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General Local Convergence Theorems about the Picard Iteration in Arbitrary Normed Fields with Applications to Super–Halley Method for Multiple Polynomial Zeros

Author

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  • Stoil I. Ivanov

    (Faculty of Physics and Technology, University of Plovdiv Paisii Hilendarski, 24 Tzar Asen, 4000 Plovdiv, Bulgaria)

Abstract

In this paper, we prove two general convergence theorems with error estimates that give sufficient conditions to guarantee the local convergence of the Picard iteration in arbitrary normed fields. Thus, we provide a unified approach for investigating the local convergence of Picard-type iterative methods for simple and multiple roots of nonlinear equations. As an application, we prove two new convergence theorems with a priori and a posteriori error estimates about the Super-Halley method for multiple polynomial zeros.

Suggested Citation

  • Stoil I. Ivanov, 2020. "General Local Convergence Theorems about the Picard Iteration in Arbitrary Normed Fields with Applications to Super–Halley Method for Multiple Polynomial Zeros," Mathematics, MDPI, vol. 8(9), pages 1-11, September.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:9:p:1599-:d:414825
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    Cited by:

    1. Watchareepan Atiponrat & Pariwate Varnakovida & Pharunyou Chanthorn & Teeranush Suebcharoen & Phakdi Charoensawan, 2023. "Common Fixed Point Theorems for Novel Admissible Contraction with Applications in Fractional and Ordinary Differential Equations," Mathematics, MDPI, vol. 11(15), pages 1-20, August.

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