Interpolation node ($x_i$)

An interpolation node is an argument value $x_i$ at which an interpolation polynomial must take a prescribed value $y_i$. The interpolation nodes specify where the constraints apply; the values specify how high the graph must pass there. In the three-point example, the interpolation nodes are $0,1,2$, while the prescribed values are $1,2,5$. Changing the interpolation nodes changes the Lagrange cardinal polynomials and the polynomial interpolation error, even if their number stays fixed.

For $n\ge 1$, 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.^[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.