Chebyshev nodes

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

Definition

For a given natural number n, Chebyshev nodes in the interval [−1, 1] are

$x_i = \cos\left(\frac{2i-1}{2n}\pi\right) \mbox{ , } i=1,\ldots,n.$

For nodes over an arbitrary interval [a, b] an affine transformation can be used:

$\tilde{x}_i = \frac{1}{2} (a+b) + \frac{1}{2} (b-a) \cos\left(\frac{2i-1}{2n}\pi\right).$

Approximation using Chebyshev nodes

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 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 ξ 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 above by 2¹⁻ⁿ. This bound is attained by the scaled Chebyshev polynomials 2¹⁻ⁿ T_n, which are also monic. (Recall that |T_n(x)| ≤ 1 for x ∈ [−1, 1].^[2]). When interpolation nodes x_i are the roots of the T_n, the interpolation error satisfies therefore

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

Notes

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

References

• Burden, Richard L.; Faires, J. Douglas: Numerical Analysis, 8th ed., pages 503–512, ISBN 0534392008.
• Stewart, Gilbert W. (1996), Afternotes on Numerical Analysis, SIAM, ISBN 978-0-89871-362-6 .