Chebyshev nodes

A specific, non-equally-spaced choice of interpolation nodes -- clustered near the ends of the interval -- that suppresses the Runge phenomenon.

The Chebyshev nodes on $[-1,1]$ for degree $n$ are $x_k=\cos\!\left(\frac{(2k+1)\pi}{2(n+1)}\right), \qquad k=0,1,\ldots,n,$ the projections onto $[-1,1]$ of $n+1$ equally spaced points on a semicircle -- deliberately bunched near $\pm1$ and sparser near the center. Interpolating at these nodes instead of equally spaced ones keeps the interpolation error bounded (and, for smooth enough $f$ , convergent) as $n$ grows, because they minimize the size of the node-product term $\prod(x-x_i)$ that drives the error formula.

Examples

For $n=1$ , the Chebyshev nodes are $\cos(\pi/4)\approx0.707$ and $\cos(3\pi/4)\approx-0.707$ -- pulled in from $\pm1$ , not sitting at the endpoints the way equally spaced nodes would.