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

Midterm 01

Oct 25, 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. Consider the following recursive equation p_{0} = 1, p_{1} = 1/3, p_{2} =, p_{n} = p_{n−1} − p_{n−2} + p_{n−3} What is the exact solution? Is it stable? Explain. If you can’t do it, do XXX for half credit.

4. Derive f” 3 point formula, error identity. then 4th order. 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.

5. Derive an approximation formula and error identity for the approximation of f′(x) using f(x − h), f(x), and f(x + 2h). If you can’t derive an error identity, derive an error bound (to get partial credit).

6. Write down Simpson’s formula for ∫_{−h}^{h} f(x)dx. Need not derive or prove anything. If you don’t remember and decide to derive, put on details.

7. Derive Gaussian quadrature for ∫_{−1}^{1} f(x)dx with n = 3. You can either take the ”maximal degree of precision’ approach or the Legendre polynomial approah. If you can’t do ‘n = 3’, do ‘n = 2’ to get half credit.

8. Implement N_{1}(h) = composite midpoint rule for ∫_{0}^{1} ^{3} \sqrt{x}dx and find the rate of convergence numerically. Based on the rate you find, apply Richardson extrapolation to get N_{2}(h).