Fast Computation of Zeros of Polynomial Systems with Bounded Degree under Finite-precision
Provides a practical finite-precision implementation of a previously theoretical algorithm, addressing computational feasibility for numerical analysts.
The paper presents a finite-precision version of an algorithm that solves Smale's 17th problem for bounded-degree polynomial systems, showing it runs in average polynomial time with polynomial precision requirements.
A solution for Smale's 17th problem, for the case of systems with bounded degree was recently given. This solution, an algorithm computing approximate zeros of complex polynomial systems in average polynomial time, assumed infinite precision. In this paper we describe a finite-precision version of this algorithm. Our main result shows that this version works within the same time bounds and requires a precision which, on the average, amounts to a polynomial amount of bits in the mantissa of the intervening floating-point numbers.