Mathematical Sciences: Complexity of Real Computation and Zeros of Polynomial Systems

斯梅尔 调查研究算法的基础上牛顿的方法找到零系统的多项式。 分析的重点是获得所需的牛顿步数的界限。 这样的界限将取决于变量的数量、多项式的次数和期望的精度。 该项目的目标是非线性方程求解的复杂性理论。 方程求解是许多数学的核心，与其现代计算理论相关的主要理论问题是其复杂性理论。 方程求解及其计算方面是工程、物理、经济和许多其他学科使用数学的主要方式。

Smale The investigator studies algorithms based on Newton's method for finding zeros of a system of polynomials. The focus of the analysis is to obtain bounds on the number of Newton steps required. Such bounds will depend on the number of variables, the degrees of the polynomials, and the desired accuracy. The goal of the project is a complexity theory of nonlinear equation solving. Equation solving is at the heart of much of mathematics and the main theoretical problem associated to its modern theory of computation is its complexity theory. Equation solving, together with its computational aspects, is a main way that mathematics is used by engineering, physics, economics, and many other disciplines.