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

Homework Assignment for Week 05

Assigned Oct 13, 2010.

1. Show that the method of secant method applied to solving f(x) = e^{x} − 1 = 0 with 0 < x_{0} < x_{1} always converges to the solution x^{∗} = 0.

Hint: This f is convex.

2. Show that the method of false position applied to solving f(x) = e^{x} − 1 = 0 with x_{0} > 0, x_{1} < 0 always converges to the solution x^{∗} = 0.

3. (Harder) What is the order of convergence for previous problem with x_{0} = 1.25643, x_{1} = −0.01?

Hint: Find a few x_{n} and see what happens. x_{0} is a root of e^{x} − 1 = 2x, chosen so that you can see some clean numbers. It is not really important.

4. Section 2.5: Problems 14, 15, 16.

Hint for problem 16: Show that

\frac{p̂_{n} − p}{p_{n} − p} = \frac{λ(δ_{n} + δ_{n+1}) − 2δ_{n} + δ_{n}δ_{n+1} − 2δ_{n}(λ − 1) − δ_{n}^{2}}{(λ − 1)^{2} + λ(δ_{n} + δ_{n+1}) − 2δ_{n} + δ_{n}δ_{n+1}}

5. Suppose Müller’s method is applied to locate a root of f(x) = x^{3} − 2 = 0, with x_{0} = 0, x_{1} = 1 and x_{2} = 2. What is x_{3}? What happens if you apply Müller’s method to solve a quadratic equation ax^{2} + bx + c = 0?

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} = 2^{−n} 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?