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

Quiz 01

Oct 01, 2010.

1. 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 = 33, a_{j} ∈ {0, 1}, −511 ≤ e ≤ 512?

2. Find first 15 digits of (100002)^{\frac{1}{3}} −(100001)^{\frac{1}{3}} . You can use any program on the computer to find the answer and then write down your answer on the answer sheet. Note: Direct evaluation only gives 10 correct digits. You will only get minimal credit for that.

3. Give the rate of convergence of the following limits

(a) : lim_{h→0} \frac{sin h}{h} (b) : lim_{n→∞} ( ln(n + 1) − ln n)

4. 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. Find a numerical solution of a root of x^{3} + 3x − 3 = 0 with absolute error less than 2^{−20} using bisection method with a_{0} = 0.5, b_{0} = 1. Note: You must explain how many iterations are needed to guarantee the specified accuracy.

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

Quiz 01

Oct 01, 2010.

1. 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 = 33, a_{j} ∈ {0, 1}, −511 ≤ e ≤ 512?

2. Find first 15 digits of (100002)^{\frac{1}{3}} −(100001)^{\frac{1}{3}} . You can use any program on the computer to find the answer and then write down your answer on the answer sheet. Note: Direct evaluation only gives 10 correct digits. You will only get minimal credit for that.

3. Give the rate of convergence of the following limits

(a) : lim_{h→0} \frac{sin h}{h} (b) : lim_{n→∞} ( ln(n + 1) − ln n)

4. 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. Find a numerical solution of a root of x^{3} + 3x − 3 = 0 with absolute error less than 2^{−20} using bisection method with a_{0} = 0.5, b_{0} = 1. Note: You must explain how many iterations are needed to guarantee the specified accuracy.