A Novel Interpolation-Based Order Six Method and its Convergence Analysis

This paper introduces an interpolationbased order six method for solving nonlinear equations. Our method offers significant improvements in accuracy, stability, and efficiency, making it valuable for computational and applied mathematics. The main goal is to address various nonlinear problems in fields such as physics, engineering, and finance. The paper explains the theoretical foundation and key principles of the method, which uses interpolation points instead of derivatives to approximate solutions. This approach enhances convergence behavior and numerical precision. We also conduct a detailed local and semilocal convergence analysis to evaluate the method’s performance. This analysis provides insights into the convergence region, radii, and error boundaries. It also assesses the method’s effectiveness in scenarios where accurate initial guesses are hard to obtain. Extensive numerical experiments on diverse test problems demonstrate the method’s superior convergence rates and error estimates, confirming its effectiveness and reliability.