p (x) = \sum_{r = 0}^{n} u_{r} \frac{π _{n, r} ( x )}{π _{n, r} ( x _{r} )}, w_{r} = \frac{1}{π _{n, r} ( x _{r} )} = \frac{1}{π _{n}^{'} ( x _{r} )} = s \neq = r \prod (x_{r} - x_{s})^{- 1} .

The interpolation nodes are distinct, so these weights are well defined. Applying the same Lagrange interpolation to the constant polynomial one gives

1 = \sum_{r} w_{r} π_{n, r} (x)

. Dividing this identity and the formula for

p

π_{n} (x)

p (x) = \frac{\sum _{r = 0}^{n} w _{r} u _{r} / ( x - x _{r} )}{\sum _{r = 0}^{n} w _{r} / ( x - x _{r} )} .

This displayed quotient is used away from the interpolation nodes; at

x = x_{r}

its removable singularity is evaluated by setting

p (x_{r}) = u_{r}

. Multiplying every weight by the same nonzero constant leaves the quotient unchanged.

Chebyshev points are generally more reliable at high degree. The Chebyshev-Lobatto nodes, or the affine images of the roots of a Chebyshev polynomial, cluster near the interval endpoints and control the amplification measured by the Lebesgue constant of interpolation. Equally spaced interpolation nodes can produce large endpoint oscillations and the Runge phenomenon. This is a comparison of numerical robustness and general approximation behaviour, not a claim that one set minimizes the error for every individual function.