A family of higher order derivative free methods for nonlinear systems with local convergence analysis

We introduce a new faster King-Werner-type derivative-free method for solving nonlinear equations. The local as well as semi-local convergence analysis is presented under weak center Lipschitz and Lipschitz conditions. The convergence order as well as the convergence radii are also provided. The radii are compared to the corresponding ones from similar methods. Numerical examples further validate the theoretical results.

Download Full-text

Higher-order derivative-free families of Chebyshev–Halley type methods with or without memory for solving nonlinear equations

Applied Mathematics and Computation ◽

10.1016/j.amc.2017.07.051 ◽

2017 ◽

Vol 315 ◽

pp. 224-245 ◽

Cited By ~ 3

Author(s):

Ioannis K. Argyros ◽

Munish Kansal ◽

Vinay Kanwar ◽

Sugandha Bajaj

Keyword(s):

Nonlinear Equations ◽

Higher Order ◽

Order Derivative ◽

Derivative Free ◽

Higher Order Derivative

Download Full-text

Development of Optimal Eighth Order Derivative-Free Methods for Multiple Roots of Nonlinear Equations

Symmetry ◽

10.3390/sym11060766 ◽

2019 ◽

Vol 11 (6) ◽

pp. 766 ◽

Cited By ~ 3

Author(s):

Janak Raj Sharma ◽

Sunil Kumar ◽

Ioannis K. Argyros

Keyword(s):

Basins Of Attraction ◽

Higher Order ◽

Order Derivative ◽

Multiple Roots ◽

Nonlinear Functions ◽

Eighth Order ◽

Multiple Zeros ◽

Derivative Free ◽

Graphical Tool ◽

Derivative Free Methods

A number of higher order iterative methods with derivative evaluations are developed in literature for computing multiple zeros. However, higher order methods without derivative for multiple zeros are difficult to obtain and hence such methods are rare in literature. Motivated by this fact, we present a family of eighth order derivative-free methods for computing multiple zeros. Per iteration the methods require only four function evaluations, therefore, these are optimal in the sense of Kung-Traub conjecture. Stability of the proposed class is demonstrated by means of using a graphical tool, namely, basins of attraction. Boundaries of the basins are fractal like shapes through which basins are symmetric. Applicability of the methods is demonstrated on different nonlinear functions which illustrates the efficient convergence behavior. Comparison of the numerical results shows that the new derivative-free methods are good competitors to the existing optimal eighth-order techniques which require derivative evaluations.

Download Full-text

Extending the Applicability of a Two-Step Chord-Type Method for Non-Differentiable Operators

Mathematics ◽

10.3390/math7090804 ◽

2019 ◽

Vol 7 (9) ◽

pp. 804

Author(s):

Ioannis K. Argyros ◽

Neha Gupta ◽

J. P. Jaiswal

Keyword(s):

Approximate Solution ◽

Convergence Analysis ◽

Banach Spaces ◽

Recurrence Relation ◽

Convergence Theorem ◽

Local Convergence ◽

Existence And Uniqueness ◽

Numerical Illustration ◽

Type Method ◽

Local Convergence Analysis

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.

Download Full-text