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| In [[numerical analysis]], '''Chebyshev nodes''' are the roots of the [[Chebyshev polynomials|Chebyshev polynomial of the first kind]], which are [[algebraic number]]s. They are often used as nodes in [[polynomial interpolation]] because the resulting interpolation polynomial minimizes the effect of [[Runge's phenomenon]].{{fact|date=December 2013}}
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| ==Definition==
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| For a given natural number ''n'', '''Chebyshev nodes''' in the interval [−1, 1] are
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| :<math>x_i = \cos\left(\frac{2i-1}{2n}\pi\right) \mbox{ , } i=1,\ldots,n.</math>
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| These are the roots of the [[Chebyshev polynomial of the first kind]] of degree n. For nodes over an arbitrary interval [''a'', ''b''] an [[affine transformation]] can be used:
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| :<math>{x}_i = \frac{1}{2} (a+b) - \frac{1}{2} (b-a) \cos\left(\frac{2i-1}{2n}\pi\right). </math>
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| ==Approximation using Chebyshev nodes==
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| The Chebyshev nodes are important in [[approximation theory]] because they form a particularly good set of nodes for [[polynomial interpolation]]. Given a function ƒ on the interval <math>[-1,+1]</math> and <math>n</math> points <math>x_1, x_2, \ldots , x_n,</math> in that interval, the interpolation polynomial is that unique polynomial <math>P_{n-1}</math> of degree <math>n-1</math> which has value <math>f(x_i)</math> at each point <math>x_i</math>. The interpolation error at <math>x</math> is
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| :<math>f(x) - P_{n-1}(x) = \frac{f^{(n)}(\xi)}{n!} \prod_{i=1}^n (x-x_i) </math>
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| for some <math>\xi</math> in [−1, 1].<ref>{{harvtxt|Stewart|1996}}, (20.3)</ref> So it is logical to try to minimize
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| :<math>\max_{x \in [-1,1]} \left| \prod_{i=1}^n (x-x_i) \right|. </math>
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| This product Π is a ''[[monic polynomial|monic]]'' polynomial of degree ''n''. It may be shown that the maximum absolute value of any such polynomial is bounded below by 2<sup>1−''n''</sup>. This bound is attained by the scaled Chebyshev polynomials 2<sup>1−''n''</sup> ''T''<sub>''n''</sub>, which are also monic. (Recall that |''T''<sub>''n''</sub>(''x'')| ≤ 1 for ''x'' ∈ [−1, 1].<ref>{{harvtxt|Stewart|1996}}, Lecture 20, §14</ref>). When interpolation nodes ''x''<sub>''i''</sub> are the roots of the ''T''<sub>''n''</sub>, the interpolation error satisfies
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| therefore
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| :<math>|f(x) - P_{n-1}(x)| \le \frac{1}{2^{n-1}n!} \max_{\xi \in [-1,1]} |f^{(n)} (\xi)|.</math>
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| <!-- To do: write about Lebesgue constant and connection with Fourier series -->
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| ==Notes==
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| <references/>
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| ==References==
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| *Burden, Richard L.; Faires, J. Douglas: ''Numerical Analysis'', 8th ed., pages 503–512, ISBN 0-534-39200-8.
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| *{{Citation | last1=Stewart | first1=Gilbert W. | title=Afternotes on Numerical Analysis | publisher=[[Society for Industrial and Applied Mathematics|SIAM]] | isbn=978-0-89871-362-6 | year=1996}}.
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| [[Category:Numerical analysis]]
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| [[Category:Algebraic numbers]]
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