Numerical Analysis I, Fall 2010 (http://www.math.nthu.edu.tw/˜ wangwc/)

Homework Assignment for Week 16

Assigned Dec 31, 2010.

1. Section 6.2: Problems 1(a), 3(a), 5(a), 31.

2. (corrected) Continue from last week’s homework problem, show that if p < −1, and c_{i} is an arbitrary point in the interval [x_{i−1}, x_{i}], then h ∑_{i=2}^{n} c_{i}^{p} − ∫_{h}^{1} x^{p}dx = O(h^{p+1}) (which implies that h ∑_{i=2}^{n} c_{i}^{p} = O(h^{p+1}), p < −1 and h ∑_{i=1}^{n} x_{i− \frac{1}{2}}^{q} −∫_{0}^{1} x^{q}dx = O(h^{q+1}), −1 < q < 1). Hint: use monotonicity of the function x^{p}. Remark: for general r ∈ R, h ∑_{i=2}^{n} x_{i− \frac{1}{2}}^{r} − ∫_{h}^{1} x^{r}dx = O(h^{min\{r+1,2\}})

3. Continue on previous problem, can you predict the order of convergence of Simpson’s rule applied to ∫_{0}^{1} x^{p}dx for 0 < p < 3 and to ∫_{2h}^{1} x^{p}dx for −1 < p < 0? Verify your answer numerically.