Division of generalized polynomials using the comrade matrix
type: Item
instance of: scholarly article
title: Division of generalized polynomials using the comrade matrix
zbMATH Open document ID: 0552.15003
DOI: 10.1016/0024-3795(84)90077-6
author: Stephen Barnett
published in: Linear Algebra and its Applications
publication date: 1984-01-01T00:00:00Z
review text: A polynomial in the generalized form is an expression \(a(x)=p_ n(x)+a_ 1p_{n-1}(x)+...+a_ np_ 0(x),\) where \(a_ i's\) are real numbers and \(\{p_ i(x)\}\) is an orthogonal polynomial basis defined by \(p_ 0(x)=1,\) \(p_ 1(x)=u_ 1x+v_ 1,\) \(p_ i(x)=(u_ ix+v_ i)p_{i-1}(x)-w_ ip_{i-2}(x),\) \(i=2,3,...\), with \(u_ i>0\), \(w_ i\geq 0\). The author shows that the greatest common divisor d(x) of polynomials a(x) and f(x) in the generalized form, the quotient a(x)/d(x), and the quotient and remainder on division of a(x) by f(x) can be determined simultaneously, without any conversion into standard power form. This is done by applying elementary row operations to the matrix f(A), where A is the comrade matrix associated with a(x). The notation and terminology follow that of the author in ibid. 41, 277-298 (1981; Zbl 0479.15012).
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 12D05
zbMATH DE Number: 3879072
zbMATH Keywords: greatest common divisor
zbMATH Keywords: generalized polynomial
zbMATH Keywords: division
zbMATH Keywords: comrade matrix
reviewed by: Mark Kleiner
MaRDI profile type: Publication
full work available at URL: 0024-3795(84)90077-6
OpenAlex ID: W1969455262
cites work: Greatest common divisor of two polynomials
cites work: Q5629775
cites work: A companion matrix analogue for orthogonal polynomials
cites work: Congenial matrices
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description: scientific article; zbMATH DE number 3879072