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shubiaohewan July 18, 2013 11:03

Help on chebyshev polynomials
 
Hi all,

I want to see the orthogonality of the chebyshev polynomials in Matlab.

The nodes I choose is the Chebyshev nodes, cos(\pi\frac{2i-1}{2n}).. i=1.....n

According to the approximation theory, I can have \int_{-1}^1 T_i(x)T_j(x)\frac{dx}{\sqrt{1-x^2}}=\frac{\pi}{2}(1+\delta_{0i})\delta_{ij}

Now, how can I numerically show that the orthogonality? What I did is
(1) I generate the chebyshev polynomials in Matlab, stored in p(:,i), i is the order 0....n
(2) sum(p(:,i).*p(:,j)), meaning that the component-wise sum of the i-th order chebyshev and j-th one.

But it turns out that some of the results work, some of them don't.

Did I do something wrong here? I didn't mention the weight function since for the chebyshev nodes, the weight function is constant pi/n. So at least, it should affect the orthogonality.

Any comments? Thanks.

shubiaohewan July 18, 2013 11:32

Sorry, please ignore this thread. Problem solved.


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