of the theory and methods of computation for Mathieu functions and modified Mathieu functions and identifies some gaps in current software capability, particularly to do with double eigenvalues of the Mathieu equation. We demonstrate how to compute Puiseux expansions of the Mathieu eigenvalues about such double eigenvalues, and give methods to compute the generalized eigenfunctions that arise there. In examining Mathieu's original contribution, we bring out that his use of anti-secularity predates that of Lindstedt. For interest, we also provide short biographies of some of the major mathematical researchers involved in the history of the Mathieu functions: Émile Mathieu, Sir Edmund Whittaker, Edward Ince, and Gertrude Blanch.

math-ph↗
arXiv2021

A Fractal Eigenvector

The recursively-constructed family of Mandelbrot matrices $M_n$ for $n=1$, $2$, $\ldots$ have nonnegative entries (indeed just $0$ and $1$, so each $M_n$ can be called a binary matrix) and have eigenvalues whose negatives $-λ= c$ give periodic orbits under the Mandelbrot iteration, namely $z_k = z_{k-1}^2+c$ with $z_0=0$, and are thus contained in the Mandelbrot set. By the Perron--Frobenius theorem, the matrices $M_n$ have a dominant real positive eigenvalue, which we call $ρ_n$. This article examines the eigenvector belonging to that dominant eigenvalue and its fractal-like structure, and similarly examines (with less success) the dominant singular vectors of $M_n$ from the singular value decomposition.

math.DS↗
arXiv2021

Generalized Standard Triples for Algebraic Linearizations of Matrix Polynomials

We define \emph{generalized standard triples} $\mathbf{X}$, $\mathbf{Y}$, and $L(z) = z\mathbf{C}_{1} - \mathbf{C}_{0}$, where $L(z)$ is a linearization of a regular matrix polynomial $\mathbf{P}(z) \in \mathbb{C}^{n \times n}[z]$, in order to use the representation $\mathbf{X}(z \mathbf{C}_{1}~-~\mathbf{C}_{0})^{-1}\mathbf{Y}~=~\mathbf{P}^{-1}(z)$ which holds except when $z$ is an eigenvalue of $\mathbf{P}$. This representation can be used in constructing so-called \emph{algebraic linearizations} for matrix polynomials of the form $\mathbf{H}(z) = z \mathbf{A}(z)\mathbf{B}(z) + \mathbf{C} \in \mathbb{C}^{n \times n}[z]$ from generalized standard triples of $\mathbf{A}(z)$ and $\mathbf{B}(z)$. This can be done even if $\mathbf{A}(z)$ and $\mathbf{B}(z)$ are expressed in differing polynomial bases. Our main theorem is that $\mathbf{X}$ can be expressed using the coefficients of the expression $1 = \sum_{k=0}^\ell e_k ϕ_k(z)$ in terms of the relevant polynomial basis. For convenience, we tabulate generalized standard triples for orthogonal polynomial bases, the monomial basis, and Newton interpolational bases; for the Bernstein basis; for Lagrange interpolational bases; and for Hermite interpolational bases. We account for the possibility of common similarity transformations.

math.NA↗
arXiv2021

Approximate GCD in Lagrange bases

For a pair of polynomials with real or complex coefficients, given in any particular basis, the problem of finding their GCD is known to be ill-posed. An answer is still desired for many applications, however. Hence, looking for a GCD of so-called approximate polynomials where this term explicitly denotes small uncertainties in the coefficients has received significant attention in the field of hybrid symbolic-numeric computation. In this paper we give an algorithm, based on one of Victor Ya. Pan, to find an approximate GCD for a pair of approximate polynomials given in a Lagrange basis. More precisely, we suppose that these polynomials are given by their approximate values at distinct known points. We first find each of their roots by using a Lagrange basis companion matrix for each polynomial, cluster the roots of each polynomial to identify multiple roots, and then "marry" the two polynomials to find their GCD. At no point do we change to the monomial basis, thus preserving the good conditioning properties of the original Lagrange basis. We discuss advantages and drawbacks of this method. The computational cost is dominated by the rootfinding step; unless special-purpose eigenvalue algorithms are used, the cost is cubic in the degrees of the polynomials. In principle, this cost could be reduced but we do not do so here.

math.NA↗