In numerical analysis, Chebyshev nodes are specific real

algebraic number An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...

s, namely the roots of the

Chebyshev polynomials of the first kind Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics. Chebyshe ...

. They are often used as nodes in polynomial interpolation because the resulting interpolation polynomial minimizes the effect of Runge's phenomenon.

Definition

For a given positive integer ''n'' the Chebyshev nodes in the interval (−1, 1) are :

x_k = \cos\left(\frac\pi\right), \quad k = 1, \ldots, n.

These are the roots of the

Chebyshev polynomial of the first kind The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric functions: The Chebyshe ...

of degree ''n''. For nodes over an arbitrary interval 'a'', ''b''an

affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally, ...

can be used: :

x_k = \frac (a + b) + \frac (b - a) \cos\left(\frac\pi\right), \quad k = 1, \ldots, n.

Approximation

The Chebyshev nodes are important in

approximation theory In mathematics, approximation theory is concerned with how function (mathematics), functions can best be approximation, approximated with simpler functions, and with quantitative property, quantitatively characterization (mathematics), characteri ...

because they form a particularly good set of nodes for polynomial interpolation. Given a function ƒ on the interval

1,+1 Onekama ( ) is a village in Manistee County in the U.S. state of Michigan. The population was 411 at the 2010 census. The village is located on the shores of Portage Lake and is surrounded by Onekama Township. The town's name is derived from "On ...

/math> and

n

points

x_1,  x_2, \ldots , x_n,

in that interval, the interpolation polynomial is that unique polynomial

P_

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_(x) = \frac \prod_^n (x-x_i)

for some

\xi

(depending on x) in minus;1, 1 So it is logical to try to minimize :

\max_ \left,  \prod_^n (x-x_i) \.

This product is a '' monic'' polynomial of degree ''n''. It may be shown that the maximum absolute value (maximum norm) of any such polynomial is bounded from below by 2^1−''n''. This bound is attained by the scaled Chebyshev polynomials 2^1−''n'' ''T''_''n'', which are also monic. (Recall that , ''T''_''n''(''x''), ≤ 1 for ''x'' ∈ minus;1, 1, Lecture 20, §14) Therefore, when the interpolation nodes ''x''_''i'' are the roots of ''T''_''n'', the error satisfies :

\left, f(x) - P_(x)\ \le \frac \max_ \left, f^ (\xi)\.

For an arbitrary interval 'a'', ''b''a change of variable shows that :

\left, f(x) - P_(x)\ \le \frac \left(\frac\right)^n \max_ \left, f^ (\xi)\.

Definition

Approximation

Notes

References

Further reading