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help me please with this question: definition of Chebyshev's polynomials it's given by $T_n(x)=\cos(n\arccos(x)) $

  1. Find by Gauss Quadrature method $\displaystyle\int_{-1}^1 \sqrt{1-x^2} \; dx$ with null error

  2. Estimate $\displaystyle\int_{-1}^1 \frac{e^x}{\sqrt{1-x^2}} \; dx$ accuracy of 5 decimal places

Thanks!

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How are the Chebyshev polynomials related to your questions? –  Antonio Vargas Apr 1 '12 at 21:15
    
@Antonio : They are used in numerical analysis to find the nodes of interpolation. OP seems confused, hence the confusing question, but a numerical analyst will probably understand the concept. This is behind be, I don't want to get back into it.. –  Patrick Da Silva Apr 1 '12 at 21:23
    
Thanks @Patrick, I agree that the OP seems confused. I wanted to find out exactly where the confusion lies, and my question was meant to prompt some thought about that. –  Antonio Vargas Apr 1 '12 at 21:25
    
What do you mean by "null error"? If you want the error to be $0$, don't use a numerical method. –  Robert Israel Apr 1 '12 at 21:37
2  
Ah: Chebyshev-Gauss quadrature. With $n=1$, the second case at en.wikipedia.org/wiki/Chebyshev%E2%80%93Gauss_quadrature gives the exact result for $\int_{-1}^1 \sqrt{1-x^2} f(x)\ dx$ where $f$ is a polynomial of degree $\le 1$. –  Robert Israel Apr 2 '12 at 0:11

1 Answer 1

up vote 1 down vote accepted

As Robert mentions, you do not want to use the Chebyshev polynomials of the first kind for your first integral; what you need are the Chebyshev polynomials of the second kind,

$$U_n(x)=\frac{\sin((n+1)\arccos\,x)}{\sqrt{1-x^2}}$$

which are orthogonal with respect to the inner product

$$\langle f,g\rangle=\int_{-1}^1 \sqrt{1-u^2}f(u)g(u)\mathrm du$$

Knowing that $\sin\,k\pi=0$ if $k$ is an integer ought to be a big hint on how to generate the nodes for Gauss-Chebyshev quadrature. From the theory, this quadrature rule is designed to give exact results for integrands of the form $\sqrt{1-x^2}p(x)$, where $p(x)$ is a polynomial, and since constant functions are effectively polynomials...


For the second one: just keep increasing the number of nodes until your error estimate (that I presume was mentioned in your textbook) gives something less than $10^{-5}$; one convenient thing about Gauss-Chebyshev is that the weights stay the same, and all one does is to change the nodes. (P.S. for what values of $u$ is $\cos\,u$ equal to zero?)

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That may be the fifth unique spelling of Chebyshev I've seen. –  Antonio Vargas Apr 3 '12 at 20:00
    
@Antonio: that, unfortunately, is a typo and not another permutation of old man Pafnuty's surname. I'll fix, and thanks for checking! –  J. M. Apr 10 '12 at 3:44

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