代数方程式に対する高次Nourein法の収束特性  [in Japanese] Convergence Property of Higher Order Nourein's Method for Algebraic Equations  [in Japanese]

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Abstract

一変数代数方程式に対する反復法について,次数の高い反復法ほど初期値に最も近い根が得られるという傾向がある.本論文では,任意次数の反復法が構成できるNourein法についてこの経験則に理論的な根拠を与える.この目的のために,反復法の近似解がそれに最も近い根に収束するような初期値の集合が議論される.しかし,一般にこのような集合の形状は複雑である.そこで,代数方程式の根を定点とするApollonius円からなる領域(Apollonius領域)をこの集合に対する表現として与え,この領域の広さを評価する.具体的に,Nourein法についてApollonius領域を導出し,次数と領域の広さについて考察した.また,この領域と代数方程式の根を母点としたVoronoi領域との関係についても言及する.In general, an iterative method of higher order has a tendency to find a solution which is the nearest to the initial value. The purpose of this paper is to give the theoretical basis to this empirical rule with Nourein's method of arbitrary order of convergence. We discuss a set of such initial values for this purpose. Though the shape of such the set is ordinarily complicated. To investigate the problem we treat a region of easier shape as alternated. That is the region made from Apollonius circles which fixed points are solutions of the algebraic equation. Furthermore the relation between this region and Voronoi region would be mentioned.

In general, an iterative method of higher order has a tendency to find a solution which is the nearest to the initial value. The purpose of this paper is to give the theoretical basis to this empirical rule with Nourein's method of arbitrary order of convergence. We discuss a set of such initial values for this purpose. Though the shape of such the set is ordinarily complicated. To investigate the problem we treat a region of easier shape as alternated. That is the region made from Apollonius circles which fixed points are solutions of the algebraic equation. Furthermore the relation between this region and Voronoi region would be mentioned.

Journal

  • IPSJ journal

    IPSJ journal 46(10), 2505-2512, 2005-10-15

    Information Processing Society of Japan (IPSJ)

Cited by:  1

Codes

  • NII Article ID (NAID)
    110002769898
  • NII NACSIS-CAT ID (NCID)
    AN00116647
  • Text Lang
    JPN
  • Article Type
    Journal Article
  • ISSN
    1882-7764
  • NDL Article ID
    7490669
  • NDL Source Classification
    ZM13(科学技術--科学技術一般--データ処理・計算機)
  • NDL Call No.
    Z14-741
  • Data Source
    CJPref  NDL  NII-ELS  IPSJ 
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