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

Homework Assignment for Week 16

1. Reading instruction for section 7.5 and section 8.1:

Section 7.5: Study the meaning and derivation of condition number. Skip the ‘iterative refinement’ part.

Section 8.1: Study the derivation of linear least square problems, p499-503. Skip the remaining part.

2. Section 7.5: Problems 2(a,c), 9, 10.

3. Use the matlab built-in command ’cond’ to find the condition numbers of the Hilbert matrices H^{(n)} in section 7.5, problem 11 for n = 5, 10, 15, 20. That is, need not find the inverse of H^{(n)}.

4. Section 8.1: Problems 2, 14.

5. Derive the continuous version of least square problem:

Give n and f(x) : [0, 1] ↦ R, find a_{0}, ⋯ a_{n} to minimize the quantity

∫_{0}^{1} (f(x) − (a_{0} + a_{1}x + ⋯ a_{n}x^{n}))^{2} dx

Derive the normal equation for the coefficient vector (a_{0}, ⋯ a_{n}).

Remark: It is a fact that, similar to its discrete counter part in section 8.1, problem 14, both linear least square problem leads to ill-conditioned matrices for large n. Therefore this approach is not recommended for n > 5. For your reference, the proper treatment for n > 5 can be found in section 8.2.