Homework

1. Show that all the roots of the first characteristic polynomial of a LMM(k) method with $\alpha_j = - \alpha_{k-j}$, for all $j$, are simple and lie on the unit circle.

2. Show that $y_{n+2}+4y_{n+1}-5y_n = h(4f_{n+1}+3f_n)$ is the explicit 2-step LMM method with highest order. Write a matlab function that implements this method to integrate the ODE $y'=y$, $y(0)=1$ on $x\in [0,1]$. Plot the resulting approximate solution for different choices of the step length h={1/10, 1/25, 1/50, …} and observe that the method is not stable.

3. Show that

• there is no LMM($k$) of order $2k+1$;
• there is a unique implicit LMM($k$) of order $2k$;
• there is a unique explicit LMM($k$) of order $2k-1$.