Numerical Analysis I, Fall 2014 (http://www.math.nthu.edu.tw/˜ wangwc/)

Homework Assignment for Week 04

1. Section 2.4: Problem 13. Also derive this formula by way of quadratic approximation of x = f^{−1}(y) at (y, x) = (f(x_{n}), x_{n}).

2. Use the code for Algorithm 2.2 to check if the standard fixed point iteration x_{n+1} = g(x_{n}) = 2 cos(x_{n}) converge to root of f(x) = x − 2 cos(x) = 0. If not, can you explain why? Can you find a constant β such that the modified fixed point iteration x_{n+1} = βx_{n} + (1 − β)g(x_{n}) converges (at least locally)?

3. Section 2.5: Problems 12(a), 14, 15, 16.

4. In Example 1 of section 2.5, the condition in Theorem 2.14 is not satisfied. Nevertheless, one could still get faster convergence of p̂_{n} than p_{n}, but of the same order. Analyze

Aitken’s Δ^{2} method to evaluate lim_{n→∞} \frac{p̂_{n} − p}{p_{n+2} − p}. Then verify your result numerically.