Ok how would we find the 14th degree Chebyshev polynomial? lets start with that
Ok so we were given this really long crazy function to be the model of a an arbitrary data set defined by vector f (function values) and vector x (data points) for
And were asked to find the best fit in the least squares sense for this model -- Find the normal equation?
where is the 14th degree Chebyshev polynomial
So I was wondering how do u find the the normal equation for something as crazy as this... or is this more of a theoretical question
to be the model of a an arbitrary data set defined by vector f (function values) and vector x (data points) for
And we're asked to find the best fit in the least squares sense for this model -- Find the normal equation?
where is the 14th degree Chebyshev polynomial (of the first kind).
You certainly can show HOW to get the normal matrix, and normal equation for this, but you have no specific values for the and no measured dependent values so you certainly can't give a numerical answer.
I'm pretty sure that should have a coefficient, in it.
I suspect that you typed instead of in the third term, which would make
On the other hand, if you merely omitted the , that would make
As for Chebyshev polynomials, Check out "Chebyshev polynomials" in Wikipedia and/or "Chebyshev polynomial" at WolframAlpha. More specifically, check out "ChebyshevT[14, x]" at WolframAlpha.
Work on that. I'll write an additional response as time permits.
yes i meant to type a c INSTEAD of the x... And thank you for your help.
The question that was posed to use was find the best fit for this model in the least squares sense. "Find the normal equation". I looked through our lecture notes for hours but I just couldn't figure it out.
I'm going to rename the coefficients to make setting up matricies more convenient, calling them for respectively. We then have:
Define the following matrix and vectors.
Vector f:
Vector b:
Matrix X, an matrix, such that: for example:
The transpose of X times X is often called the Normal matrix. It is a 4x4 matrix.
Notice that
We also need a vector, y, of observed values, each corresponding to
The normal equations resulting from this are:
Solve this for b to get the least squares result.