Local Convergence for a Frozen Family of Steffensen-Like Methods under Weak Conditions

The semi-local convergence analysis of a well defined and efficient two-step Chord-type method in Banach spaces is presented in this study. The recurrence relation technique is used under some weak assumptions. The pertinency of the assumed method is extended for nonlinear non-differentiable operators. The convergence theorem is also established to show the existence and uniqueness of the approximate solution. A numerical illustration is quoted to certify the theoretical part which shows that earlier studies fail if the function is non-differentiable.

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

Mathematics ◽

10.3390/math9161855 ◽

2021 ◽

Vol 9 (16) ◽

pp. 1855 ◽

Cited By ~ 1

Author(s):

Petko D. Proinov ◽

Maria T. Vasileva

Keyword(s):

Iterative Methods ◽

Local Convergence ◽

Semilocal Convergence ◽

High Order ◽

Convergence Theorems ◽

New Family ◽

Special Cases ◽

Iteration Functions ◽

Iteration Function ◽

Local Convergence Analysis

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.

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Application of a Generalized Secant Method to Nonlinear Equations with Complex Roots

Axioms ◽

10.3390/axioms10030169 ◽

2021 ◽

Vol 10 (3) ◽

pp. 169

Author(s):

Avram Sidi

Keyword(s):

Nonlinear Equations ◽

Local Convergence ◽

Real Root ◽

Numerical Procedure ◽

Secant Method ◽

Convergence Theory ◽

Simple Complex ◽

Real Roots ◽

Complex Roots ◽

Appl Math

The secant method is a very effective numerical procedure used for solving nonlinear equations of the form f(x)=0. In a recent work (A. Sidi, Generalization of the secant method for nonlinear equations. Appl. Math. E-Notes, 8:115–123, 2008), we presented a generalization of the secant method that uses only one evaluation of f(x) per iteration, and we provided a local convergence theory for it that concerns real roots. For each integer k, this method generates a sequence {xn} of approximations to a real root of f(x), where, for n≥k, xn+1=xn−f(xn)/pn,k′(xn), pn,k(x) being the polynomial of degree k that interpolates f(x) at xn,xn−1,…,xn−k, the order sk of this method satisfying 1<sk<2. Clearly, when k=1, this method reduces to the secant method with s1=(1+5)/2. In addition, s1<s2<s3<⋯, such that limk→∞sk=2. In this note, we study the application of this method to simple complex roots of a function f(z). We show that the local convergence theory developed for real roots can be extended almost as is to complex roots, provided suitable assumptions and justifications are made. We illustrate the theory with two numerical examples.

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