Editing Chebyshev nodes (section)

{{Short description|Roots of the Chebyshev polynomials of the first kind}}
[[File:Chebyshev nodes from the circle.png|thumb|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 {{mvar|x}}-axis. {{math|2''n''}} equispaced points on the circle project onto {{mvar|n}} Chebyshev zeros or {{mvar|n+1}} Chebyshev extrema. (Here {{math|1=''n'' = 5}}.)]] 
[[File:Chebyshev nodes.png|thumb|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 number]]s used as nodes for [[polynomial interpolation]] and [[numerical integration]]. They are the [[Projection (linear algebra)|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 {{tmath|n}} ''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 [[Zero of a function|zeros]] of a [[Chebyshev polynomial]] of the first kind, {{tmath|T_n}}. The corresponding {{tmath|n+1}} ''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 [[Maximum and minimum|extrema]] of {{tmath|T_n}}, which are also the zeros of a Chebyshev polynomial of the second kind, {{tmath|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.<ref>{{harvnb|Trefethen|2013|pp=7}}</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>{{harvnb|Fink|Mathews|1999|pp=236–238}}</ref> They can be easily converted to a representation as a weighted sum of Chebyshev polynomials using the [[fast Fourier transform]].