# Exercises

On this page you will find exercise sets and solution proposals.

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.

Sessions in week | Exercise set | Solution suggestion | MATLAB code |
---|---|---|---|

36 | Exercise 1 | Solution 1 | poisson1.m poisson2.m |

40 | Exercise 2 | Solution 2 | |

41 | Exercise 3 | Solution 3 | |

42 | Exercise 4, version 2^{1)} | Solution 4 | |

45 | Exercise 5 | Solution 5 | |

Exercise 6 | Solution 6 |

## Additional exercises

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.