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

Homework Assignment for Week 02

Assigned Sep 25, 2009. Due Oct 02, 2009.

1. Section 1.2: Problems 3, 8, 16, 19, 20, 21, 22.

Extra part (c) of Problem 20: compare Problem 20 (a) with evaluating log(2) using (1.29) directly. In order to control the error within 5×10^{−5}, how many terms of Taylor polynomial does it take in both methods, respectively?

Hint for Problem 21 (b): π = 8 ∗ (π/8).

2. The proof of (1.28) for α ∈ ℝ is not straight forward. Try analyzing R_{n}(x) directly and you’ll see why. Nevertheless, it can be proved through the following steps:

(a) Verify that

(j + 1) ( \begin{matrix}α \\ j + 1 \end{matrix} ) + j ( \begin{matrix}α \\ j \end{matrix} ) = α ( \begin{matrix}α \\ j \end{matrix} )

(b) Show that

f(x) = ∑_{j=0}^{∞} ( \begin{matrix}α \\ j \end{matrix} ) x^{j} = 1 + αx + \frac{α(α − 1)}{2!}x^{2} + \frac{α(α − 1)(α − 2)}{3!}x^{3} + ⋯

is well defined (that is, the series converges) on |x| < 1, f(0) = 1 and

(1 + x)f′(x) = αf(x), |x| < 1.

(c) Show that f(x) = (1 + x)^{α} on |x| < 1.

3. Section 1.3: Problems 3, 7, 8, 9.

4. (Programming)

Section 1.3: Problem 5. Matlab has the built-in function erf. Denote by erfh your polynomial approximation. Also plot erf(x)−erfh(x) over [−b, b] on a separate graph. It may be helpful to study the program plot sint and modify it accordingly.

If you don’t know yet, \sqrt{π} is “sqrt(pi)” in Matlab.