Use of Orthogonal Polynomials

• We have mentioned that the system of normal equations for a polynomial fit is illconditioned when the degree is high. Even for a cubic least-squares polynomial, the condition number of the coefficient matrix can be large.
• In one experiment, a cubic polynomial was fitted to 21 data points. When the data were put into the coefficient matrix of Eq. 7, its condition number (using 2-norms) was found to be 22,000!.
• This means that small differences in the $y$-values will make a large difference in the solution. In fact, if the four right-hand-side values are each changed by only 0.01 (about 0.1%), the solution for the parameters of the cubic were changed significantly, by as much as 44%!
• However, if we fit the data with orthogonal polynomials (A sequence of polynomials is said to be orthogonal with respect to the interval [a,b] if $\int_a^b P_n^{*}(x) P_m(x)dx=0~when~ n\neq m$) such as the Chebyshev polynomials. The condition number of the coefficient matrix is reduced to about 5 and the solution is not much affected by the perturbations.

Example: The following data:
R/C: 0.73, 0.78, 0.81, 0.86, 0.875, 0.89, 0.95, 1.02, l.03, 1.055, 1.135, 1.14, 1.245, 1.32, 1.385, 1.43, 1.445, 1.535, 1.57, 1.63, 1.755;
$V_\theta /V_\infty$: 0.0788, 0.0788, 0.064, 0.0788, 0.0681, 0.0703, 0.0703, 0.0681, 0.0681, 0.079, 0.0575, 0.0681, 0.0575, 0.0511, 0.0575, 0.049, 0.0532, 0.0511, 0.049, 0.0532,0.0426:
Let $x = R/C$ and $y=V_\theta/V_\infty$, We would like our curve to be of the form

$$g(x)=\frac{A}{x}(1-e^{-\lambda x^2})$$

and our least-squares equation becomes

$$S = \sum_{i=1}^{21}(Y_i-\frac{A}{x_i}(1-e^{-\lambda x_i^2}))^2$$

Setting $S_\lambda=S_A=0$ gives the following equations:

$$\begin{array}{l} \sum_{i=1}^{21}(\frac{1}{x_i})(1-e^{-\lamb... ...^2})(Y_i-\frac{A}{x_i}(1-e^{-\lambda x_i^2}))=0 \\ \end{array}$$

When this system of nonlinear equations is solved, we get

$$g(x)=\frac{0.07618}{x}(1-e^{-2.30574x^2})$$

For these values of $A$ and $\lambda, S = 0.00016$. The graph of this function is presented in Figure 8.

Figure 8: The graph of $V_\theta /V_\infty$ vs $R/C$.

An algorithm for obtaining a least-squares polynomial: