Given a system of polynomial equations and inequations with coefficients in the field of rational numbers, we show how to compute a geometric resolution of the set of common roots of the system over the field of complex numbers. A geometric resolution consists of a primitive element of the algebraic extension defined by the set of roots, its minimal polynomial and the parametrizations of the coordinates. Such a representation of the solutions has a long history which goes back to Leopold Kronecker and has been revisited many times in computer algebra. We introduce a new generation of probabilistic algorithms where all the computations use only univariate or bivariate polynomials. We give a new codification of the set of solutions of a positive dimensional algebraic variety relying on a new global version of Newton’s iterator. Roughly speaking the complexity of our algorithm is polynomial in some kind of degree of the system, in its height, and linear in the complexity of evaluation

� � � We estimate the probability that a given number of projective Newton steps applied to a linear homotopy of a system of n homogeneous polynomial equations in n +1 complex variables of fixed degrees will find all the roots of the system. We also extend the framework of our analysis to cover the classical implicit function theorem and revisit the condition number in this context. Further complexity theory is developed. 1. Introduction. 1A. Bezout’s Theorem Revisited. Let f: � n+1 → � n be a system of homogeneous polynomials f =(f1,...,fn), deg fi = di, i=1,...,n. The linear space of such f is denoted by H (d) where d = (d1,...,dn). Consider the

this paper is to show that the problem of finding approximately a zero of a polynomial system of equations can be solved in polynomial time, on the average. The number of arithmetic operations is bounded by cN

In this paper we take a new look at smoothing Newton methods for solving the nonlinear complementarity problem (NCP) and the box constrained variational inequalities (BVIs). Instead of using an infinite sequence of smoothing approximation functions, we use a single smoothing approximation function and Robinson's normal equation to reformulate the NCP and the BVIs as an equivalent nonsmooth equation H(u, x) = 0, where H : # 2n # # 2n , u # # n is a parameter variable and x # # n is the original variable. The central idea of our smoothing Newton methods is that we construct a sequence {z k = (u k , x k )} such that the mapping H(·) is continuously di#erentiable at each z k and may be non-di#erentiable at the limiting point of {z k }. We prove that three most often used Gabriel-More smoothing functions can generate strongly semismooth functions, which play a fundamental role in establishing superlinear and quadratic convergence of our new smoothing Newton methods....

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this paper asserts:

We propose a continuation method for a class of nonlinear complementarity problems(NCPs), including the NCP with a P 0 and R 0 function and the monotone NCP with a feasible interior point. The continuation method is based on a class of Chen-Mangasarian smooth functions. Unlike many existing continuation methods, the method follows the non-interior smoothing paths, and as a result, an initial point can be easily constructed. In addition, we introduce a procedure to dynamically update the neighborhoods associated with the smoothing paths, so that the algorithm is both globally convergent and locally superlinearly convergent under suitable assumptions. Finally, a hybrid continuation-smoothing method is proposed and is shown to have the same convergence properties under weaker conditions. 1 Introduction Let F : R n ! R n be a continuously differentiable function. The nonlinear complementarity problem, denoted by NCP(F ), is to find a vector (x; y) 2 R n \Theta R n such that F (x)...

Abstract--We substantially improve the known algorithms for approximating all the complex zeros of an n th degree polynomial p(x). Our new algorithms save both Boolean and arithmetic sequential time, versus the previous best algorithms of SchSnhage [1], Pan [2], and Neff and Reif [3]. In parallel (NC) implementation, we dramatically decrease the number of processors, versus the parallel algorithm of Neff [4], which was the only NC algorithm known for this problem so far. Specifically, under the simple normalization assumption that the variable x has been scaled so as to confine the zeros of p(x) to the unit disc {x: Ix [ < 1}, our algorithms (which promise to be practically effective) approximate all the zeros of p(x) within the absolute error bound 2-b, by using order of n arithmetic operations and order of (b + n)n 2 Boolean (bitwise) operations (in both cases up to within polylogarithmic factors). The algorithms allow their optimal (work preserving) NC parallelization, so that they can be implemented by using polylogarithmic time and the orders of n arithmetic processors or (b + n)n 2 Boolean processors. All the cited bounds on the computational complexity are within polylogarithmic factors from the optimum (in terms of n and b) under both arithmetic and Boolean models of computation (in the Boolean case, under the additional (realistic) assumption that n = O(b)).

There recently has been much interest in non-interior continuation/smoothing methods for solving linear/nonlinear complementarity problems. We describe extensions of such methods to complementarity problems defined over the cone of block-diagonal symmetric positive semidefinite real matrices. These extensions involve the ChenMangasarian class of smoothing functions and the smoothed Fischer-Burmeister function. Issues such as existence of Newton directions, boundedness of iterates, global convergence, and local superlinear convergence will be studied. Preliminary numerical experience on semidefinite linear programs is also reported. Key words. Semidefinite complementarity problem, smoothing function, non-interior continuation, global convergence, local superlinear convergence. 1 Introduction There recently has been much interest in semidefinite linear programs (SDLP) and, more generally, semidefinite linear complementarity problems (SDLCP), which are extensions of LP and LCP, respecti...

Abstract. Complexity theoretic aspects of continuation methods for the solution of square or underdetermined systems of polynomial equations have been studied by various authors. In this paper we consider overdetermined systems where there are more equations than unknowns. We study Newton’s method for such a system. I.