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

Notes

All Chebyshev nodes are contained in the interval 1. To get nodes over an arbitrary interval b a linear 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. In order to make the following construction easier we restrict ourself to the interval 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
\|f\|_{\infty} := \max \lbrace\, |f(x)| : x \in -1,1\, \rbrace.
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
f(x) - p(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!} \prod_{i=0}^n (x-x_i)
for some \xi in −1, 1. So it is logical to try to minimize
\max_{x \in -1,1} \prod_{i=0}^n (x-x_i).
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
\|f - p\|_{\infty} \le \frac{1}{2^n(n+1)!} \sup_{\xi \in -1,1} |f^{(n+1)} (\xi)|.

 

<< PreviousWord BrowserNext >>
estriol
amon ereb
charles causley
barry melrose
long wall
wchl
carlos gomes jnior
arfenhouse
amon lhaw
side project
negro american league
amon rdh
eastern colored league
negro national league (the first)
fatal hilarity
george walker
taurus
negro national league (the second)
the shredder
whaley house
cross generation ship
bunraku
twin cities public television
father divine
negro league world series
aeolian skinner
elisabeth (biblical person)
duchy of the archipelago
comrade
delta iv rocket
treaty of accession 2003
university of copenhagen faculty of theology
university of copenhagen faculty of law
germania (book)
university of copenhagen faculty of social sciences
north manitou island
choctaw, mississippi
civil rights act of 1875
university of copenhagen faculty of health sciences
university of copenhagen faculty of humanities
university of copenhagen department of chemistry
raoul heertje
many waters
dane rudhyar
Copyright 2005-2009 OnPedia.com. All Rights Reserved