Chebyshev nodes

In numerical analysis, Chebyshev nodes are specific real algebraic numbers, namely the roots of the Chebyshev polynomials of the first kind. 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 of degree ''n''. For nodes over an arbitrary interval 'a'', ''b''an affine transformation 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 because they form a particularly good set of nodes for polynomial interpolation. Given a function ƒ on the interval 1,+1/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
:$\left, f(x) - P_(x)\ \le \frac \left(\frac\right)^n \max_ \left, f^ (\xi)\.$

Notes

References

*.

Further reading

*Burden, Richard L.; Faires, J. Douglas: ''Numerical Analysis'', 8th ed., pages 503–512, .

{{Algebraic numbers

Numerical analysis
Algebraic numbers