Parallel Methods for Solving Polynomial Problems on Distributed Memory Multicomputers

Abstract

We give a group of parallel methods for solving polynomial related problems and their implementations on a distributed memory multicomputer. These problems are 1. the evaluation of polynomials, 2. the multiplication of polynomials, 3. the division of polynomials, and 4. the interpolation of polynomials. Mathematical analyses are given for exploiting the parallelisms of these operations. The related parallel methods supporting the solutions of these polynomial problems, such as FFT, Toeplitz linear systems and others are also discussed. We present some experimental results of these parallel methods on the Intel hypercube. polynomials based on the Horner’s rule is discussed in section 2. The experimental results on the Intel hypercube are also presented. The parallelism of the polynomial multiplication is exploited by transferring the problem to a set of special FFT series functions, on which the operations can be perfectly distributed among different processors. Section 3 gives the mathematical analyses and parallel method of the polynomial multiplication. The polynomial division problem is solved based on parallel solutions for Toeplitz triangular linear systems and the parallel polynomial multiplication, and is discussed in section 4. Section 5 addresses a parallel method for the Lagrange piecewise cubic polynomial interpolation. Finally, we give a summary and future work in the last section.