Exercises on the bisection method

 

Exercises on the bisection method


Numerical analysis > Exercises on the bisection method

Exercise 1 edit

  • Write a Octave/MATLAB function for the bisection method. The function takes as arguments the function  , the extrema of the interval   and  , the tolerance   and the maximum number of iterations.
  • Consider the function   in  .
    1. How many roots are there in this interval?
    2. Theoretically, how many iterations are needed to find a solution?
    3. With  , how many iterations are needed? Does the numerical result satisfy this condition?
    4. With  , how many iterations are needed? Does the numerical result satisfy this condition?

Exercise 2 edit

  • Consider the function   in  .
    1. Show the existence and uniqueness of the root  .
    2. Given the tolerance  , how many iterations are needed?
    3. Consider the restriction of the interval to  . 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   e consider it as the exact solution. Then considering  , draw a logarithmic plot to represent the average error and the actual error. Comment.

Exercise 3 edit

Show that the sequence defined by the bisection method with   we have

 .