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A Neural Root Finder of Polynomials Based on Root Moments

dshuang{at}iim.ac.cn, Hefei Institute of Intelligent Machines, Chinese Academy of Sciences, Hefei, Anhui 230031, China, and AIMtech Center, Department of Computer Science, City University of Hong Kong, Kowloon Tong, Hong Kong

Horace H.S. Ip

cship{at}cityu.edu.hk, Institute of Intelligent Machines, Chinese Academy of Sciences, Hefei, Anhui 230031, China

Zheru Chi

enzheru{at}inet.poly.edu.hk, Center for Multimedia Signal Processing, Hong Kong Polytechnic University, Hong Kong

This letter proposes a novel neural root finder based on the root moment method (RMM) to find the arbitrary roots (including complex ones) of arbitrary polynomials. This neural root finder (NRF) was designed based on feedforward neural networks (FNN) and trained with a constrained learning algorithm (CLA). Specifically, we have incorporated the a priori information about the root moments of polynomials into the conventional backpropagation algorithm (BPA), to construct a new CLA. The resulting NRF is shown to be able to rapidly estimate the distributions of roots of polynomials. We study and compare the advantage of the RMM-based NRF over the previous root coefficient method–based NRF and the traditional Muller and Laguerre methods as well as the mathematica roots function, and the behaviors, the accuracies of the resulting root finders, and their training speeds of two specific structures corresponding to this FNN root finder: the log and the – FNN. We also analyze the effects of the three controlling parameters { P₀,_p,} with the CLA on the two NRFs theoretically and experimentally. Finally, we present computer simulation results to support our claims.