Chebyshev nodes
|
|
In the mathematical subfield of 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 problem of Runge's phenomenon.
Definition
For a given n, the n Chebyshev nodes are
- <math>x_i = \cos\left(\frac{2i-1}{2n}\pi\right) \mbox{ , } i=1,\ldots,n.<math>
Notes
All Chebyshev nodes are contained in the interval [−1, 1]. To get nodes over an arbitrary interval [a, b] a linear transformation can be used.
- <math>\tilde{x}_i = \frac{1}{2} (a+b) + \frac{1}{2} (b-a) \cos\left(\frac{2i-1}{2n}\pi\right). <math>
Approximation using Chebyshev nodes
The Chebyshev nodes are important in approximation theory because they form a particularly good set of nodes for polynomial interpolation.
In order to make the following construction easier we restrict ourself to the interval [−1, 1]. Generalizing to any interval [a, b] is straightforward by scaling the Chebyshev polynomials.
Given a function f on [−1, 1], we want to find a polynomial of some given degree, say n, which approximates f well in the maximum norm or Chebyshev norm which is defined as
- <math>\|f\|_{\infty} := \max \lbrace\, |f(x)| : x \in [-1,1]\, \rbrace.<math>
Such a polynomial p can be constructed by polynomial interpolation: we pick n + 1 points x0, ..., xn in the interval [−1, 1], and then we let p be the unique polynomial which coincides with f on these points.
The interpolation error for polynomial interpolation is
- <math>f(x) - p(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!} \prod_{i=0}^n (x-x_i) <math>
for some <math>\xi<math> in [−1, 1]. So it is logical to try to minimize
- <math>\max_{x \in [-1,1]} \prod_{i=0}^n (x-x_i). <math>
The product Π (x − xi) is a polynomial of degree n + 1 with leading coefficient 1 (such a polynomial is said to be monic). It turns out that the maximum norm of any such polynomial is greater than or equal to 2−n. Furthermore, the scaled Chebyshev polynomials 2−n Tn+1 are monic and attain equality, because |Tn+1(x)| ≤ 1 for x ∈ [−1, 1].
Thus when using the roots of the Tn+1 polynomial as the interpolation nodes xi we can bound the interpolation error as
- <math>\|f - p\|_{\infty} \le \frac{1}{2^n(n+1)!} \sup_{\xi \in [-1,1]} |f^{(n+1)} (\xi)|.<math>it:Nodi di Chebyshev