Multiplication-free evaluation of polynomials via a Stochastic Bernstein Representation

摘要

A new multiplication-free method for the evaluation of multidimensional polynomials is proposed. The method is based on a Stochastic Bernstein Representation (SBR) and utilizes a random number generator, a locally encoded data structure, and a system table. The SBR is shown to be capable of representing any polynomial function and of approximating any continuous function arbitrarily closely. An error bound analysis is performed using a large deviations technique. A variant of the SBR, which has been used by others to balance an inverted pendulum, is also analyzed.