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

Quiz 05

Dec 27, 2011.

1. Derive the normal equation for the following overdetermined system and find the solution. Here n is small so you don’t need to worry about the condition number. If you can’t derive but memorized it, then solve it directly with partial credit.

\begin{matrix}a_{1} + 2a_{2} = 0 \\ 2a_{1} − a_{2} = 1 \\ 3a_{1} − a_{2} = 5 \end{matrix}

2. The sequence a_{n} = 1 + \frac{1}{n^{2}} converges to its limit of order α. Find α. Give an example of a quadratically convergent sequence.

3. Give a convergent fixed point iteration (and not other methods) for solving f(x) = x − 2 cos x = 0. Explain why your formulae converges. You can use the fact that the solution x^{∗} ≈ 1.03, f′(x^{∗}) ≈ 1 + 2 · sin(1.03) ≈ 2.71. Write the iteration formula and need not implement.

4. Write a pseudo-code for solving f(x) = x − 2 cos x = 0 with Newton’s method till |f(x_{n})| < 10^{−10}. Do the same for secant method. Need not implement them.

5. Implement the secant method from last problem and report your x_{k}, k = 0, ⋯, n.

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

Quiz 05

Dec 27, 2011.

1. Derive the normal equation for the following overdetermined system and find the solution. Here n is small so you don’t need to worry about the condition number. If you can’t derive but memorized it, then solve it directly with partial credit.

\begin{matrix}a_{1} + 2a_{2} = 0 \\ 2a_{1} − a_{2} = 1 \\ 3a_{1} − a_{2} = 5 \end{matrix}

2. The sequence a_{n} = 1 + \frac{1}{n^{2}} converges to its limit of order α. Find α. Give an example of a quadratically convergent sequence.

3. Give a convergent fixed point iteration (and not other methods) for solving f(x) = x − 2 cos x = 0. Explain why your formulae converges. You can use the fact that the solution x^{∗} ≈ 1.03, f′(x^{∗}) ≈ 1 + 2 · sin(1.03) ≈ 2.71. Write the iteration formula and need not implement.

4. Write a pseudo-code for solving f(x) = x − 2 cos x = 0 with Newton’s method till |f(x_{n})| < 10^{−10}. Do the same for secant method. Need not implement them.

5. Implement the secant method from last problem and report your x_{k}, k = 0, ⋯, n.