Numerical Analysis I, Fall 2020 (http://www.math.nthu.edu.tw/˜ wangwc/)

Homework Assignment for Week 05

Goal: Study both theoretical and practical aspects of Newton’s method and its extensions.

1. Section 2.3: Problems 13(a), 17(c), 18

Hint for problem 18: The expression can be simplified. Apply trigonometric identity to \frac{1}{2} − \frac{1}{2} cos 2x and combine with the remaining terms. Can you accelerate the convergence?

2. Section 2.4: Problems 7(a), 8, 9, 10, 12, 13, 14.

Remark: Problem 13 can be derived from quadratic approximation of x = f^{−1}(y) at (y, x) = (f(x_{n}), x_{n}) (as shown in the slides for the linear approximation, Newton’s method). Try to derive it this way.

3. Section 2.4: If f(x) = (x−x^{∗})^{m}q(x), m > 1 and q is smooth with q(x^{∗}) ≠ 0. Let μ(x) =

\frac{f(x)}{f′(x)}. Show that f′(x) = (x − x^{∗})^{m−1}q_{1}(x) with q_{1}(x^{∗}) ≠ 0 and μ(x) = (x − x^{∗})q_{2}(x)

with q_{2}(x^{∗}) ≠ 0.