Generalized solutions of Polynomial and Transcendental Equations by the strong method of (Generalized Iterative Method approximation of Roots (GRIM) )

TL;DR

This paper introduces a new iterative fixed point method, called GRIM, for solving polynomial and transcendental equations in Banach spaces, with proven convergence and enhanced speed via Newton's method, demonstrated through various examples.

Contribution

The paper develops a generalized iterative method (GRIM) with convergence theory for nonlinear equations, extending previous approaches by categorizing roots and integrating Newton's method for faster solutions.

Findings

Convergence of the GRIM method is established in Banach spaces.

The method effectively solves polynomial and transcendental equations.

Numerical examples demonstrate the method's efficiency and accuracy.

Abstract

In this paper, we explain a new Iterative Method-Fixed Point and develop its convergence theory for finding approximate solutions of nonlinear equations in the setting of Banach spaces. First, we discuss the convergence analysis of our method by separating the equation into functions from which it is derived and the remaining part of the equation, according to the Generalized Theorem[1] for solving polynomial and transcendental equations. Solving an equation, either polynomial or transcendental, is solved only by categorizing the roots and not by some procedure of searching in individual intervals and this way was followed in the past by all existing methods. But the methods developed in the past are limited to isolated intervals without general acceptance. Finally, the method is strengthened by Newton's method to speed up the finding of the roots. At the end of the analysis, we give several examples of the application of the method.