Exercises on the bisection method

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

Exercise 2

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

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