# Exercises

Note that the exercises are optional and should not be handed in. I do, however, recommend to try to solve them, because they will be part of the pensum.

These exercises won't be discussed in the exercise sessions, and we won't publish any solution proposals.

### Basic iterative methods

• Assume that the matrix $A$ is such that the Jacobi method converges for all initial values. Show that in this case also the weighted Jacobi method converges for all weights $0 < \omega \le 1$.
• Consider the matrix $A=\begin{pmatrix} 1 & a & a \\ a & 1 & a \\ a & a & 1 \end{pmatrix}.$ Show that the Gauss-Seidel method for the solution of the linear system $Ax=b$ converges as long as $-1/2 < a < 1$, while the Jacobi method does not converge for $1/2 < a < 1$.

### General projection methods

• Exercise 5.1, page 145 in YS.
• Exercise 5.2, page 146 in YS.

### Krylov space methods

• Assume that the matrix $A$ has singular values in the interval $(1,2)$. Estimate, how many steps of the algorithm in problem 1, exercise 4, are necessary in order to reduce the error by a factor of $10^{-6}$.
• Assume that we use the CG method in order to solve a problem $Ax=b$ with $A$ being an SPD matrix, and that all the eigenvalues of $A$ are contained in $(1,1.1)\cup (2,2.2)$. How many iterations are necessary in order to reduce the error by a factor of $10^{-3}$?
1)
In the original version, there has been an error in the formulation of the algorithm in problem 1.