Past exam of the approximation theory and finite element methods course of the University of Oxford / 2009 / A / 1 / c / Solution

For uniformly spaced nodes,There are $n-s$ negative factors, coming from $j>s$. Thus the weights fixed by part (b) satisfy this formula exactly. The printed sign $(-1{)}^s$ agrees for even $n$; for odd $n$ it differs by the common factor $-1$. Rescaling all weights by $(-1{)}^n$ gives the printed convention without changing the barycentric interpolation formula.

For even $n$, the required magnitude isStirling's formula givesThus central weights are exponentially larger in magnitude than endpoint weights. Their alternating signs can cause severe cancellation in polynomial interpolation at equispaced nodes; they reflect the unfavorable high-degree interpolation geometry associated with Runge's phenomenon. A barycentric evaluation formula improves the implementation but does not remove the intrinsic sensitivity or guarantee convergence for every smooth function. The comparison with the bounded relative weights at Chebyshev-Lobatto nodes is decisive.