We propose a new algorithm for the classical and still practically important problem of approximating the zeros z j of an n-th degree polynomial p(x) within the error bound 2 ?b max j jz j j. The algorithm uses O((n 2 log n) log(bn)) arithmetic operations and comparisons for approximating all the n zeros and O((kn log n) log(bn)) for approximating the k zeros lying in a xed domain (disc or squa...

The aim of this paper is to state initial conditions for the safe and fast convergence of the simultaneous method of Weierstrass’ type for finding simple zeros of algebraic polynomial. This conditions are computationally verifiable and they depend only on the available data polynomial coefficients, its degree and initial approximations z 1 , . . . , z (0) n to the zeros. It is shown that under ...

In this paper we study some iterative methods for simultaneous approximation of polynomial zeros. We give new semilocal convergence theorems with error bounds for Ehrlich’s and Nourein’s iterations. Our theorems generalize and improve recent results of Zheng and Huang [J. Comput. Math. 18 (2000), 113– 122], Petković and Herceg [J. Comput. Appl. Math. 136 (2001), 283–307] and Nedić [Novi Sad J. ...

Isolated multiple zeros or clusters of zeros of analytic maps with several variables are known to be difficult to locate and approximate. This article is in the vein of the α-theory, initiated by M. Shub and S. Smale in the beginning of the eighties. This theory restricts to simple zeros, i.e., where the map has corank zero. In this article we deal with situations where the analytic map has cor...

In this article we extend Alon’s Nullstellensatz to functions which have multiple zeros at the common zeros of some polynomials g1, g2, . . . , gn, that are the product of linear factors. We then prove a punctured version which states, for simple zeros, that if f vanishes at nearly all, but not all, of the common zeros of g1(X1), . . . , gn(Xn) then every residue of f modulo the ideal generated...

Let pk(x) = x +■■■ , k e N0 , be the polynomials orthogonal on [-1, +1] with respect to the positive measure da . We give sufficient conditions on the real numbers p , j = 0, ... , m , such that the linear combination of orthogonal polynomials YfLo^jPn-j has n simple zeros in (—1,-1-1) and that the interpolatory quadrature formula whose nodes are the zeros of Yfj=oßjPn-j has positive weights.

In this article we extend Alon’s Nullstellensatz to functions which have multiple zeros at the common zeros of some polynomials g1, g2, . . . , gn, that are the product of linear factors. We then prove a punctured version which states, for simple zeros, that if f vanishes at nearly all, but not all, of the common zeros of g1(X1), . . . , gn(Xn) then every residue of f modulo the ideal generated...

We define an infinite set of families of graphs, which we call p-wheels and denote (Wh) n , that generalize the wheel (p = 1) and biwheel (p = 2) graphs. The chromatic polynomial for (Wh) n is calculated, and remarkably simple properties of the chromatic zeros are found: (i) the real zeros occur at q = 0, 1, ...p+ 1 for n− p even and q = 0, 1, ...p+ 2 for n− p odd; and (ii) the complex zeros al...

Let φ and f be functions in the Laguerre–Pólya class. Write φ(z) = e−αzφ1(z) and f (z) = e−βzf1(z), where φ1 and f1 have genus 0 or 1 and α,β 0. If αβ < 1/4 and φ has infinitely many zeros, then φ(D)f (z) has only simple real zeros, where D denotes differentiation.  2004 Elsevier Inc. All rights reserved.

It is shown that the complex Ginzburg-Landau (CGL) equation on the real line admits nontrivial 2-periodic vortex solutions that have 2n simple zeros (\vortices") per period. The vortex solutions bifurcate from the trivial solution and inherit their zeros from the solution of the linearized equation. This result rules out the possibility that the vortices are determining nodes for vortex solutio...