# Exercises on the bisection method

Exercises on the bisection method

## Exercise 1 edit

- Write a Octave/MATLAB function for the bisection method. The function takes as arguments the function , the extrema of the interval and , the tolerance and the maximum number of iterations.
- Consider the function in .
- How many roots are there in this interval?
- Theoretically, how many iterations are needed to find a solution?
- With , how many iterations are needed? Does the numerical result satisfy this condition?
- With , how many iterations are needed? Does the numerical result satisfy this condition?

## Exercise 2 edit

- Consider the function in .
- Show the existence and uniqueness of the root .
- Given the tolerance , how many iterations are needed?
- Consider the restriction of the interval to . In this case how many iterations are needed?
- With the aid pf the Octave/MATLAB function of exercise 1, compute the root of the function.
- Compute the solution with precision e consider it as the exact solution. Then considering , draw a logarithmic plot to represent the average error and the actual error. Comment.

## Exercise 3 edit

Show that the sequence defined by the bisection method with we have

- .