Numerical Analysis I, Fall 2020 (http://www.math.nthu.edu.tw/˜ wangwc/)

Homework Assignment for Week 17

1. Section 8.1: Problems 2, 14.

2. 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: The matrix corresponding to this linear system is ill-conditioned for large n (why?). The discrete counter part, problem 14, is similarly ill-conditioned for large n. The proper way of solving these problems numerically for large n, say n > 5, is to incorporate the Gram-Schmidt procedure to select an orthogonal basis of polynomial functions in place of 1, x, x^{2}, · · ·. See section 8.2 if you are interested to find out the details.