| course |
course_year |
question_number |
tags |
title |
year |
Numerical Analysis |
II |
149 |
II |
2006 |
Numerical Analysis |
|
1.II.38C |
2006 |
(a) Define the Jacobi method with relaxation for solving the linear system $A x=b$.
(b) Let $A$ be a symmetric positive definite matrix with diagonal part $D$ such that the matrix $2 D-A$ is also positive definite. Prove that the iteration always converges if the relaxation parameter $\omega$ is equal to 1 .
(c) Let $A$ be the tridiagonal matrix with diagonal elements $a_{i i}=1$ and off-diagonal elements $a_{i+1, i}=a_{i, i+1}=1 / 4$. Prove that convergence occurs if $\omega$ satisfies $0<\omega \leqslant 4 / 3$. Explain briefly why the choice $\omega=1$ is optimal.
[You may quote without proof any relevant result about the convergence of iterative methods and about the eigenvalues of matrices.]