Discuss the relationships between the Bisection Method fixed

Discuss the relationships between the Bisection Method, fixed-point methods, and Newton\'s Method How fas, do the methods converge to the solution to the problem, Which method is best anl der what circumstances? What are the differences and similarities between the methods?

Solution

The three methods are used to approximate a root of a real-valued function.

- Convergence of the methods:

In general the order of convergence of the Newton\'s method is quadratic, but this method doesn\'t guarantee the convergence to the solution.

For the bisection method the convergence is always guaranteed, but this one converges linearly, so it is slower than the Newton\'s method.

For the fixed-point method there are several theories to guaranteed the existence of the fixed- point; the speed of convergence can be increased using different methods.

- Which method is best and under what circumstances?

The Newton\'s method is the most efficient but since there is not a theorem of global convergence, it is necessary to start of and initial approximation sufficiently close to the root. In this case, the Newton\'s method is the best.

- What are the differences and similarities between the methods?

For applying any method the function must be continuous around the root. In the bisection and fixed poind methods we need an interval to start the iterations. On the contrary, the Newton\'s method starts in a point.