Need to fit polynomial using chebyshev polynomial basis

Question

I have been fitting linear least-squares polynomials to data using the polyfit function in matlab. From what I read, this uses standard polynomial basis (monomial basis). I have read that using Chebyshev polynomial basis to fit leads to greater numerical stability so I would like to do this. Does matlab have this option?

Answer 1

I will assume here that you want Chebyshev polynomials of the first kind. As far as I know, Matlab does not have this inbuilt. It is easy to code yourself though. Chebyshev polynomials are only defined on [-1,1] so first you must map your x data to this range. Then use the recurrence relation for generating Chebyshev polynomials http://en.wikipedia.org/wiki/Chebyshev_polynomials#Definition

T_(n+1)(x) = 2xT_(n)x - T_(n-1)(x)

If x are your abscissae and y your data points generate your observation matrix A (this is the equivalent of the Vandermonde matrix for monomial basis) for a degree n polynomial fit using:

n = degree;
m = length(x);
%% Generate the z variable as a mapping of your x data range into the 
%% interval [-1,1]

z = ((x-min(x))-(max(x)-x))/(max(x)-min(x));

A(:,1) = ones(m,1);
if n > 1
   A(:,2) = z;
end
if n > 2
  for k = 3:n+1
     A(:,k) = 2*z.*A(:,k-1) - A(:,k-2);  %% recurrence relation
  end
end

then you can just solve the linear system for your solution parameters (approximation coefficients) b using matrix divide

b = A \ y

What you have to remember here is that when you evaluate your approximation you must map the value to the interval [-1,1] before evaluating it. The coefficients will only be valid on the initial range of x you provide for the approximation. You will not be able to evaulate the approximation (in a valid sense) outside of your initial x range. If you want to do that you should use a wider interval than your data has, that way when you map points interior to that using the transform, your points will always lie in [-1,1] and therefore evaluation of your approximation is valid.

I have not used matlab for a while so new versions may actually have an inbuilt function that will do all of this for you. It was not the case when I last used it and if all else fails the above will allow you to generate least-squares polynomial approximations using chebyshev basis (first kind)