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

Midterm 01

Oct 29, 2010.

1. (8 pts) How many ”bits” does it take to store floating point numbers of the form ±1.a_{1}a_{2} ⋯ a_{s} × 2^{e} with s = 23, a_{j} ∈ {0, 1}, −127 ≤ e ≤ 128? What is the largest number of this form?

2. (10 pts) Evaluate

p(x) = 1 + x^{2} + \frac{x^{4}}{2!} + \frac{x^{6}}{3!} + \frac{x^{8}}{4!} + \frac{x^{10}}{5!}

as efficient as possible. You may give your answer either in the form of a loop, or any other expression as long as it is clear enough. How many multiplications are needed?

3. (10 pts) Solve for x^{2} − 1900x + 1 = 0 to 15 correct digits. Every digit counts. Explain how you find your answer (direct evaluation using ’calculator’ will receive no credits).

4. (10 pts) Is the following algorithm stable or not? p_{0} = 1, p_{1} = 1/3, p_{n} = \frac{10}{3}p_{n−1} − p_{n−2}. Explain (with mathematical reasoning, not numerical observation). The true solution is p_{n}^{e} = (\frac{1}{3})^{n}.

5. (8+4 pts)

(a) Give a convergent fixed point iteration for solving f(x) = x+2 sin(x)−0.01. Just give the formulae, no numerical values needed.

(b) Give an upper bound for the number of steps it takes to reach |x_{n} − x^{∗}| < 10^{−6} with x_{0} = 1.

6. (6+8+6 pts) Suppose that g : ℝ ↦ ℝ, g ∈ C^{2}(ℝ) and |g′| < 1/2.

(a) Prove that the equation x = g(x) has a unique solution.

(b) Prove (in detail) that the iteration x_{n+1} = g(x_{n}) always converges to the solution.

(c) Give formulae of Steffensen’s method for this problem.

7. (4+8 pts)

(a) Give the formula of Newton’s method for solving x^{2} − 2x + 1 = 0.

(b) Find the order of convergence and prove your answer.

8. (4+8 pts)

(a) Apply Aitken’s Δ^{2} method to the sequence p_{n} = cos \frac{1}{n}.

(b) Find lim_{n→∞} \frac{p̂_{n} − p}{p_{n} − p}.

9. (6 pts) Suppose Müller’s method is applied to locate a root of f(x) = x^{3} − 2 = 0, with x_{0} = 1, x_{1} = 2 and x_{2} = 3. What is x_{3}?