Nearest singular polynomials
type: Item
instance of: scholarly article
title: Nearest singular polynomials
zbMATH Open document ID: 0917.65055
author: Lihong Zhi
author: Wenda Wu
published in: Journal of Symbolic Computation
publication date: 1999-03-22T00:00:00Z
review text: The following problem is considered: Given a monic polynomial \(f(x)\) \[ f= x^m+ \sum^m_{j=1} f_j\cdot x^{m-j},\quad f_j\in\mathbb{C}. \] Find a monic polynomial \(h(x)\) \[ h= (x- c)^k\cdot \Biggl(x^{m-k}+ \sum^{m- k}_{j= 1} \Phi_j\cdot x^{m- k-j}\Biggr),\quad c,\Phi_j\in \mathbb{C},\;k\geq 2 \] such that \(\| f- h\|^2\to\) minimum, where \(\| p(x)\|^2= \sum^k_{j= 0}| p_j|^2\) for any \(p(x)= \sum^k_{j= 0} p_j\cdot x^{k-j}\), \(p_j\in\mathbb{C}\). Some recursive relations between the polynomials determining the multiple zeros for consecutive \(k\)'s are presented. Two numerical examples are given.
Mathematics Subject Classification ID: 65K05
Mathematics Subject Classification ID: 30C10
Mathematics Subject Classification ID: 90C20
zbMATH DE Number: 1268447
zbMATH Keywords: nearest singular polynomials
zbMATH Keywords: quadratic programming
zbMATH Keywords: multiple zeros
zbMATH Keywords: numerical examples
reviewed by: Hans Benker
MaRDI profile type: Publication
full work available at URL: jsco.1998.0233
OpenAlex ID: W1967980374
DOI: 10.1006/JSCO.1998.0233
description: scientific article; zbMATH DE number 1268447