Lab 7

In this lab we will explore relaxation techniques to help iterative methods converge

Exercise 1: A non-convergent fixed-point method

Consider the fixed-point method given by $g(x) = -x^4+2$ .

1) Plot $g(x)$ to see that it has a unique fixed point for $x \geq 0$ .

2) Choose the initial guess $x_0 = .5$ for the fixed point and see that the iteration does not converge.

3) Consider the associated relaxation method

$$ g_w(x) = w g(x) + (1-w)x. $$

Find a value of $w$ so that this fixed point method converges to a fixed point of $g(x)$ .

Consider the fixed-point method given by $g(x) = \cos(x)$ . The corresponding fixed-point iteration is (globally) linearly convergent.

1) Find $w$ so that the relaxation method

% $$ g_w(x) = w g(x) + (1-w)x,$ $

converges faster to the fixed point of $g(x)$ .

2) With this choice of $w$ , is the method still globally convergent?

Let

A = [-2 1 0 0 0; 1 -2 1 0 0; 0 1 -2 1 0; 0 0 1 -2 1; 0 0 0 1 -2];
b = [.5, .5, .5, .5, .5]';

Define $T = I -A$ and consider the fixed point method given by $g(x) = Tx + b$ .

1) Show that this method does not converge.

2) Consider the associated relaxation method $g_w(x) = w g(x) + (1-w)x.$ Can you choose $w$ so that the method converges?