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On the approximate zero of Newton method.

Zheng-da Huang1

  • 1Department of Mathematics, Zhejiang University, Hangzhou 310028, China. huangzd@css.zju.edu.cn

Journal of Zhejiang University. Science
|March 27, 2003
PubMed
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This study introduces a new criterion for Chen

Area of Science:

  • Numerical Analysis
  • Applied Mathematics

Background:

  • Newton's method is a fundamental iterative technique for solving nonlinear equations.
  • Existing convergence criteria for approximate zeros can be overly restrictive.
  • Weak Lipschitz continuity offers a more practical condition than standard Lipschitz continuity.

Purpose of the Study:

  • To develop a robust judgment criterion for identifying Chen's approximate zeros.
  • To establish error estimates for the convergence of Newton's method under new conditions.
  • To relax the conditions required for analyzing the convergence of Newton's method.

Main Methods:

  • Dominating sequence techniques are employed to derive the judgment criterion.
  • Analysis is based on the properties of a dominating function with a single positive zero.

Related Experiment Videos

  • The operator is assumed to be weak Lipschitz continuous.

    • Main Results:

    • A novel criterion is established to guarantee points as Chen's approximate zeros.
    • Demonstration that Chen's approximate zeros are not necessarily Smale's approximate zeros.
    • Derivation of error estimates indicating the convergent order for practical software stopping criteria (e.g., /f(x)/ < epsilon).

    Conclusions:

    • The proposed criterion is more relaxed and easier to check than traditional Lipschitz conditions.
    • The findings provide a better understanding of convergence properties for Newton's method.
    • The results are applicable to solving nonlinear partial derivative and integration equations.