Chebyshev nodes

In numerical analysis, Chebyshev nodes (also called Chebyshev points or a Chebyshev grid) are a set of specific algebraic numbers used as nodes for polynomial interpolation and numerical integration. They are the projection of a set of equispaced points on the unit circle onto the real interval $[-1,1]$ , the circle's diameter.

The ⁠ $n$ ⁠ Chebyshev nodes of the first kind, also called the Chebyshev–Gauss nodes^[1] or Chebyshev zeros, are the zeros of a Chebyshev polynomial of the first kind, ⁠ $T_{n}$ ⁠. The corresponding ⁠ $n+1$ ⁠ Chebyshev nodes of the second kind, also called the Chebyshev–Lobatto nodes^[2] or Chebyshev extrema, are the extrema of a Chebyshev polynomial of the first kind, which are also the zeros of a Chebyshev polynomial of the second kind, ⁠ $U_{n-1}$ ⁠, along with the two endpoints of the interval. Both types of numbers are commonly referred to as Chebyshev nodes or Chebyshev points in literature.^[3] Polynomial interpolants constructed from these nodes minimize the effect of Runge's phenomenon.^[4] Chebyshev nodes are named after Pafnuty Chebyshev, who first introduced Chebyshev polynomials.

Definition

Chebyshev nodes of both kinds from $n=2$ to $n=50$ .

For a given positive integer $n$ , the ⁠ $n$ ⁠ Chebyshev nodes of the first kind are given by

$x_{k}=\cos {\frac {{\bigl (}k+{\tfrac {1}{2}}{\bigr )}\pi }{n}},\quad k=0,\ldots ,n-1.$

This is the projection of ⁠ $2n$ ⁠ equispaced points on the unit circle onto the circle's diameter, consisting of the roots of ⁠ $T_{n}$ ⁠, the Chebyshev polynomial of the first kind with degree ⁠ $n$ ⁠.

The ⁠ $n+1$ ⁠ Chebyshev nodes of the second kind are given by

$x_{k}=\cos {\frac {k\pi }{n}},\quad k=0,\ldots ,n.$

This is also the projection of ⁠ $2n$ ⁠ equispaced points on the unit circle onto the circle's diameter, but this time including the points at the endpoints of the interval, each of which is only the projection of one point on the circle rather than two. These points are also the extrema of ⁠ $T_{n}$ ⁠ in the interval ⁠ $[-1,1]$ ⁠, the places where it takes the value ⁠ $\pm 1$ ⁠.^[5] The interior points among the nodes, not including the endpoints, are also the zeros of ⁠ $U_{n-1}$ ⁠, a Chebyshev polynomial of the second kind, a rescaling of the derivative of ⁠ $T_{n}$ ⁠.

Both kinds of nodes are always symmetric about zero, the midpoint of the interval.

For nodes over an arbitrary interval $[a,b]$ an affine transformation from $[-1,1]$ can be used: ${\tilde {x}}_{k}={\tfrac {1}{2}}(a+b)+{\tfrac {1}{2}}(b-a)x_{k}.$

Approximation

The Chebyshev nodes are important in approximation theory because they form a particularly good set of nodes for polynomial interpolation. Given a function $f$ on the interval $[-1,+1]$ and $n$ points $x_{1},x_{2},\ldots ,x_{n},$ in that interval, the interpolation polynomial is that unique polynomial $P_{n-1}$ of degree at most $n-1$ which has value $f(x_{i})$ at each point $x_{i}$ . The interpolation error at $x$ is $f(x)-P_{n-1}(x)={\frac {f^{(n)}(\xi )}{n!}}\prod _{i=1}^{n}(x-x_{i})$ for some $\xi$ (depending on $x$ ) in $[-1, 1]$ .^[6] So it is logical to try to minimize $\max _{x\in [-1,1]}{\biggl |}\prod _{i=1}^{n}(x-x_{i}){\biggr |}.$

