Newtons Method Assignment Help
Steps in finding the roots of an equation.
- The first task is to verify if the equation is differentiable. An equation that is not differentiable will not yield the results. Therefore, you cannot go any further with the process of finding the solution if it’s not differentiable.
- Once you verified its differential, get its derivative.
- Guess the initial value to use.
- Use newton’s iteration formula to get the next value, which is a better approximation of the root.
- Then iterate the process from step 2 until you find the roots of the equation.
Major problems with newton’s method
1. Problems in finding the derivatives
Note that without finding the derivative, there might be no convergence. The effect of this is that we may not find the roots at all. Depending on the equation that we have, we might find it easy to arrive at the derivative or very complicated. At times, equations do not have a derivative at all. In real life, the analytical expression of the derivative might be extensive.
2. Failure to converge
Newton’s method has its own challenges too. Most of the time, we believe the root-finding method will converge. But it’s not so in most cases. Here are some of the instances when the method might not converge.
- The initial starting point is crucial. If you have a bad starting point, there is a chance that the method might not converge. It might be that the value with which you start does not lie in the interval where the method converges. In such a case, the bisection method could be the best method. Another issue arises when the starting point is a stationary point. Sometimes the starting point enters an infinite circle, which prevents the method from converging.
- Derivative issues. Of course, from calculus, we know that a function that is non-differential will not converge. But the effect of this is largely felt if the function used is not differentiable.