Exercises on the bisection method/Solution

Solution of the exercises on the bisection method

Exercise 1

 * The following is a possible implementation of the bisection method with Octave/MATLAB:

For point 4 we have
 * The solution of the points 1, 2 e 3 can be found in the example of the bisection method.
 * $$k\geq\log_2 3\cdot 10^20\pi\approx 69.8$$,

so we would need at least 70 iterations. The chance of convergence with such a small precision depends on the calculatord: in particular, with Octave, the machine precision is roughly $$2\cdot 10^{-16}$$. For this reason it does not make sense to choose a smaller precision. The number of iterations, if we don't specify a maximum number, would be infinite.

Exercise 2

 * 1) To verify the existence of a root $$\alpha$$ we need to show that the hypothesis roots theorem are satisfied. The first hypothesis requires $$f$$ to be continuous. Obviosly this is a continuous function since it is sum of two continuous functions. The second hypotheses requires the function to have oppiste signs at the interval extrema, and in fact we find
 * $$f(-2)\approx -3.86, \quad f(0)=1 \quad \implies f(a)f(b)<0$$.
 * NumericalAnalysis BisectionExercise 2.png
 * To show the uniqueness of the root we need to prove that the function $$f$$ is monotone and in fact
 * $$f'(x)=e^x-2x > 0 \quad \forall x \in [-2, 0]$$.
 * 1) The number of iterations need is given by
 * $$k\geq\log_2\frac{b-a}{\epsilon}=\log_2 2 \cdot 10^8 \approx 25.8$$,
 * and so we have $$k\geq 26$$.
 * 1) The interval $$[-2,-1]$$ does not contain aany root as the second hypotesis of the roots theorem fails, in fact
 * $$f(-2)\approx -3.86, \quad f(-1)=-0.63 \quad \implies f(a)f(b)>0$$.
 * 1) $$\alpha \approx -0.7035$$
 * 2) In the plot we show in red the average errorand in blu the actual error. From the graph, it is clear that the actual error is not a monotone function. Moreover, note that the global behavior of both curves is the same, clarifying the term average error for $$\displaystyle e_k$$.

Exercise 3
For the solution look at the convergence analysis in the bisection method page.