INTRODUCTION;
Nonlinear equations are common in mathematics, engineering, and science. Unlike linear equations, they often require numerical methods for solutions. In this post, we will explore three iterative techniques to solve nonlinear equations: The Bisection Method, Newton’s Raphson Method and the Secant Method.
Problem Statement
We are given a nonlinear (Transcendental) equation to solve using these three methods. The full solution, including step-by-step calculations, is provided in the attached image.
1. Bisection Method; The Bisection Method is a simple and reliable approach that repeatedly divides an interval containing the root until t3he desired accuracy is achieved.
Steps:
i) Select two initial points (interval), (a) and (b), such that f(a) and f(b) have opposite signs.
ii). Compute the midpoint (c), such that c = (a+b)/2 and evaluate f(c).
iii) Replace either (a) or (b) with (c) depending on the
sign of f(c), ensuring that the new interval still contains the root.
iv) Repeat the process until the error is within an
acceptable range.
Given; x - e^(-x)
Following the steps below, then we have
