= Chebyshev-Lobatto node
{c}
{title2=$x_k=\cos(k\pi/n),\quad k=0,\ldots,n$}
= Chebyshev extreme point
{c}
{synonym}
For $n\geq1$, the $n+1$ <Chebyshev-Lobatto nodes> are the <extrema> of the <Chebyshev polynomial> $T_n$ on $[-1,1]$, including both endpoints: $T_n(x_k)=(-1)^k$. Imagine $n+1$ equally spaced angles on a <semicircle> and project their points onto the horizontal axis. The <interpolation nodes> crowd near $-1$ and $1$ because equal angular steps have smaller horizontal projections there. Reversing the order gives the same set. These are distinct from the <roots of a polynomial> $\cos((2k+1)\pi/(2n+2))$ of $T_{n+1}$, which omit the endpoints. Driscoll and Braun call this endpoint-including family Chebyshev points of the second kind or Chebyshev extreme points.https://fncbook.com/python/stability-1/[Driscoll and Braun, Fundamentals of Numerical Computation, §9.3.2, equation 9.3.2]{ref}
On an interval $[a,b]$, use $x_k=(a+b)/2+(b-a)\cos(k\pi/n)/2$. The <Runge phenomenon> example compares <polynomial interpolation> at eleven <Chebyshev-Lobatto nodes> with eleven equally spaced <interpolation nodes>. The distinction matters whenever sampling locations are a choice rather than fixed observations.
Back to article page
