Chebyshev nodes

File:Chebyshev nodes from the circle.png

Chebyshev zeros (solid dots, red lines) and extrema (hollow squares, blue lines) are the projection of two sets of equispaced points on the unit circle onto the Template:Mvar-axis. Template:Math equispaced points on the circle project onto Template:Mvar Chebyshev zeros or Template:Mvar Chebyshev extrema. (Here Template:Math.)

File:Chebyshev nodes.png

The Chebyshev zeros (solid dots) are roots of a Chebyshev polynomial of the first kind (red). The Chebyshev extrema (hollow squares) are roots of a Chebyshev polynomial of the second kind (blue), and also the extrema (crosses) of a Chebyshev polynomial of the first kind.

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 <math>[-1, 1]</math>, the circle's diameter.

There are two kinds of Chebyshev nodes. The Template:Tmath Chebyshev nodes of the first kind, also called the Chebyshev–Gauss nodes<ref>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.</ref> or Chebyshev zeros, are the zeros of a Chebyshev polynomial of the first kind, Template:Tmath. The corresponding Template:Tmath Chebyshev nodes of the second kind, also called the Chebyshev–Lobatto nodes<ref>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.</ref> or Chebyshev extrema, are the extrema of Template:Tmath, which are also the zeros of a Chebyshev polynomial of the second kind, Template:Tmath, along with the two endpoints of the interval. Both types of numbers are commonly referred to as Chebyshev nodes or Chebyshev points in literature.<ref>Template:Harvnb</ref> They are named after 19th century Russian mathematician Pafnuty Chebyshev, who first introduced Chebyshev polynomials.

Unlike some other interpolation nodes, the Chebyshev nodes "nest": the existing nodes are retained when doubling the number of nodes, reducing computation for each grid refinement by half. Polynomial interpolants constructed from Chebyshev nodes minimize the effect of Runge's phenomenon.<ref>Template:Harvnb</ref> They can be easily converted to a representation as a weighted sum of Chebyshev polynomials using the fast Fourier transform.

DefinitionEdit

File:ChebyshevNodes2.jpg

Chebyshev nodes of both kinds from <math>n=2</math> to <math>n=50</math>.

For a given positive integer <math>n</math>, the Template:Tmath Chebyshev nodes of the first kind are given by

<math display="block">x_k = \cos\frac{\bigl(k+\tfrac12\bigr)\pi}{n}, \quad k = 0, \ldots, n-1.</math>

This is the projection of Template:Tmath equispaced points on the unit circle onto the interval Template:Tmath, the circle's diameter. These points are also the roots of Template:Tmath, the Chebyshev polynomial of the first kind with degree Template:Tmath.

The Template:Tmath Chebyshev nodes of the second kind are given by

<math display="block">x_k = \cos\frac{k\pi}{n}, \quad k = 0, \ldots, n.</math>

This is also the projection of Template:Tmath equispaced points on the unit circle onto Template:Tmath, this time including 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 Template:Tmath in Template:Tmath, the places where it takes the value Template:Tmath.<ref>Template:Harvnb</ref> The interior points among the nodes, not including the endpoints, are also the zeros of Template:Tmath, a Chebyshev polynomial of the second kind, a rescaling of the derivative of Template:Tmath.

For nodes over an arbitrary interval <math>[a,b]</math> an affine transformation from <math>[-1,1]</math> can be used: <math display="block">\tilde{x}_k = \tfrac12(a + b) + \tfrac12(b - a) x_k.</math>

PropertiesEdit

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

ExamplesEdit

The node sets for the first few integers <math>n</math> are: <math display="block">\begin{align} \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{align}</math>

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.

ApproximationEdit

The Chebyshev nodes are important in approximation theory because they form a particularly good set of nodes for polynomial interpolation. Given a function Template:Math on the interval <math>[-1,+1]</math> and <math>n</math> points <math>x_1, x_2, \ldots , x_n,</math> in that interval, the interpolation polynomial is that unique polynomial <math>P_{n-1}</math> of degree at most <math>n - 1</math> which has value <math>f(x_i)</math> at each point <math>x_i</math>. The interpolation error at <math>x</math> is <math display="block">f(x) - P_{n-1}(x) = \frac{f^{(n)}(\xi)}{n!} \prod_{i=1}^n (x-x_i) </math> for some <math>\xi</math> (depending on Template:Mvar) in Template:Closed-closed.<ref>Template:Harvnb</ref> So it is logical to try to minimize <math display="block">\max_{x \in [-1,1]} \biggl| \prod_{i=1}^n (x-x_i) \biggr|. </math>

This product is a monic polynomial of degree Template:Mvar. It may be shown that the maximum absolute value (maximum norm) of any such polynomial is bounded from below by Template:Math. This bound is attained by the scaled Chebyshev polynomials Template:Math, which are also monic. (Recall that Template:Math for Template:Math.<ref>Template:Harvnb</ref>) Therefore, when the interpolation nodes Template:Math are the roots of Template:Math, the error satisfies <math display="block">\left|f(x) - P_{n-1}(x)\right| \le \frac{1}{2^{n - 1}n!} \max_{\xi \in [-1,1]} \left| f^{(n)} (\xi) \right|.</math> For an arbitrary interval [a, b] a change of variable shows that <math display="block">\left|f(x) - P_{n-1}(x)\right| \le \frac{1}{2^{n - 1}n!} \left(\frac{b-a}{2}\right)^n \max_{\xi \in [a,b]} \left|f^{(n)} (\xi)\right|.</math>

Modified even-order nodesEdit

Some applications for interpolation nodes, such as the design of equally terminated passive Chebyshev filters, cannot use even-order Chebyshev nodes directly due to the lack of a root at 0. Instead, the Chebyshev nodes can be moved toward zero, with a double root at zero directly, using a transformation:<ref name=":02">Template:Cite book</ref>

<math display=block>\tilde{x}_k = \operatorname{sgn}(x_k)\sqrt{\frac{x_k^2-x_{n/2}^2}{1-x_{n/2}^2}}</math>

For example, Chebyshev nodes of the first kind of order 4 are <math>{0.9239,0.3827,-0.3827,-0.9239}</math>, with <math>x_{n/2} = 0.382683</math>. Applying the transformation yields new nodes <math>{0.910180, 0, 0, -0.910180}</math>. The modified even-order nodes now include zero twice.

NotesEdit

Template:Reflist

ReferencesEdit

Template:Refbegin

Template:Refend