Intra-Basis Multiplication of Polynomials Given in Various Polynomial Bases

TL;DR

This paper develops formulas and techniques for multiplying and dividing polynomials directly within various polynomial bases, including orthogonal, Bernstein, and Lagrange bases, without changing bases, and applies these methods to discretize elliptic problems.

Contribution

It introduces basis-preserving multiplication and division formulas for diverse polynomial bases and generalizes the long division algorithm to degree-graded bases.

Findings

Formulas for intra-basis polynomial multiplication in various bases

Techniques for intra-basis polynomial division and generalization of long division

Application to discretize elliptic problems with stochastic coefficients

Abstract

Multiplication of polynomials is among key operations in computer algebra which plays important roles in developing techniques for other commonly used polynomial operations such as division, evaluation/interpolation, and factorization. In this work, we present formulas and techniques for polynomial multiplications expressed in a variety of well-known polynomial bases without any change of basis. In particular, we take into consideration degree-graded polynomial bases including, but not limited to orthogonal polynomial bases and non-degree-graded polynomial bases including the Bernstein and Lagrange bases. All of the described polynomial multiplication formulas and techniques in this work, which are mostly presented in matrix-vector forms, preserve the basis in which the polynomials are given. Furthermore, using the results of direct multiplication of polynomials, we devise techniques for intra-basis polynomial division in the polynomial bases. A generalization of the well-known "long division" algorithm to any degree-graded polynomial basis is also given. The proposed framework deals with matrix-vector computations which often leads to well-structured matrices. Finally, an application of the presented techniques in constructing the Galerkin representation of polynomial multiplication operators is illustrated for discretization of a linear elliptic problem with stochastic coefficients.