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. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Numerical analysis is the study of algorithms for the problems of continuous mathematics (as distinguished from discrete mathematics). ... In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev (Пафнутий Чебышёв), are special polynomials. ... In the mathematical subfield of numerical analysis, polynomial interpolation is the interpolation of a given data set by a polynomial. ... In the mathematical subfield of numerical analysis, interpolation is a method of constructing new data points from a discrete set of known data points. ... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ... The red curve is the Runge function, the blue curve is a 5th-degree polynomial, while the green curve is a 9th-degree polynomial. ...

Definition

For a given n, the n Chebyshev nodes are

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. In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...

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 mathematics, approximation theory is concerned with how functions can be approximated with other, simpler, functions, and with characterising in a quantitative way the errors introduced thereby. ... In the mathematical subfield of numerical analysis, polynomial interpolation is the interpolation of a given data set by a polynomial. ...

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 In mathematical analysis, the uniform norm assigns to real- or complex-valued functions f the nonnegative number This norm is also called the supremum norm or the Chebyshev norm. ...

Such a polynomial p can be constructed by polynomial interpolation: we pick n + 1 points x₀, ..., x_n 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

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

The product Π (x − x_i) 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 T_n+1 are monic and attain equality, because |T_n+1(x)| ≤ 1 for x ∈ [−1, 1].

Thus when using the roots of the T_n+1 polynomial as the interpolation nodes x_i we can bound the interpolation error as