Euclidean remainders for generalized polynomials
type: Item
instance of: scholarly article
title: Euclidean remainders for generalized polynomials
zbMATH Open document ID: 0654.65038
DOI: 10.1016/0024-3795(88)90128-0
author: Stephen Barnett
published in: Linear Algebra and its Applications
publication date: 1988-01-01T00:00:00Z
review text: The paper deals with methods for computations with real polynomials in generalized form, i.e. with polynomials written as linear combinations of the elements of an orthogonal basis \(\{p_ i(\lambda)\}\) for \({\mathbb{R}}[\lambda]\) satisfying the relations \(p_ 0(\lambda)=1\), \(p_ 1(\lambda)=\alpha_ 1\lambda +\beta_ 1\), and \(p_ i(\lambda)=(\alpha_ i\lambda +\beta_ i)p_{i-1}(\lambda)- \gamma_{i-1}p_{i-2}(\lambda)\) for \(i\geq 2\), where \(\alpha_ i>0\) and \(\gamma_ i\geq 0\). Namely, if \(a(\lambda)=p_ n(\lambda)+a_ 1p_{n- 1}(\lambda)+...+a_ np_ 0(\lambda)\) and \(b(\lambda)=b_ 0p_ m(\lambda)+b_ 1p_{m-1}(\lambda)+...+b_ mp_ 0(\lambda)\) with \(b_ 0\neq 0\) and \(m<n\), the author shows how to apply the Euclidean algorithm for finding gcd of a(\(\lambda)\) and b(\(\lambda)\) without ever resorting to the ordinary representation of any of the polynomials involved in terms of the power basis \(\{\) \(\lambda\) \(i\}\). For this, the author introduces the comrade matrix A of the polynomial a(\(\lambda)\) and then reduces the block matrix [b(A),I], where I is the n-by-n identity matrix of the same size as A, to a quasitriangular form by elementary row operations. As a byproduct, solutions to a set of linear congruences are obtained.
Mathematics Subject Classification ID: 65H05
Mathematics Subject Classification ID: 12D05
Mathematics Subject Classification ID: 65F30
zbMATH DE Number: 4068174
zbMATH Keywords: Euclidean remainders
zbMATH Keywords: greatest common divisor of polynomials
zbMATH Keywords: real polynomials in generalized form
zbMATH Keywords: Euclidean algorithm
zbMATH Keywords: comrade matrix
zbMATH Keywords: linear congruences
reviewed by: Mark Kleiner
MaRDI profile type: Publication
full work available at URL: 0024-3795(88)90128-0
OpenAlex ID: W1978251933
cites work: Greatest common divisor of two polynomials
cites work: Q3309525
cites work: Division of generalized polynomials using the comrade matrix
cites work: Multiplication of Generalized Polynomials, with Applications to Classical Orthogonal Polynomials
cites work: Q5340128
cites work: Continued-fraction expansions for ratios of generalized polynomials
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description: scientific article; zbMATH DE number 4068174