Numerical Analysis I, Fall 2010 (http://www.math.nthu.edu.tw/˜ wangwc/)

Homework Assignment for Week 04

Assigned Oct 05, 2010.

1. Section 2.2: Problems 2(b), 7, 12, 16, 24. Hand in part (a) of problem 12 by Oct 15.

2. The iteration x_{n+1} = g(x_{n}) almost for sure does not converge when |g′| > 1 near the solution x_{∗}. An easy remedy is to give a rough estimate on g′(x_{∗}), then perform the iteration

x_{n+1} = αx_{n} + (1 − α)g(x_{n}).

Show that this iteration, if convergent, gives a root to x = g(x). The advantage of the new iteration is that, you can choose an α according to the approximate value of g′(x_{∗}), so that the new iteration converges. Apply this trick to the equation x = g(x) = 1 − 2x + 0.2 sin x.

3. Section 2.3: Problems 17. Hand in by Oct 20.