Help on chebyshev polynomials

shubiaohewan · July 18, 2013, 11:03

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.

July 18, 2013, 11:03	Help on chebyshev polynomials	#1
shubiaohewan New Member Join Date: Apr 2013 Posts: 15 Rep Power: 13	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.

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July 18, 2013, 11:32		#2
shubiaohewan New Member Join Date: Apr 2013 Posts: 15 Rep Power: 13	Sorry, please ignore this thread. Problem solved.