# Simultaneous factorization of a polynomial by rational approximation

Carsten Carstensen, Tetsuya Sakurai

Research output: Contribution to journalArticle

### Abstract

In this note we present a numerical method to approximate some relatively prime factors of a polynomial simultaneously. Our approach gives methods of arbitrary order; Grau's method (Carstensen, 1992; Grau, 1971) is obtained as the second order method which is Durand-Kerner's method when we have linear factors. For linear factors our approach yields the simultaneous methods introduced in Sakurai et al. (1991). We prove local convergence and estimate the R-order of the total step version as well as the single step version of the methods. We derive an algorithm and present numerical examples which confirm the convergence behavior theoretically predicted. © 1995.

Original language English 165-178 14 Journal of Computational and Applied Mathematics 61 2 Published - 31 Jul 1995

### Fingerprint

Rational Approximation
Factorization
Polynomial
Simultaneous Methods
Relatively prime
Prime factor
Local Convergence
Numerical Methods
Numerical Examples
Arbitrary
Estimate

### Keywords

• Factorization of polynomials
• Polynomial zeros
• Rational approximation
• Simultaneous methods
• Single step methods

### Cite this

Carstensen, Carsten ; Sakurai, Tetsuya. / Simultaneous factorization of a polynomial by rational approximation. In: Journal of Computational and Applied Mathematics. 1995 ; Vol. 61, No. 2. pp. 165-178.
title = "Simultaneous factorization of a polynomial by rational approximation",
abstract = "In this note we present a numerical method to approximate some relatively prime factors of a polynomial simultaneously. Our approach gives methods of arbitrary order; Grau's method (Carstensen, 1992; Grau, 1971) is obtained as the second order method which is Durand-Kerner's method when we have linear factors. For linear factors our approach yields the simultaneous methods introduced in Sakurai et al. (1991). We prove local convergence and estimate the R-order of the total step version as well as the single step version of the methods. We derive an algorithm and present numerical examples which confirm the convergence behavior theoretically predicted. {\circledC} 1995.",
keywords = "Factorization of polynomials, Polynomial zeros, Rational approximation, Simultaneous methods, Single step methods",
author = "Carsten Carstensen and Tetsuya Sakurai",
year = "1995",
month = "7",
day = "31",
language = "English",
volume = "61",
pages = "165--178",
journal = "Journal of Computational and Applied Mathematics",
issn = "0377-0427",
publisher = "Elsevier",
number = "2",

}

Simultaneous factorization of a polynomial by rational approximation. / Carstensen, Carsten; Sakurai, Tetsuya.

In: Journal of Computational and Applied Mathematics, Vol. 61, No. 2, 31.07.1995, p. 165-178.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Simultaneous factorization of a polynomial by rational approximation

AU - Carstensen, Carsten

AU - Sakurai, Tetsuya

PY - 1995/7/31

Y1 - 1995/7/31

N2 - In this note we present a numerical method to approximate some relatively prime factors of a polynomial simultaneously. Our approach gives methods of arbitrary order; Grau's method (Carstensen, 1992; Grau, 1971) is obtained as the second order method which is Durand-Kerner's method when we have linear factors. For linear factors our approach yields the simultaneous methods introduced in Sakurai et al. (1991). We prove local convergence and estimate the R-order of the total step version as well as the single step version of the methods. We derive an algorithm and present numerical examples which confirm the convergence behavior theoretically predicted. © 1995.

AB - In this note we present a numerical method to approximate some relatively prime factors of a polynomial simultaneously. Our approach gives methods of arbitrary order; Grau's method (Carstensen, 1992; Grau, 1971) is obtained as the second order method which is Durand-Kerner's method when we have linear factors. For linear factors our approach yields the simultaneous methods introduced in Sakurai et al. (1991). We prove local convergence and estimate the R-order of the total step version as well as the single step version of the methods. We derive an algorithm and present numerical examples which confirm the convergence behavior theoretically predicted. © 1995.

KW - Factorization of polynomials

KW - Polynomial zeros

KW - Rational approximation

KW - Simultaneous methods

KW - Single step methods

UR - http://www.scopus.com/inward/record.url?scp=0029425701&partnerID=8YFLogxK

M3 - Article

VL - 61

SP - 165

EP - 178

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

IS - 2

ER -