Euclid algorithm, orthogonal polynomials, and generalized Routh-Hurwitz algorithm
type: Item
instance of: scholarly article
title: Euclid algorithm, orthogonal polynomials, and generalized Routh-Hurwitz algorithm
zbMATH Open document ID: 0879.12001
DOI: 10.1016/0024-3795(94)00342-4
author: Yves V. Genin
published in: Linear Algebra and its Applications
publication date: 1998-01-18T00:00:00Z
review text: This paper, which is based on control-systems engineering, begins with a review of the classical Euclidean algorithm to find the gcd of two univariate polynomials over a field. The algorithm is said to be regular if all the quotient polynomials have degree one, otherwise it is singular. In the case of real coefficients, the relationship of the algorithm to orthogonal polynomials and Sturm's theorem is outlined. The main thrust of the paper is to lift the restriction to real coefficients. Let a bar denote complex conjugation. Let \(P(p)\) be any complex polynomial of degree \(n\) and define \[ \begin{aligned} P_0(p) & =\textstyle {{1\over 2}} \bigl[P(p) +(-1)^n \overline P(-\overline p) \bigr], \\ P_1(p) & = \textstyle {{1\over 2}} \bigl[P(p) -(-1)^n \overline P(-\overline p) \bigr], \end{aligned} \] which are often called the paraeven and paraodd parts of \(P(p)\). The leading coefficient of \(P_0(p)\) may be assumed to be pure real so that \(\deg P_0= n>\deg P_1\). Then the generalized Routh-Hurwitz algorithm is defined to be the Euclidean algorithm applied to \(P_0(p)\), \(P_1(p)\), i.e. \(P_{i-1} (p)= q_i(p) P_i(p) + P_{i+1} (p)\) for \(i=1,2, \dots, n'\), with \(n'\leq n\) and \(P_{n'+1} (p)=0\), extended as follows to handle singular cases. If there exists some value of \(i\), say \(i_0\), for which \(\deg P_{i_0} >0\) and \(P_{i_0+1} (p)=0\) then replace the ``missing'' polynomial by \(P_{i_0+1} (p): =dP_{i_0} (p)/ dp\) and continue the algorithm. Denote by \(i_0,i_1, \dots, i_{s-1}\) the sequence of indices for which this occurs. The quotient polynomials can be expanded as \(q_i(p)= -j \sum^{\mu_i}_{s=0} q_{i,s} (jp)^{\mu_i-s}\), where \(j= \sqrt {-1}\), \(\mu_i= \deg q_i\) and all the coefficients \(q_{i,s}\) are real numbers. Only the leading coefficients \(q_{i,0}\) are used in the following. Let \(N(P) \), \(N_0(P)\), \(\overline N(P)\) denote respectively the number of zeros of \(P(p)\) with \(\text{Re} p>0\), \(=0\), \(<0\). Then the following theorem i…
Mathematics Subject Classification ID: 12D10
Mathematics Subject Classification ID: 30C15
zbMATH DE Number: 951795
zbMATH Keywords: Euclidean algorithm
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: Sturm's theorem
zbMATH Keywords: complex polynomial
zbMATH Keywords: paraeven
zbMATH Keywords: paraodd
zbMATH Keywords: generalized Routh-Hurwitz algorithm
zbMATH Keywords: zero distributions
reviewed by: Francis Wright
MaRDI profile type: Publication
full work available at URL: 0024-3795(94)00342-4
OpenAlex ID: W2002647180
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description: scientific article; zbMATH DE number 951795