| course |
course_year |
question_number |
tags |
title |
year |
Numerical Analysis |
II |
103 |
II |
2010 |
Numerical Analysis |
|
Paper 1, Section II, A |
2010 |
(a) State the Householder-John theorem and explain its relation to the convergence analysis of splitting methods for solving a system of linear equations $A x=b$ with a positive definite matrix $A$.
(b) Describe the Jacobi method for solving a system $A x=b$, and deduce from the above theorem that if $A$ is a symmetric positive definite tridiagonal matrix,
$$A=\left[\begin{array}{ccccc}
a_{1} & c_{1} & & & \\
c_{1} & a_{2} & c_{2} & & \mathbf{0} \\
& \ddots & \ddots & \ddots & \\
\mathbf{0} & & c_{n-2} & a_{n-1} & c_{n-1} \\
& & & c_{n-1} & a_{n}
\end{array}\right]$$
then the Jacobi method converges.
[Hint: At the last step, you may find it useful to consider two vectors $x=\left(x_{1}, x_{2}, \ldots, x_{n}\right)$ and $y=\left((-1) x_{1},(-1)^{2} x_{2}, \ldots,(-1)^{n} x_{n}\right)$.]