Numerical Analysis I, Fall 2009 (http://www.math.nthu.edu.tw/˜ wangwc/)

Homework Assignment for Week 07

Assigned Oct 28, 2009. Due Nov 06, 2009.

1. Section 4.2: Problems 1, 4, 8, 14, 16, 18.

Hint for problem 8: read problem 9.

For problem 16, find the largest error in the intervals by sampling with four times as many points. For example, if you interpolated with n = 10, then sample on x_{i}, i = 0, 1, ⋯, 40 to find the largest interpolation error and report your result. Summarize your result in table(s). Need not hand in the code.

For problem 18: try it first, then change the left hand side from f(x) − P_{1}(x) to f(x) − P_{2}(x) and change the right hand side accordingly. Hand in the P_{2} version.

2. Section 4.3: Problems 5, 6, 9, 15.

3. Section 4.5: Problems 1, 3, 6, 7, 8, 10.

4. (Programming)

Section 4.3: Problem 20.

5. Challenge of the week with extra credit. Extension may be possible upon request, pending on your progress.

Consider solving the system of equations

\begin{matrix}e^{x} − 1 + 0.1y & − α = 0 \\ sin(0.1x + m sin y) & − β = 0 \end{matrix}  (1)

(a) It is clear that (x, y) = (0, 0) is a solution for α = β = 0. For general (α, β) with α^{2} + β^{2} small enough, what condition is needed for the existence and uniqueness of solution to equation (1) in the neighborhood of (0, 0)? Does equation (1) satisfy this condition when m = 1 and m = −1?

(b) What would be the secant method for (1) with m = 1? Hint: In the 1D case, we solve for f(x) = 0 with f : ℝ ↦ ℝ. The linear approximation of f is determined by (x_{n−1}, f(x_{n−1})) and (x_{n}, f(x_{n})). In case of (1), each equation in (1) is to be approximated by a linear approximation. How many points in ℝ^{3} uniquely determines a plane?

(c) (Programming) Solve equation (1) with m = 1, (α, β) = (0.1, 0.1) using Newton’s iteration.

(d) (Programming)

Solve equation (1) with m = 1, (α, β) = (0.1, 0.1) using fixed point iteration.

(e) (Programming) Solve equation (1) with m = −1, (α, β) = (0.1, 0.1) using fixed point iteration.