On the local convergence of Sakurai-Torii-Sugiura method for simultaneous approximation of polynomial zeros

In this paper, we establish two local convergence theorems that provide initial conditions and error estimates to guarantee the Q-convergence of an extended version of Chebyshev–Halley family of iterative methods for multiple polynomial zeros due to Osada (J. Comput. Appl. Math. 2008, 216, 585–599). Our results unify and complement earlier local convergence results about Halley, Chebyshev and Super–Halley methods for multiple polynomial zeros. To the best of our knowledge, the results about the Osada’s method for multiple polynomial zeros are the first of their kind in the literature. Moreover, our unified approach allows us to compare the convergence domains and error estimates of the mentioned famous methods and several new randomly generated methods.

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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

Mathematics ◽

10.3390/math8091599 ◽

2020 ◽

Vol 8 (9) ◽

pp. 1599

Author(s):

Stoil I. Ivanov

Keyword(s):

Error Estimates ◽

Local Convergence ◽

A Priori ◽

A Posteriori Error Estimates ◽

Sufficient Conditions ◽

Posteriori Error ◽

Picard Iteration ◽

Polynomial Zeros ◽

Convergence Theorems ◽

Simple And Multiple Roots

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.

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On the convergence of Halley’s method for simultaneous computation of polynomial zeros

Journal of Numerical Mathematics ◽

10.1515/jnma-2015-0026 ◽

2015 ◽

Vol 23 (4) ◽

Cited By ~ 11

Author(s):

Petko D. Proinov ◽

Stoil I. Ivanov

Keyword(s):

Convergence Theorem ◽

Local Convergence ◽

Semilocal Convergence ◽

Polynomial Zeros ◽

Convergence Theorems ◽

Halley’S Method ◽

Halley's Method ◽

Semilocal Convergence Theorem

AbstractIn this paper we study the convergence of Halley’s method as a method for finding all zeros of a polynomial simultaneously. We present two types of local convergence theorems as well as a semilocal convergence theorem for Halley’s method for simultaneous computation of polynomial zeros.

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