Question about weights using Chebyshev polynomials as quadrature

[confused_engineer]

Hello everyone.

I am studying this article since I am interested in optimization. The article makes use of Clenshaw–Curtis quadrature scheme to discretize the integral part of the cost function to a finite sum using Chebyshev polynomials.

The article differentiates between the case of odd and even number of collocation points. In equation 27 and 28 (fourth page), the case of N even is discussed. If N is even, then the N+1 collocation points, including 0, form a vector of odd length.

Then weights are calculated as w_s=w_N-s=... for s=1, 2, ..., N/2. Meanwhile, w₀ and w_N are calculated on a different way.

This in turn means that one of the elements is calculated twice, as shown in the following example:

N=6; N+1=7; N/2=3; s=1, 2, 3; N-s=6-1=5, 6-2=4, 6-3=3.

As you can see, the fourth element of the vector, number 3, appears two times. I find this weird. Can someone please tell me if I am understanding the article wrong or if this is intended to happen?

Thanks for reading.
Regards.

 

[fresh_42]

I do not see where the problem is. For $N=6$ we have $w_0=w_6=1/35$ and then formulas for $w_1=w_5$, $w_2=w_4$, and $w_3$. Since $s=3= N-s=6-3$ and the formula for $w_3$ only depends on the given number $N$ and $s=3$, where is it defined twice?

 

[confused_engineer]

I do not see where the problem is. For $N=6$ we have $w_0=w_6=1/35$ and then formulas for $w_1=w_5$, $w_2=w_4$, and $w_3$. Since $s=3= N-s=6-3$ and the formula for $w_3$ only depends on the given number $N$ and $s=3$, where is it defined twice?

First of all, thanks for answering my question.

I am confused because the article tallks about ω_s and ω_N-s and I find extrange that following the paper one arrives to ω₃=ω₃=4/N*... for this particular example.

Therefore, I understand from your answer that ω₃=ω₃=4/N*... is the weight of the fourth element of the vector. I thought that since I had ω₃=ω₃, the middle element might have twice the weight.

Sorry if I haven't expressed myself clearly.
Thanks again for your answer.

 

[fresh_42]

No, it was only a way to avoid writing an extra line for $\omega_{N/2}$. $N/2 = N-N/2$ for even $N$ so it is only defined once.

 

[confused_engineer]

No, it was only a way to avoid writing an extra line for $\omega_{N/2}$. $N/2 = N-N/2$ for even $N$ so it is only defined once.

Then is all clear. Thank you very much.