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 $| T n (x)| \leq 1$ for $x \in [-1, 1]$ .^[7]) Therefore, when the interpolation nodes $x i$ are the roots of $T n$ , the error satisfies $\left|f(x)-P_{n-1}(x)\right|\leq {\frac {1}{2^{n-1}n!}}\max _{\xi \in [-1,1]}\left|f^{(n)}(\xi )\right|.$ For an arbitrary interval [a, b] a change of variable shows that $\left|f(x)-P_{n-1}(x)\right|\leq {\frac {1}{2^{n-1}n!}}\left({\frac {b-a}{2}}\right)^{n}\max _{\xi \in [a,b]}\left|f^{(n)}(\xi )\right|.$

Examples

The node sets for the first few integers $n$ are: ${\begin{aligned}{\text{roots}}(T_{0})&=\{\},&{\text{roots}}(U_{0})&=\{\},&{\text{extrema}}(T_{1})&=\{-1,+1\},\\{\text{roots}}(T_{1})&=\{0\},&{\text{roots}}(U_{1})&=\{0\},&{\text{extrema}}(T_{2})&=\{-1,0,+1\},\\{\text{roots}}(T_{2})&=\{-1/{\sqrt {2}},+1/{\sqrt {2}}\},&{\text{roots}}(U_{2})&=\{-1/2,+1/2\},&{\text{extrema}}(T_{3})&=\{-1,-1/2,+1/2,+1\}\\\end{aligned}}$

While these sets are sorted by ascending values, the defining formulas given above generate the Chebyshev nodes in reverse order from the greatest to the smallest.

Even order modified Chebyshev nodes

Many applications for Chebyshev nodes, such as the design of equally terminated passive Chebyshev filters, cannot use Chebyshev nodes directly, due to the lack of a root at 0. However, the Chebyshev nodes may be modified into a usable form by translating the roots down such that the lowest roots are moved to zero, thereby creating two roots at zero of the modified Chebyshev nodes.^[8]

The even order modification translation is:

$X_{k}Even={\sqrt {\frac {X_{k}^{2}-X_{n/2}^{2}}{1-X_{n/2}^{2}}}}{\text{ For }}n>2$

The sign of the ${\sqrt {}}$ function is chosen to be the same as the sign of $X_{k}$ .

For example, the Chebyshev nodes for a 4th order Chebyshev function are, {0.92388,0.382683,-0.382683,-0.92388}, and $X_{n/2}^{2}$ is $0.382683^{2}$ , or 0.146446. Running all the nodes through the translation yields $X_{k}Even$ to be {0.910180, 0, 0, -0.910180}.

The modified even order Chebyshev nodes now contains two nodes of zero, and is suitable for use in designing even order Chebyshev filters with equally terminated passive element networks.

Notes

^ The name Chebyshev–Gauss nodes comes from the use of Chebyshev zeros in numerical integration, which can be seen as a variant of Gaussian quadrature.
^ The name Chebyshev–Lobatto nodes comes from Rehuel Lobatto, who made a variant of Gaussian quadrature, known as Lobatto quadrature, whose nodes included the ends of the interval, a feature shared by the Chebyshev extrema.
^ Trefethen 2013, pp. 7
^ Fink & Mathews 1999, pp. 236–238
^ Trefethen 2013, pp. 14
^ Stewart 1996, (20.3)
^ Stewart 1996, Lecture 20, §14
^ Saal, Rudolf (January 1979). Handbook of Filter Design (in English and German) (1st ed.). Munich, Germany: Allgemeine Elektricitäts-Gesellschaft. pp. 25, 26, 56–61, 116, 117. ISBN 3-87087-070-2.

References

Fink, Kurtis D.; Mathews, John H. (1999). Numerical Methods using MATLAB (3rd ed.). Upper Saddle River NJ: Prentice Hall.
Stewart, Gilbert W. (1996). Afternotes on Numerical Analysis. SIAM. ISBN 978-0-89871-362-6.
Trefethen, Lloyd N. (2013), Approximation Theory and Approximation Practice, SIAM