2013-107.md

course	course_year	question_number	tags	title	year
Numerical Analysis	II	107	II 2013 Numerical Analysis	Paper 1, Section II, 40C	2013

Let

$$A(\alpha)=\left(\begin{array}{ccc} 1 & \alpha & \alpha \\ \alpha & 1 & \alpha \\ \alpha & \alpha & 1 \end{array}\right), \quad \alpha \in \mathbb{R}$$

(i) For which values of $\alpha$ is $A(\alpha)$ positive definite?

(ii) Formulate the Gauss-Seidel method for the solution $\mathbf{x} \in \mathbb{R}^{3}$ of a system

$$A(\alpha) \mathbf{x}=\mathbf{b}$$

with $A(\alpha)$ as defined above and $\mathbf{b} \in \mathbb{R}^{3}$. Prove that the Gauss-Seidel method converges to the solution of the above system whenever $A$ is positive definite. [You may state and use the Householder-John theorem without proof.]

(iii) For which values of $\alpha$ does the Jacobi iteration applied to the solution of the above system converge?