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

## Exercise 2

• Consider the function $\displaystyle f(x)=e^{x}-x^{2}$ in $\displaystyle [-2,0]$ .
1. Show the existence and uniqueness of the root $f(\alpha )=0$ .
2. Given the tolerance $\epsilon =10^{-8}$ , how many iterations are needed?
3. Consider the restriction of the interval to $\displaystyle [-2,-1]$ . 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 $\epsilon =10^{-15}$ e consider it as the exact solution. Then considering $\epsilon =10^{-8}$ , 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 $k\geq 0$ we have

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