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What about Chebyshev polynomials of the second kind? Can they be evaluated quickly?

Yes, they can be evaluated quickly, just replace${\displaystyle x}$ with${\displaystyle 2x}$ in the expression for${\displaystyle b_0}$. As a matter of fact, the Clenshaw algorithm is a general method for evaluation of a linear combination of functions that can be expressed using a recurrence formula. If you like, I will describe the general algorithm including the references to relevant literature.--Zdeněk Wagner (talk)10:27, 23 November 2007 (UTC)[reply]

That would be nice—Precedingunsigned comment added by76.165.224.41 (talk)23:17, 24 April 2008 (UTC)[reply]

I can't follow this page - is a paragraph or equation missing? What do the${\displaystyle b_n}$ actually relate to?

The last equation (${\displaystyle p\left(x\right)=b_0}$ ) can't be right

I would like to know:

Given polynomial, with coefficients${\displaystyle b_n}$:

${\displaystyle p\left(x\right)=\sum _{n=0}^N{b}_n{x}^n}$

how do we find the coefficients${\displaystyle a_n}$ for the sum of Chebyshev polynomials

${\displaystyle p\left(x\right)=\sum _{n=0}^N{a}_n{T}_n\left(x\right)}$

- is that what it was meant to do?

If I find out I will come back and offer a fix --Robertpaynter (talk)17:34, 28 October 2008 (UTC)[reply]

It is quite easy. Look at the definition ofChebyshev polynomials.${\displaystyle T_n}$ is thus a polynomial of${\displaystyle n}$-th order. If you combine the above written equation, you get the set of${\displaystyle N+1}$ linear equation for${\displaystyle N+1}$ unknown coefficients${\displaystyle a_n}$. In practice you never obtain Chebyshev expansion that way. There are several kinds of polynomials that can be defined by recurrence relations. When determining polynomial series of such polynomials, you often take advantage of their orthogonality or you obtain them as a solution of a differential equation of some kind etc. Now you need to evaluate the function value of such an expansion for some${\displaystyle x}$. The direct use of the expansion in such polynomials is computationally demanding because you have to evaluate all polynomials and then sum them. Moreover, their coefficients are often large with opposite signs so that there is high risk of round-off errors. The Clenshaw formula allows you to evaluate the function value by a recurrence relation without evaluating the polynomials. It is therefore much faster and almost always numerically stable. The equation given here is just a special form for evaluating the series of Chebyshev polynomials of the first kind, but the algorithm can be used for any polynomial that obeys some recurrence relation. When I have some more time, I will rewrite this chapter generally including detailed derivation and references to literature. --Zdeněk Wagner (talk)14:44, 12 November 2008 (UTC)[reply]

This article is like an answer from the Microsoft User Support:short, correct and absolutly useless. It is just for people who have known it allready better before. Now they have learned that there is another suffisticated way to explain it complicated. SeeAlan Greenspan's Methode: "If you understand it, I haven't explained it properly." CBa--79.219.230.24 (talk)10:21, 29 January 2011 (UTC)[reply]

There's an error in the Chebyshev section; the equations listed are only covered to degree 2 for some reason.

Arghman (talk)17:47, 30 September 2012 (UTC)[reply]

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