= Chebyshev node
{c}
{title2=$x_j=\cos((2j-1)\pi/(2N))$}
A standard set of $N$ Chebyshev nodes on $[-1,1]$ consists of the zeros of the <Chebyshev polynomial> $T_N$:
$$
x_j=\cos\frac{(2j-1)\pi}{2N},\qquad j=1,\ldots,N.
$$
They are equally spaced in angle before projection onto the interval, so the resulting real nodes cluster near the endpoints. An affine transformation gives nodes on any nondegenerate finite interval.
For $N=3$, the nodes are $\sqrt3/2,0,-\sqrt3/2$. Their monic nodal polynomial is
$$
\prod_{j=1}^N(x-x_j)=2^{1-N}T_N(x),
$$
whose maximum absolute value on the interval is $2^{1-N}$. This controls the nodal-product factor in the smooth-function <polynomial interpolation error> formula, explaining why endpoint clustering can be useful.
The related <Chebyshev-Lobatto nodes> use $\cos(j\pi/N)$, $j=0,\ldots,N$, and include both endpoints. They are a different node set; the zero-node and endpoint-inclusive conventions should not be silently interchanged.
Back to article page
