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Intra-Basis Multiplication of Polynomials Given in Various Polynomial Bases

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Intra-Basis Multiplication of Polynomials Given in Various Polynomial Bases Intra-Basis Multiplication of Polynomials Given in Various Polynomial Bases S. Karamia, M. Ahmadnasabb, M. Hadizadehd, A. Amiraslanic,d aDepartment of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan, Iran bDepartment of Mathematics, University of Kurdistan, Sanandaj, Iran cSchool of STEM, Department of Mathematics, Capilano University, North Vancouver, BC, Canada dFaculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran 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 tech- niques 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 repre- sentation of polynomial multiplication operators is illustrated for discretization of a linear elliptic problem with stochastic coefficients. Keywords: Polynomial arithmetic, Degree-graded & non degree-graded polynomial bases, Polynomial multiplication, Stochastic Galerkin matrices, Polynomial division 2010 MSC: 15B99, 08A40, 33C47, 80M10 arXiv:2108.01558v1 [math.NA] 3 Aug 2021 1. Introduction There is an important and ongoing thread of research into polynomial computation using polyno- mial bases other than the standard monomial power basis. Such problems have interesting applica- tions in several areas including approximation theory [10, 20, 21, 22], cryptography [26], coding [37] and computer science [18]. The investigations in [1, 14, 36] specifically involve computation with polynomials expressed in the Lagrange basis, or, in other words, polynomial computation directly by values. ∗Corresponding author Email address: [email protected] (A. Amiraslani) Preprint submitted to Linear Algebra and its Applications August 4, 2021 A key motivation for this interest in polynomial computation using alternatives to the power basis is that conversion between bases can be unstable (more commonly so from other bases to the monomial basis) and the instability increases with the degree. (See e.g. [23]). The complications arising from such computations motivate explorations of hybrid symbolic-numeric techniques for polynomial computation that are relevant to researchers interested in computer algebra and numerical analysis. For instance, investigations in [8] and [24] show that working directly in the Lagrange polynomial basis is both numerically stable and efficient, more stable and efficient than had been heretofore credited widely in the numerical analysis community. These and similar results strengthen the motivation for examining algorithms for direct manipulation of polynomials given in polynomial bases other than the standard monomial power basis. Figure 1 summarizes this idea which we intend to use here as well. This work attempts the dashed road. The symbols P , S, PM and SM stand for the \Problem" to be solved in an arbitrary basis, the \Solution" in the same basis, the \Polynomial after conversion to Monomial basis", and the \Solution in the Monomial basis", respectively. P 99K S + * PM ) SM Figure 1 This work is mainly geared toward implementing methods for direct multiplication of polynomials given in polynomial bases. To this end, we look at two important families of polynomial bases: degree-graded polynomial bases such as orthogonal polynomial bases and Newton basis, and non- degree-graded polynomial bases including the Lagrange and Bernstein polynomial bases. From there and as an important application, we devise techniques for direct division of polynomials given in polynomial bases. That includes a generalization of the well-known \long division" method to degree-graded polynomial bases. Most of the techniques implemented in this work are based on matrix-vector representations. This enables us to use some known and straightforward techniques from linear algebra which make things a lot easier to follow and, if necessary, extend. Due to the importance and applications of Chebyshev basis in various disciplines such as Electri- cal Engineering, direct multiplication of polynomials in that orthogonal polynomial basis has been studied fairly exhaustively. The paper [6] deals with the operation of direct multiplication of polyno- mials in Chebyshev form using two approaches: a direct multiplication of polynomials in Chebyshev form as well as another approach based on the discrete cosine transform (DCT). In [28], the same problem has been also addressed through the well-known Karatsuba's algorithm. This result is inter- esting since it provides a more practicable algorithm, avoiding conversions between bases. Notably, [20] completely reduces the multiplication in Chebyshev form to the one in monomial basis. More generally, in [34], the authors present a general method that allows the derivation of the expansion coefficients of the product of two orthogonal functions provided the generating function is known. Besides, writing the product of two polynomials Pn(x) and Pm(x) of degrees m and n, respectively, given in an orthogonal polynomial basis as the linear expansion of polynomials in the same basis is a well-known problem called "the linearization of the product". The goal is to find the coefficients Pn+m gmnk such that Pn(x)Pm(x) = k=0 gnmkPk(x). Some methods and approaches for computing the gnmk have been developed so far. For classical families of orthogonal polynomial bases, explicit expressions have been obtained, usually in terms of generalized hypergeometric series, using impor- 2 tant characterizing properties: recurrence relations, generating functions, orthogonality weights, etc. (see. e.g. [34, 4, 29, 35] and references therein). We intend to find direct multiplication of polynomials in degree-graded as well as non-degree- graded polynomial bases. In particular and in comparison with [34], all we need to know here about an orthogonal polynomial basis such as but not limited to the Chebyshev polynomial basis (and other degree-graded polynomial bases for that matter) are the coefficients of its \three-term recurrence relation". However, to the best of our knowledge, there has not been any comprehensive study in the literature on direct multiplication of polynomials in various polynomial bases. As such, it is important to implement algorithms to handle direct multiplication on polynomials in such bases. This is for instance of great interest to rigorous computing which aims to guarantee numerical approximation of functions through certified bounds on the results. There are also some prominent works on direct division of polynomials given in polynomial bases which we briefly mention here. In particular, [5] uses the \Comrade matrix" to simultaneously determine the greatest common divisor as well as the quotient and remainder of the division of two polynomials given in a degree-graded polynomial basis (the author only mentions orthogonal bases, but the idea can actually be extended to other degree-graded polynomial bases as well). The paper [11], extends the classical Euclidean division algorithm to the Bernstein polynomial basis using an algorithm based on a simple ring isomorphism that converts the division problems from the Bernstein polynomial basis to an equivalent problem in the monomial basis. Moreover, [30] devises algorithms similar to the classical long division method for direct division of polynomials in non-degree-graded bases (i.e., the Lagrange and Bernstein polynomial bases). Note that the aim of the current study is to provide a direct division approach that does not require basis conversion, therefore our proposed techniques here are all based on the presented direct multiplication methods in matrix-vector forms. The structure of this paper is as follows. We discuss direct multiplication of polynomials in degree-graded polynomial bases in Section 2. To this end, the multiplication of a basis element of a degree-graded polynomial basis by a basis vector of the same polynomial basis with a given degree is represented in a matrix-vector form. Section 3 concerns the direct polynomial multiplication formulas in more famous non-degree-graded polynomial bases, i.e. the Bernstein and Lagrange polynomials bases. To derive those formulas, we show how to rewrite a polynomial given in a non-degree-graded polynomial basis of certain degree as a polynomial in the same basis with a higher (or possibly lower) degree using \lifting" (or possibly \dropping") matrices. In Section 4, using the obtained results in previous sections, we present intra-basis polynomial division techniques in degree-graded
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