Abstract. A unitarily invariant projective framework is introduced to analyze the complexity of path-following methods for the eigenvalue problem. A condition number, and its relation to the distance to ill-posedness, is given. A Newton map appropriate for this context is defined, and a version of Smale's γ-Theorem is proven. The main result of this paper bounds the complexity of path-following methods in terms of the length of the path in the condition metric.

Smale's 17th problem asks for an algorithm which finds an approximate zero of polynomial systems in average polynomial time (see Smale [17]). The main progress on Smale's problem is Beltrán-Pardo [6] and Bürgisser-Cucker [9]. In this paper we will improve on both approaches and we prove an important intermediate result. Our main results are Theorem 1 on the complexity of a randomized algorithm which improves the result of [6], Theorem 2 on the average of the condition number of polynomial systems *

Abstract. We prove that points in the sphere associated with roots of random polynomials via the stereographic projection are surprisingly well-suited with respect to the minimal logarithmic energy on the sphere. That is, roots of random polynomials provide a fairly good approximation to elliptic Fekete points.

a b s t r a c tIn this paper we define a new condition number adapted to directionally uniform perturbations in a general framework of maps between Riemannian manifolds. The definitions and theorems can be applied to a large class of problems. We show the relation with the classical condition number and study some interesting examples.

We describe algorithms for computing eigenpairs (eigenvalueeigenvector pairs) of a complex n × n matrix A. These algorithms are numerically stable, strongly accurate, and theoretically efficient (i.e., polynomialtime). We do not believe they outperform in practice the algorithms currently used for this computational problem. The merit of our paper is to give a positive answer to a long-standing open problem in numerical linear algebra. So the problem of devising an algorithm [for the eigenvalue problem] that is numerically stable and globally (and quickly!) convergent remains open.Our initial quotation followed these words in Demmel's text. It asked for an algorithm which will be numerically stable and for which, convergence, and if possible small complexity bounds, can be established. Today, 17 years after Demmel's text, this demand retains all of its urgency: it is not known if any of the standard numerical linear algebra algorithms satisfies the properties above. For example:• The unshifted QR algorithm terminates with probability 1 but is probably infinite average cost if approximations to the eigenvectors are to be output (see [29]).• The QR algorithm with Rayleigh Quotient shift fails for open sets of real input matrices (see [6,7]).