| course |
course_year |
question_number |
tags |
title |
year |
Numerical Analysis |
II |
150 |
II |
2007 |
Numerical Analysis |
|
2.II.38C |
2007 |
(a) State the Householder-John theorem and explain how it can be used to design iterative methods for solving a system of linear equations $A x=b$.
(b) Let $A=L+D+U$ where $D$ is the diagonal part of $A$, and $L$ and $U$ are, respectively, the strictly lower and strictly upper triangular parts of $A$. Given a vector $b$, consider the following iterative scheme:
$$(D+\omega L) x^{(k+1)}=(1-\omega) D x^{(k)}-\omega U x^{(k)}+\omega b$$
Prove that if $A$ is a symmetric positive definite matrix, and $\omega \in(0,2)$, then the above iteration converges to the solution of the system $A x=b$.