Question about the roots of Chebyshev polynomials

[confused_engineer]

TL;DR

I need to calculate the roots of the Chebyshev polynomials, but the results do not correspond with the theory

Hello everyone. I am trying to construct an optimization problem using Chebyshev pseudospectral method as described in this article. For that, I need to calculate the zeros of the Chebyshev polynomial of any order. In the article is sugested to do it as

t_k=cos(πk/N) k=0, ..., N

However, if I assign the value 2 to N, the nodes are -1, 0 and 1.

The Chebyshev polynomial of order 3 is 4t³-3t and it's roots are -√3/2, 0 and √3/2, this doesn't correspond with the results of the expression above.

In my model I am using the first expression, the cosine, to find the collocation points instead of using the roots of the polynomial of order N. The solver returns an error or an incorrect optimal solution based on the number of colocation points. Can someone please tell me if I am doing it right by choosing my collocation points as the cosine or if I should try to find the roots of the polynomials instead?

Any answer is appreciated.
Thanks for reading.
Regards.

 

[member 587159]

The article does not claim these are the roots of the Chebyshev polynomials, but interpolation points (as far as I understand, the article claims that these are the points where these polynomials attain extrema).

 

[confused_engineer]

First of all thanks for your answer.

If I understand propperly, the node points are the roots of the polynomial of order N? but then, what is the use of the cosine?

I have plotted the values if the cosine for N=60 and these values cluster arround the endpoints of the interval [-1, 1], I think that is the reason why I have confused the cosine with the nodes.

Regards.