Jean-Claude Yakoubsohn

On the Complexity of Isolating Real Roots and Computing with Certainty the Topological Degree

In this contribution the isolation of real roots and the computation of the topological degree in two dimensions are considered and their complexity is analyzed. In particular, we apply Stengers degree computational method by splitting properly the boundary of the given region to obtain a sequence of subintervals along the boundary that forms a sufficient refinement. To this end, we properly approximate the function using univariate polynomials. Then we isolate each one of the zeros of these polynomials on the boundary of the given region in various subintervals so that these subintervals form a sufficiently refined boundary.

On Location and Approximation of Clusters of Zeros: Case of Embedding Dimension One

AbstractIsolated multiple zeros or clusters of zeros of analytic maps with several variables are known to be difficult to locate and approximate. This paper is in the vein of the α-theory, initiated by M. Shub and S. Smale in the beginning of the 1980s. This theory restricts to simple zeros, i.e., where the map has corank zero. In this paper we deal with situations where the analytic map has corank one at the multiple isolated zero, which has embedding dimension one in the frame of deformation theory. These situations are the least degenerate ones and therefore most likely to be of practical significance. More generally, we define clusters of embedding dimension one. We provide a criterion for locating such clusters of zeros and a fast algorithm for approximating them, with quadratic convergence. In the case of a cluster with positive diameter our algorithm stops at a distance of the cluster which is about its diameter.

On Location and Approximation of Clusters of Zeros of Analytic Functions

AbstractAt the beginning of the 1980s, M. Shub and S. Smale developed a quantitative analysis of Newtons method for multivariate analytic maps. In particular, their α-theory gives an effective criterion that ensures safe convergence to a simple isolated zero. This criterion requires only information concerning the map at the initial point of the iteration. Generalizing this theory to multiple zeros and clusters of zeros is still a challenging problem. In this paper we focus on one complex variable function. We study general criteria for detecting clusters and analyze the convergence of Schroders iteration to a cluster. In the case of a multiple root, it is well known that this convergence is quadratic. In the case of a cluster with positive diameter, the convergence is still quadratic provided the iteration is stopped sufficiently early. We propose a criterion for stopping this iteration at a distance from the cluster which is of the order of its diameter.

Numerical analysis of a bisection-exclusion method to find zeros of univariate analytic functions

We state precise results on the complexity of a classical bisection-exclusion method to locate zeros of univariate analytic functions contained in a square. The output of this algorithm is a list of squares containing all the zeros. It is also a robust method to locate clusters of zeros. We show that the global complexity depends on the following quantities: the size of the square, the desired precision, the number of clusters of zeros in the square, the distance between the clusters and the global behavior of the analytic function and its derivatives. We also prove that, closed to a cluster of zeros, the complexity depends only on the number of zeros inside the cluster. In particular, for a polynomial which has d simple roots separated by a distance greater than sep, we will prove the bisection-exclusion algorithm needs O(d3 log(d/sep)) tests to isolate the d roots and the number of squares suspected to contain a zero is bounded by 4d. Moreover, always in the polynomial case, we will see the arithmetic complexity can be reduced to O(d2(log d)2 log(d/sep)) using ⌈log d⌉ steps of the Graeffe iteration.

Finding a Cluster of Zeros of Univariate Polynomials

A method to compute an accurate approximation for a zero cluster of a complex univariate polynomial is presented. The theoretical background on which this method is based deals with homotopy, Newtons method, and Rouches theorem. First the homotopy method provides a point close to the zero cluster. Next the analysis of the behaviour of the Newton method in the neighbourhood of a zero cluster gives the number of zeros in this cluster. In this case, it is sufficient to know three points of the Newton sequence in order to generate an open disk susceptible to contain all the zeros of the cluster. Finally, an inclusion test based on a punctual version of the Rouche theorem validates the previous step. A specific implementation of this algorithm is given. Numerical experiments illustrate how this method works and some figures are displayed.

Computing the real roots of a polynomial by the exclusion algorithm

We describe a new algorithm for localizing the real roots of a polynomialP(x). This algorithm determines intervals on whichP(x) does not possess any root. The remainder set contains the real roots ofP(x) and can be arbitrarily small.

Finding zeros of analytic functions: a theory for secant type methods

Abstract We present new results concerning the convergence of secant type methods with only conditions at a point. The radius of robustness of these methods is given, and we apply it to the study of the complexity of homotopy methods for approximating roots. In particular, we say how to use the secant type method to get an approximate zero relative to the Newton method.

Localization of an algebraic hypersurface by the exclusion algorithm

We describe a new algorithm for the localization of an algebraic hypersurfaceV inRn or Cn. This algorithm computes a decreasing sequence of closed sets whose intersection isV. In the particular case of an hypersurface without any point at infinity, the notion of the asymptotic cone is used to determine a compact set containing this hypersurface. We give also a numerical version of this algorithm.

Approximating the zeros of analytic functions by the exclusion algorithm

We give a practical version of the exclusion algorithm for localizing the zeros of an analytic function and in particular of a polynomial in a compact of ℂ. We extend the real exclusion algorithm to a Jordan curve and give a method which excludes discs without any zero. The result of this algorithm is a set of discs arbitrarily small which contains the zeros of the analytic function.

A subdivision method for computing nearest gcd with certification

A new subdivision method for computing the nearest univariate gcd is described and analyzed. It is based on an exclusion test and an inclusion test. The exclusion test in a cell exploits Taylor expansion of the polynomial at the center of the cell. The inclusion test uses Smales @a-theorems to certify the existence and unicity of a solution in a cell. Under the condition of simple roots for the distance minimization problem, we analyze the complexity of the algorithm in terms of a condition number, which is the inverse of the distance to the set of degenerate systems. We report on some experimentation on representative examples to illustrate the behavior of the algorithm.