在不同多项式基中给定多项式的基内乘法
Intra-Basis Multiplication of Polynomials Given in Various Polynomial Bases
摘要：
多项式的乘法是计算机代数中的关键运算之一，它在开发其他常用多项式运算（如除法、求值/插值和因式分解）的技术中起着重要作用。在这项工作中，我们提出的公式和技术的多项式乘法表示在各种著名的多项式基地没有任何变化的基础。特别地，我们考虑度分次多项式基，包括但不限于正交多项式基和非度分次多项式基，包括伯恩斯坦和拉格朗日基。所有描述的多项式乘法公式和技术，在这项工作中，这主要是在矩阵向量的形式，保持在其中的多项式给出的基础。此外，利用多项式直接相乘的结果，我们设计了在多项式基中的基内多项式除法的技术。将著名的“长除法“算法推广到任意阶分次多项式基。所提出的框架处理矩阵向量计算，这往往会导致良好的结构矩阵。最后，说明了所提出的技术在构造多项式乘法算子的Galerkin表示中的应用，用于离散具有随机系数的线性椭圆问题。
英文摘要：
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.
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