Influence of the multiplicity of the roots on the basins of attraction of Newton’s method

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• 作者：José M. Gutiérrez (1)
  Luis J. Hernández-Paricio (1)
  Miguel Mara?ón-Grandes (1)
  M. Teresa Rivas-Rodríguez (1)
  
• 关键词：Newton’s method ; Basin of attraction ; Multiplicity of a root ; Subdivisions on the sphere ; Measure algorithms ; 65H04 ; 65Y99 ; 54H20
• 刊名：Numerical Algorithms
• 出版年：2014
• 出版时间：July 2014
• 年：2014
• 卷：66
• 期：3
• 页码：431-455
• 全文大小：1,376 KB
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• 作者单位：José M. Gutiérrez (1)
  Luis J. Hernández-Paricio (1)
  Miguel Mara?ón-Grandes (1)
  M. Teresa Rivas-Rodríguez (1)
  
  1. Departamento de Matemáticas y Computación, Universidad de La Rioja, c/ Luis de Ulloa s/n, 26004, Logro?o, Spain
  
• ISSN：1572-9265

文摘

In this work, we develop and implement two algorithms for plotting and computing the measure of the basins of attraction of rational maps defined on the Riemann sphere. These algorithms are based on the subdivisions of a cubical decomposition of a sphere and they have been made by using different computational environments. As an application, we study the basins of attraction of the fixed points of the rational functions obtained when Newton’s method is applied to a polynomial with two roots of multiplicities m and n. We focus our attention on the analysis of the influence of the multiplicities m and n on the measure of the two basins of attraction. As a consequence of the numerical results given in this work, we conclude that, if m > n, the probability that a point in the Riemann Sphere belongs to the basin of the root with multiplicity m is bigger than the other case. In addition, if n is fixed and m tends to infinity, the probability of reaching the root with multiplicity n tends to zero.