How to write new algorithm of root finding by combining 2 or 3 standard algorithms(bisection, fixed, etc)

Question

I just learned about Bisection Method, Fixed-Point Iteration Method, Newton- Raphson Method, and Secant Method. My prof wants us to be able to write new Algorithm of root finding by coming 2 or 3 standard algorithms to solve a particular equation numerically.

I have no idea what to do.

Could you guys help me out this problem? and Could you give me example?

Thank you so much.

Answer 1

Consider the algorithms that you have carefully.

The fixed point iteration converges slowly and under only under very special circumstances. The bisection method converges slowly, but convergence is ensured as soon as you establish a bracket, i.e. an interval which contains the root. The secant method converges rapidly, provided your initial guess is close to a root. Newton's method converges even faster, but requires you to compute the derivative and you still have to start close to a root. They all have strengths and weaknesses. You have to make a combination which has all their strengths and none of their weaknesses.

Imagine an algorithm which starts with a bracket, makes it suitably small and then selects a reasonably good approximation which is then used to initialize a rapidly converging algorithm.

When you refine you implementation make sure that you always maintain a bracket around your root.

Answer 2

Hope you are well, did you find a new algorithm which maybe derived from Newton Raphson or any exists methods ? Tell me how you begin ? because i have to make a new algorithm for finding root problem. if you did your algorithm, can you help me how to begin ? all regards