Use of Orthogonal Polynomials

• We have mentioned that the system of normal equations for a polynomial fit is ill-conditioned 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. 3, its condition number (using 2-norms) was found to be 22000!.
• 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 such as the Chebyshev polynomials.
• A sequence of polynomials is said to be orthogonal with respect to the interval [a,b], if $\int_a^b P_n^{*}(x) P_m(x)dx=0~when~ n\neq m$.
• The condition number of the coefficient matrix is reduced to about 5 and the solution is not much affected by the perturbations.