# Numerical Analysis/stability of RK methods/quizzes

1 the A-stability is characterized by:

 all the points in the left half of the complex plane . it a special case of the absolute stability. inludes all $\{z\in \mathbb {C} |\mathrm {Re} (z)<0\}$ , all of the above

2 Euler method is:

 A-stable not A-stable

3 the absolute stability region for RK4 is greater than the A-stability region for the same method:

 false True True for certain values of $b_{j}$ .

4 A numerical method is stable if :

 small change in the initial condition will produce small change in subsequent steps. small change in the initial condition will produce huge change in subsequent steps. big change in the initial conditions produce oscillatory solution at the end.

5 $\lambda =1$ is always a root of the characteristic polynomial of the multistep method:

 True False usually not the case

6 the root of the C.P. can be real or complex,and the method still be stable:

 True False

7 A multi- step method is strongly stable if :

 $\lambda =1$ is the only root of magnitude 1 . all other roots has magnitude <1 the first and the second sentence just one root inside the unit circle if it has more than one .

8 if more than one root has magnitude equal to 1, and the others are less than one, the method is

 strongly stable. A-stable weakly stable. Unstable.

9 The numerical method is unstable if

 $\lambda >1$ for at least one root $Re(\lambda )>1$ for at least one root $Re(\lambda )\leqslant 1$ for at least one root the firs and the second sentences.

10 the absolute stabity region for the explicit Euler's method is

 unit cicle in the complex plane, its center shifted to the left, by one unit. unit cicle in the complex plane, its center shifted to the rigt, by one unit. the whole left side of the complex plane. the method is unstable.

11 None of the RK methods is A-Stable:

 True False

12 All explicit methods are :

 A-Stable Not A-Stable. Unstable. None of the above

13 implicit multistep methods are A-stable if the have order at most

 1 2 3 they are always A-stable.