How Does the Initial Guess Affect Newton's Method? A Mathematical Investigation

Abstract

Newton’s Method is a numerical technique used to find approximate solutions of equations that may be difficult to solve algebraically. The method uses the derivative of a function to repeatedly improve an initial estimate of a root.

This investigation examines how the choice of the initial guess affects the behaviour of Newton’s Method. The function f(x)= x³ - x was selected because it has three real roots, -1, 0 and 1. Ten different initial guesses between -2 and 2 were tested. For each starting value, successive approximations were calculated, and the root reached and number of iterations were recorded.

The results show that changing the initial guess can affect both the root reached and the number of iterations required for convergence. For example, the initial guesses -0.5 and 0.5 converged to different roots despite being equally close to zero. The number of iterations varied from 0 to 6 among the tested starting values. These results demonstrate that the initial guess can be an important factor in the behaviour of Newton’s Method.