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Chebyshev nodes

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Title: Chebyshev nodes  
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Subject: Runge's phenomenon, Polynomial interpolation, Algebraic numbers, Remez algorithm, Lagrange polynomial
Collection: Algebraic Numbers, Numerical Analysis
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Chebyshev nodes

In numerical analysis, Chebyshev nodes are the roots of the Chebyshev polynomial of the first kind, which are algebraic numbers. They are often used as nodes in polynomial interpolation because the resulting interpolation polynomial minimizes the effect of Runge's phenomenon.

Contents

  • Definition 1
  • Approximation 2
  • Notes 3
  • References 4
  • Further reading 5

Definition

For a given natural number n, Chebyshev nodes in the interval (−1, 1) are

x_k = \cos\left(\frac{2k-1}{2n}\pi\right) \mbox{ , } k=1,\ldots,n.

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:

{x}_k = \frac{1}{2} (a+b) + \frac{1}{2} (b-a) \cos\left(\frac{2k-1}{2n}\pi\right) \mbox{ , } k=1, \ldots, n.

Approximation

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 [-1,+1] and n points x_1, x_2, \ldots , x_n, in that interval, the interpolation polynomial is that unique polynomial P_{n-1} of degree at most n-1 which has value f(x_i) at each point x_i. The interpolation error at x is

f(x) - P_{n-1}(x) = \frac{f^{(n)}(\xi)}{n!} \prod_{i=1}^n (x-x_i)

for some \xi in [−1, 1].[1] So it is logical to try to minimize

\max_{x \in [-1,1]} \left| \prod_{i=1}^n (x-x_i) \right|.

This product Π is a monic polynomial of degree n. It may be shown that the maximum absolute value of any such polynomial is bounded below by 21−n. This bound is attained by the scaled Chebyshev polynomials 21−n Tn, which are also monic. (Recall that |Tn(x)| ≤ 1 for x ∈ [−1, 1].[2]). Therefore, when interpolation nodes xi are the roots of Tn, the interpolation error satisfies

\left|f(x) - P_{n-1}(x)\right| \le \frac{1}{2^{n - 1}n!} \max_{\xi \in [-1,1]} \left|f^{(n)} (\xi)\right|.

Notes

  1. ^ Stewart (1996), (20.3)
  2. ^ Stewart (1996), Lecture 20, §14

References

  • Stewart, Gilbert W. (1996), Afternotes on Numerical Analysis, .  

Further reading

  • Burden, Richard L.; Faires, J. Douglas: Numerical Analysis, 8th ed., pages 503–512, ISBN 0-534-39200-8.
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