A simple equation such as x^2+5x+6=0 can be easily solved but an equation (e^x-1)/x+10*e^x=sinx may not be solved analytically.
We, therefore, need to use numerical techniques to find the solution. To solve a nonlinear equation we have many multiple methods such as the Bisection method, fixed-point iteration method, secant method, newtons method.
Each of the above mention methods has its rate of convergence and weaknesses. Such as bisection method is based on Intermediate value theorem and it tells that a root exists between two points but it does not tell about the number of roots that exist in that interval, also the order of convergence of bisection method is 1 thus it would take a long time to each to the solution when compared to a method of higher order of convergence such as newtons method.
In the case of a fixed-point iteration method you need to generate multiple equations and all of them might not converge and it also converges quite slowly. Secant method has a faster rate of convergence than linear methods and does not require to calculate the derivative of the function but chances are that it may not converge and if the derivative of the function at the root is zero then this approach is likely to have difficulty.
Newton’s method is quite effective as it approaches the root quadratically and it is quite simple to implement but this method fails if the derivative of the function at the root tends to zero and it is choosing a starting point that does not converge to the desired root.
By proposing changes in the newtons method, methods to apply on multiple root equations have also been used through the newtons method loses its quadratic order of convergence.
So, which method would you apply to the above-mentioned equation and does the equation pose any problem when using the newtons method?
For any Discussion Comment below or Go to the Next Chapter Initial Value Problems for ODEs: One-Step Methods