A general class of derivative free optimal root finding methods based on rational interpolation.

Area of Science:

• Numerical Analysis
• Computational Mathematics
• Algorithm Development

Background:

• Iterative methods are crucial for approximating solutions to complex mathematical problems.
• Existing methods often require derivative information or complex initial steps, limiting their applicability.
• The pursuit of higher-order convergence and derivative-free approaches remains a key research area.

Purpose of the Study:

• To introduce a novel general class of derivative-free iterative methods.
• To achieve an optimal order of convergence of 2(n-1) for these new methods.
• To develop methods that bypass the need for Newton's iterate in the initial step.

Main Methods:

• Construction of a new class of iterative methods using rational interpolants.
• Analysis of the order of convergence for the proposed methods.
• Derivation of special cases from the general class.
• Implementation of numerical computations to validate performance.

Main Results:

• A new general class of derivative-free n-point iterative methods was successfully constructed.
• The methods achieve an optimal order of convergence of 2(n-1).
• The proposed schemes do not require Newton's iterate in the first step.
• Numerical results demonstrate the efficiency of the new methods.

Conclusions:

• The developed derivative-free iterative methods are efficient and effective.
• These methods represent a significant improvement and offer better alternatives to existing techniques.
• The simplification in the initial steps enhances practical usability.