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A New Family of High-Order Ehrlich-Type Iterative Methods

Author

Listed:
  • Petko D. Proinov

    (Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 24 Tzar Asen, 4000 Plovdiv, Bulgaria)

  • Maria T. Vasileva

    (Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 24 Tzar Asen, 4000 Plovdiv, Bulgaria)

Abstract

One of the famous third-order iterative methods for finding simultaneously all the zeros of a polynomial was introduced by Ehrlich in 1967. In this paper, we construct a new family of high-order iterative methods as a combination of Ehrlich’s iteration function and an arbitrary iteration function. We call these methods Ehrlich’s methods with correction . The paper provides a detailed local convergence analysis of presented iterative methods for a large class of iteration functions. As a consequence, we obtain two types of local convergence theorems as well as semilocal convergence theorems (with computer verifiable initial condition). As special cases of the main results, we study the convergence of several particular iterative methods. The paper ends with some experiments that show the applicability of our semilocal convergence theorems.

Suggested Citation

  • Petko D. Proinov & Maria T. Vasileva, 2021. "A New Family of High-Order Ehrlich-Type Iterative Methods," Mathematics, MDPI, vol. 9(16), pages 1-25, August.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:16:p:1855-:d:609161
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    References listed on IDEAS

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    1. Proinov, Petko D., 2016. "A general semilocal convergence theorem for simultaneous methods for polynomial zeros and its applications to Ehrlich’s and Dochev–Byrnev’s methods," Applied Mathematics and Computation, Elsevier, vol. 284(C), pages 102-114.
    2. Proinov, Petko D. & Ivanov, Stoil I. & Petković, Miodrag S., 2019. "On the convergence of Gander’s type family of iterative methods for simultaneous approximation of polynomial zeros," Applied Mathematics and Computation, Elsevier, vol. 349(C), pages 168-183.
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