Scholarlink

<p>Schröder’s theorem studied the convergence of Newton’s method for the complex quadratic function, showing that on either side of the perpendicular bisector of the roots, Newton’s method converges to the root on that particular side. This paper rigorously proves that Schröder’s theorem also holds for a new variant of Newton’s method—named Backtracking New Q-Newton’s method (BNQN). However, the orbit behavior of the BNQN method differs from Newton’s method on the perpendicular bisector of the roots. The motivation for our rigorous proof comes from a remarkable experimental discovery that the basins of attraction of the BNQN method of polynomials of any degree seem to have piecewise smooth boundaries, while for Newton’s method of polynomials of degrees larger than two are usually fractal.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Schröder’s theorem for backtracking new Q-Newton’s method

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2026年5月15日

查看全文
  • John Erik Fornæss,
  • Mi Hu,
  • Tuyen Trung Truong,
  • Takayuki Watanabe

摘要

Schröder’s theorem studied the convergence of Newton’s method for the complex quadratic function, showing that on either side of the perpendicular bisector of the roots, Newton’s method converges to the root on that particular side. This paper rigorously proves that Schröder’s theorem also holds for a new variant of Newton’s method—named Backtracking New Q-Newton’s method (BNQN). However, the orbit behavior of the BNQN method differs from Newton’s method on the perpendicular bisector of the roots. The motivation for our rigorous proof comes from a remarkable experimental discovery that the basins of attraction of the BNQN method of polynomials of any degree seem to have piecewise smooth boundaries, while for Newton’s method of polynomials of degrees larger than two are usually fractal.