Numerical Analysis I, Fall 2010 (http://www.math.nthu.edu.tw/˜ wangwc/)

Homework Assignment for Week 05

Assigned Oct 13, 2010.

1. Section 2.3: Problems 16, 19, 23(c).

Hint for problem 16: see problem 4 below.

2. Section 2.4: Problems 8, 9, 10, 12, 14.

3. Read Example 2 and Example 3 of section 2.4 on page 79-81.

4. Suppose that f(x) has a zero of multiplicity m at x^{∗}, compute g′(x^{∗}) where g(x) = x − f(x)/f′(x). What does the result tell you about the order of convergence for Newton’s method applied to computing a multiple root? What happens when m becomes larger?

5. Suppose that f(0) = f′(0) = 0, f′′(x) > 0 for all x ∈ ℝ. Show that x = 0 is the unique root of f(x) = 0 and Newton’s method converges globally to 0, regardless of x_{0}. What is the order of convergence?

6. Show that the iteration x_{n+1} = g(x_{n}) converges locally (that is, if x_{0} is close enough to a solution) with cubic order (α = 3 in Definition 2.6, page 75), provided g′(x^{∗}) = g′′(x^{∗}) = 0 at a root x^{∗}, x^{∗} = g(x^{∗}). An application to solving f(x) = 0 is given in problem 13, section 2.4.