Numerical Analysis I, Fall 2014 (http://www.math.nthu.edu.tw/˜ wangwc/)

Quiz 03

Nov 14, 2014.

1. Give (need not derive) an approximation of f′′(x) from f(x − h), f(x) and f(x + h). Then derive an error identity of the form f′′(x) − f′′_{h}(x) = C_{1}f^{(n)}(ξ)h^{n+k}.

2. Derive a fourth order approximation of f′(x) from f(x), f(x ± h), f(x ± 2h), f(x ± 3h), · · ·.

3. At least how many points among f(x), f(x±h), f(x±2h), f(x±3h), ⋯ are needed to approximate f^{(4)}(x)? What is the order of the truncation error p in f^{(4)}(x) − f_{h}^{(4)}(x) = O(h^{p})? Explain. You could give your answer with or without explicitly finding the coefficients of f(x), f(x ± h), f(x ± 2h), f(x ± 3h), · · ·.

4. Suppose that M = N_{1}(h) + K_{1}h^{2} + K_{2}h^{4} + +K_{3}h^{6} + ⋯ and N_{1}(h), N_{1}(h/2), N_{1}(h/4) are given. Construct N_{3}(h) using an extrapolation table or line by line derivation. What is the order of accuracy for N_{3}(h)?

Numerical Analysis I, Fall 2014 (http://www.math.nthu.edu.tw/˜ wangwc/)

Quiz 03

Nov 14, 2014.

1. Give (need not derive) an approximation of f′′(x) from f(x − h), f(x) and f(x + h). Then derive an error identity of the form f′′(x) − f′′_{h}(x) = C_{1}f^{(n)}(ξ)h^{n+k}.

2. Derive a fourth order approximation of f′(x) from f(x), f(x ± h), f(x ± 2h), f(x ± 3h), · · ·.

3. At least how many points among f(x), f(x±h), f(x±2h), f(x±3h), ⋯ are needed to approximate f^{(4)}(x)? What is the order of the truncation error p in f^{(4)}(x) − f_{h}^{(4)}(x) = O(h^{p})? Explain. You could give your answer with or without explicitly finding the coefficients of f(x), f(x ± h), f(x ± 2h), f(x ± 3h), · · ·.

4. Suppose that M = N_{1}(h) + K_{1}h^{2} + K_{2}h^{4} + +K_{3}h^{6} + ⋯ and N_{1}(h), N_{1}(h/2), N_{1}(h/4) are given. Construct N_{3}(h) using an extrapolation table or line by line derivation. What is the order of accuracy for N_{3}(h)?