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

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