Numerical Analysis I, Fall 2017 (http://www.math.nthu.edu.tw/˜ wangwc/)

Homework Assignment for Week 05

1. Section 10.1: Problems 3, 4.

2. As in the scalar case, where knowing the approximate value of f′(x_{∗}) would help to design a fixed point iteration for solving the equation f(x) = 0 by finding a suitable equivalent form x = g(x) by introducing a free parameter α. Try apply this technique to the following system of equations

\begin{aligned}1x_{1} + 2x_{2} + 0.03 ∗ sin(x_{1} + x_{2}) & = 4 \\ 5x_{1} + 6x_{2} + 0.07 ∗ cos(x_{1} − x_{2}) & = 8 \end{aligned}

and find a convergent fixed point iteration. If the equivalent form is not obvious to you, to use the trick as in the scalar case, but take α to be a 2 × 2 matrix.

3. Section 10.2: Problems 7(a,b), 14.