Abstract
It is crucial to achieve high-order convergence for nonlinear equations in numerical analysis and scientific computing. Newton-Raphson and other conventional techniques frequently converge slowly. High-order algorithms offer quicker convergence and less expensive processing.This abstract outlines important ideas and innovations, such as improved Newtonian methods for better convergence. To hasten convergence, these techniques make use of enhanced Jacobian matrix approximations and higher-order derivatives. Additionally, it covers useful topics like dealing with stability problems and computing higher-order derivatives, which could lead to efficiency and accuracy gains in scientific and engineering applications. Keywords: Non- linear Equations, High Order Convergence, Numerical Methods, Jacobian Matrix, High Order Derivatives
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