Math 6010, Fall 2016, Project on Orthogonal Polynomials
(Due: Monday December 12, 2016)
1- Introduction:
Set up a fourth-order polynomial regression model, using the Forsythe-Hayes algorithm,
though you are not asked to solve the model.
2- Your Project:
Consider the data
\(
x_i := \frac{i}{100},
\)
for $i=1,\ldots,100$.
Find polynomials, \(\varphi_0,\ldots,\varphi_4\) such that:
- Each \(\varphi_j\) is a \(j\)th order polynomials; and
- The \(\varphi_j\)'s are orthonormal over our data; i.e.,
for all \(0\le k,\ell\le 4\),
\[
\sum_{i=1}^{100} \varphi_k(x_i)\varphi_\ell(x_i)= \begin{cases}
1&\text{if $k=\ell$},\\
0&\text{if $k\neq \ell$}.
\end{cases}
\]
The report-writing portion of this project is straight-forward. Just write out
what you did carefully, attach a copy of your program, and display careful formulas for
the 5 polynomials. The coefficients of these polynomials should be stated explicitly in
numerical terms.