Numerical Analysis I, Fall 2011 (http://www.math.nthu.edu.tw/˜ wangwc/)

Homework Assignment for Week 15

Assigned Dec 23, 2011.

1. Section 2.3: Problems 17(b,c). This problem will be included in quiz 5.

2. Section 2.3: Problems 16, 19.

3. 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.

4. Section 2.4: Problems 8, 9, 10, 14.

5. Section 2.5: Problems 14, 15.

6. Find out p̂_{n} explicitly for Aitken’s Δ^{2} method for the following sequence: a_{n} = 1/n, b_{n} = 1/n^{2}, c_{n} = α^{n}, 0 < α < 1 and d_{n} = 2^{−2^{n}}. Note the first two are linearly convergent sequence but does not satisfy the assumption of Theorem 2.13. The last one is a quadratically convergent sequence. Are the convergence accelerated in these cases?