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

Homework Assignment for Week 10

The linear combination ∑_{i=−k}^{k} c_{i}f(x_{0} + ih) is a pth order approximation of f^{(q)}(x_{0}) if

∑_{i=−k}^{k} c_{i}f(x_{0} + ih) = f^{(q)}(x_{0}) + O(h^{p})

1. Section 4.1:

Apply the round-off error instability calculation (end of section 4.1) to second order approximation of f′′(x_{0}). Find the critical h^{∗} that minimizes e(h). Express both h^{∗} and e(h^{∗}) in terms of the machine ε as O(ε^{p}) and find p for both h^{∗} and e(h^{∗}).

2. Section 4.1:

Repeat the last problem for fourth order approximation of f′(x_{0}).

3. Section 4.1, 4.2:

Derive five-point formula for f′′′(x_{0}) and and f^{(4)}(x_{0}), respectively using f(x_{0}), f(x_{0} ± h) and f(x_{0} ± 2h). You can either use the method of undetermined coefficients (combined with Taylor expansion around x_{0}), or apply similar procedure of Richardson extrapolation on three-point formula for f′(x_{0}) and f′′(x_{0}), respectively .

4. Section 4.2: Problems 5, 10, 12.

5. For section 4.2, problem 12, also show K_{1} = 0 alternatively by assuming the expansion

e = (\frac{2 + h}{2 − h})^{\frac{1}{h}} + C_{1}h^{p_{1}} + ⋯

and find p_{1} numerically using N(h) with h = 0.02 and 0.01.

6. Section 4.3: Problems 14, 19, 20, 22, 23, 24.