Numerical Analysis I, Fall 2017 (http://www.math.nthu.edu.tw/˜ wangwc/)

Midterm 01

Oct 24, 2017.

1. (10 pts) The half precision format uses 16 bits to store a binary floating point number of the form ±1.a_{1}a_{2} ⋯ a_{t} × 2^{e} where a_{j} ∈ \{0, 1\}, −14 ≤ e ≤ 15. Find t and \underline{derive} an upper bound for relative error caused by \underline{rounding}. Express your final answer as a real number, but need not convert it to decimal expression.

2. (10 pts) Given p_{0}, p_{1} and p_{2}, the general solution to the recursion formula p_{n} = \frac{10}{3}p_{n−1} − 3p_{n−2} + \frac{2}{3}p_{n−3} is p_{n} = c_{1}1^{n} + c_{2}2^{n} + c_{3}(\frac{1}{3})^{3} (need not show this). Find all (c_{1}, c_{2}, c_{3}) ≠ (0, 0, 0) such that the above iteration is unstable in relative error. \underline{Explain}.

3. (10 pts) The first few iteration (p_{i}, f(p_{i})), i = 0, 1, 2, 3, 4 of method of false position for some equation f(x) = 0 is given by

(0, 2), (3, 1), (2, 2), (1, 1), (\frac{2}{3}, \frac{2}{9})

Find p_{5} (4 digits will do). \underline{Explain}.

4. (15 pts) Use any method to find a solution of \sqrt{1 + 0.9x} − \sqrt{1 − 0.8x} = 1.0 × 10^{−10} to 15 correct digits. You need to prevent loss of accuracy. Standard methods only gives you about 5 correct digits (and 1/3 partial credits).

5. (10+5 pts) It is known that the unique solution to f(x) = x + 3 sin(x) − 0.01 = 0 is located near x = 0.

(a) Find a fixed point iteration that will converge for any x_{0} ∈ [−\frac{1}{2}, \frac{1}{2}]. Show that your method satisfies the assumptions of a relevant Theorem, but need not prove the Theorem again. You can use the numerical values of sin(\frac{1}{2}), cos(\frac{1}{2}), exp(\frac{1}{2}), etc. in your proof.

(b) Find an N (need not be optimal) such that |x_{n} − x^{∗}| < 10^{−30} for all n ≥ N with x_{0} = 0 (assuming a higher precision floating point arithmetic is used).

6. (15 pts) Give a cubically convergent method to solve for e^{x} − 1 = 0. Give the formula and prove that it is cubically convergent (locally). If you cannot do it, do the same for a locally quadratically convergent method for partial credit.

7. (10 pts) Derive Aitken’s Δ^{2} acceleration method.

Hint: the starting point is to assume p_{n} converges to p linearly.

8. (15 pts) Use any method to solve the nonlinear system of equations

sin(x) + \frac{2y}{1 + x} = 0.01, 5x + sin(\frac{6y}{1 + y^{2}} ) = 0.02.

Write your answer in the format of ’format long e’. Hint: the solution is near (0, 0) where sin x ≈ x, \frac{y}{1+x} ≈ y, etc. to leading orders.