Numerical analysis > 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

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