Chebyshev nodes

Here we plot the Chebyshev nodes of the first kind and the second kind, both for n = 8. For both kinds of nodes, we first plot the points equi-distant on the upper half unit circle in blue. Then the blue points are projected down to the x-axis. The projected points, in red, are the Chebyshev nodes.

In numerical analysis, Chebyshev nodes are a set of specific real algebraic numbers, used as nodes for polynomial interpolation. They are the projection of equispaced points on the unit circle onto the real interval the diameter of the circle.

The Chebyshev nodes of the first kind, also called the Chebyshev zeros, are the zeros of the Chebyshev polynomials of the first kind. The Chebyshev nodes of the second kind, also called the Chebyshev extrema, are the extrema of the Chebyshev polynomials of the first kind, which are also the zeros of the Chebyshev polynomials of the second kind. Both of these sets of numbers are commonly referred to as Chebyshev nodes in literature.[1] Polynomial interpolants constructed from these nodes minimize the effect of Runge's phenomenon.[2]

  1. ^ Trefethen 2013, pp. 7
  2. ^ Fink & Mathews 1999, pp. 236–238