We explore the computation of roots of polynomials via eigenvalue problems. In particular, we look at the case when the leading coefficient is relatively very small. We argue that the companion matrix algorithm (used, for instance, by the Matlab roots function) is inaccurate in this case. The accuracy problem is addressed by using matrix pencils instead. This improvement can be predicted from the backward error bound of Edelman and Murakami (for companion matrices) versus the bound of Van Dooren and Dewilde (for pencils). We then show how to extend the accurate algorithm to B'ezier polynomials and present computational experiments. 1 Introduction Computing the roots of a univariate polynomial is a fundamental problem that arises in many applications. The focus of this paper is on polynomials where the leading coefficient is much smaller than some of the other coefficients. Such polynomials occur frequently in geometric applications like mesh generation and graphics. The reason is that...

A new version of the Graeffe’s algorithms for finding all the roots of univariate complex polynomials is proposed. It is obtained from the classical algorithm by a process analogous to renormalization of dynamical systems. This iteration is called Renormalized Graeffe Iteration. It is globally convergent, with probability 1. All quantities involved in the computation are bounded, once the initial polynomial is given (with probability 1). This implies remarkable stability properties for the new algorithm, thus overcoming known limitations of the classical Graeffe algorithm. If we start with a degree-d polynomial, each renormalized Graeffe iteration costs O(d 2) arithmetic operations, with memory O(d). A probabilistic global complexity bound is given. The case of univariate real polynomials is briefly discussed. A numerical implementation of the algorithm presented herein allowed us to solve random polynomials of degree up to 1000. 1

Abstract — In this paper, we study the problem of optimal trajectory generation for a team of mobile robots tracking a moving target using distance and bearing measurements. Contrary to previous approaches, we explicitly consider limits on the robots ’ speed and impose constraints on the minimum distance at which the robots are allowed to approach the target. We first address the case of a single sensor and show that although this problem is non-convex with non-convex constraints, in general, its optimal solution can be determined analytically. Moreover, we extend this approach to the case of multiple sensors and propose an iterative algorithm, Gauss-Seidel-relaxation (GSR), for determining the set of feasible locations that each sensor should move to in order to minimize the uncertainty about the position of the target. Extensive simulation results are presented demonstrating that the performance of the GSR algorithm, whose computational complexity is linear in the number of sensors, is indistinguishable of that of a grid-based exhaustive search, with cost exponential in the number of sensors, and significantly better than that of a random, towards the target, motion strategy. I.

The representation of general distributions or measured data by phase-type distributions is an important and non-trivial task in analytical modeling. Although a large number of different methods for fitting parameters of phase-type distributions to data traces exist, many approaches lack efficiency and numerical stability. In this paper, a novel approach is presented that fits a restricted class of phase-type distributions, namely mixtures of Erlang distributions, to trace data. For the parameter fitting an algorithm of the expectation maximization type is developed. The paper shows that these choices result in a very efficient and numerically stable approach which yields phase-type approximations for a wide range of data traces that are as good or better than approximations computed with other less efficient and less stable fitting methods. To illustrate the effectiveness of the proposed fitting algorithm, we present comparative results for our approach and two other methods using six benchmark traces and two real traffic traces as well as quantitative results from queueing analysis. Keywords: Performance and dependability assessment/analytical and numerical techniques, design of tools for performance/dependability assessment, traffic modeling, hyper-Erlang distributions.

. The pseudozero set of a system f of polynomials in n complex variables is the subset of C n which is the union of the zero - sets of all polynomial systems g that are near to f in a suitable sense. This concept is made precise and general properties of pseudozero sets are established. In particular it is shown that under wide circumstances, the pseudozero set is a semialgebraic set. Also, estimates are given for the size of the projections of pseudozero sets into coordinate directions. Several examples are presented illustrating some of the general theory developed here. Finally, algorithmic ideas are proposed for solving multivariate polynomials. 1. Introduction 1.1. Summary. The pseudozero set of a general polynomial in a single variable was investigated in [18]. Our purpose here is to extend some ideas from that work to systems of polynomials in several variables, with special attention to the case in which the zero set of the system consists of finitely many points. Recall th...

Abstract. Experimental observations of univariate rootfinding by generalized companion matrix pencils expressed in the Lagrange basis show that the method can sometimes be numerically stable. It has recently been proved that a new condition number, defined for points on a set containing the interpolation points, is never larger than the rootfinding condition number for the Bernstein polynomial (which is itself optimal in a certain sense); and computation shows that sometimes it can be much smaller. These results also hold for the matrix polynomial case, when we are not looking for polynomial roots but rather for eigenvalues where the matrix polynomial is singular. This current paper explores the influence of the geometry of the interpolation nodes on the conditioning of the rootfinding and eigenvalue problems.

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One of the most popular methods for computing the zeros of a polynomial in power form is to determine the eigenvalues of the associated companion matrix. This approach, which commonly is referred to as the companion matrix method, however, can yield poor accuracy. We demonstrate that when approximations of the zeros are available, their accuracy often can be improved by computing the eigenvalues of a modi ed companion matrix associated with the polynomial expressed in a basis of Newton polynomials de ned by the available approximations of the zeros. The latter approximations can be determined in many ways, e.g., by the companion matrix method.

We provide a partial convergence analysis of an iterative polynomial root-finding algorithm [7, 14]. The iteration in the algorithm is based upon floating-point eigenvalue computation. The algorithm is quite efficient, particularly for approximating the zeroes of ill-conditioned, high-degree polynomials. A previous analysis of the algorithm [7] depended upon a technical assumption, that eigenvalue computation is componentwise backwards stable for a particular class of matrices. We present a new analysis that eliminates this assumption, using instead the usual normwise backwards stability of eigenvalue computation and certain additional genericity conditions. 1

The computation of zeros of polynomials is a classical computational problem. This paper presents two new zero finders that are based on the observation that, after a suitable change of variable, any polynomial can be considered a member of a family of Szegö polynomials. Numerical experiments indicate that these methods generally give higher accuracy than computing the eigenvalues of the companion matrix associated with the polynomial.