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

Quiz 01

Oct 04, 2011.

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 = 11, a_{j} ∈ {0, 1}, −6 ≤ e ≤ 7? Write down the floating number representation (a finite sequence of 0, 1) of −0.625.

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

3. Give the rate of convergence for ∑_{n=1}^{∞} \frac{1}{n^{2}} = \frac{π^{2}}{6} either analytically or numerically.

4. Consider the following recursive equation p_{0} = 1, p_{1} = 1/3, p_{n} = \frac{10}{3}p_{n−1} − p_{n−2}. What is the exact solution? Is it stable? Explain.

5. At least how many points among f(x), f(x ± h), f(x ± 2h), f(x ± 3h), ⋯ are needed to approximate f′(x) to O(h^{4})? Explain.

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

Quiz 01

Oct 04, 2011.

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 = 11, a_{j} ∈ {0, 1}, −6 ≤ e ≤ 7? Write down the floating number representation (a finite sequence of 0, 1) of −0.625.

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

3. Give the rate of convergence for ∑_{n=1}^{∞} \frac{1}{n^{2}} = \frac{π^{2}}{6} either analytically or numerically.

4. Consider the following recursive equation p_{0} = 1, p_{1} = 1/3, p_{n} = \frac{10}{3}p_{n−1} − p_{n−2}. What is the exact solution? Is it stable? Explain.

5. At least how many points among f(x), f(x ± h), f(x ± 2h), f(x ± 3h), ⋯ are needed to approximate f′(x) to O(h^{4})? Explain.