# Exercises on the bisection method

Exercises on the bisection method

## Exercise 1

• Write a Octave/MATLAB function for the bisection method. The function takes as arguments the function ${\displaystyle f}$ , the extrema of the interval ${\displaystyle a}$  and ${\displaystyle b}$ , the tolerance ${\displaystyle \epsilon }$  and the maximum number of iterations.
• Consider the function ${\displaystyle \displaystyle f(x)=\cos x}$  in ${\displaystyle \displaystyle [0,3\pi ]}$ .
1. How many roots are there in this interval?
2. Theoretically, how many iterations are needed to find a solution?
3. With ${\displaystyle \epsilon =10^{-10}}$ , how many iterations are needed? Does the numerical result satisfy this condition?
4. With ${\displaystyle \epsilon =10^{-20}}$ , how many iterations are needed? Does the numerical result satisfy this condition?

## Exercise 2

• Consider the function ${\displaystyle \displaystyle f(x)=e^{x}-x^{2}}$  in ${\displaystyle \displaystyle [-2,0]}$ .
1. Show the existence and uniqueness of the root ${\displaystyle f(\alpha )=0}$ .
2. Given the tolerance ${\displaystyle \epsilon =10^{-8}}$ , how many iterations are needed?
3. Consider the restriction of the interval to ${\displaystyle \displaystyle [-2,-1]}$ . In this case how many iterations are needed?
4. With the aid pf the Octave/MATLAB function of exercise 1, compute the root of the function.
5. Compute the solution with precision ${\displaystyle \epsilon =10^{-15}}$  e consider it as the exact solution. Then considering ${\displaystyle \epsilon =10^{-8}}$ , draw a logarithmic plot to represent the average error and the actual error. Comment.

## Exercise 3

Show that the sequence defined by the bisection method with ${\displaystyle k\geq 0}$  we have

${\displaystyle |\alpha -x_{k}|\leq {\frac {b-a}{2^{k+1}}}}$ .