Past exam of the mathematics course of the University of Cambridge / 2020 / ib / Paper 1 / 18C (Numerical Analysis)

Define the cardinal polynomialsThey satisfy ${\ell }_k(x_i)={\delta }_{ik}$. Hence evaluating the proposed polynomial interpolation formula at $x_i$ gives $p_n(x_i)=a_i$, so the required coefficients areThe polynomial ${\sum }_k{f}_k{\ell }_k$ has degree at most $n$ and takes every prescribed value. Its uniqueness follows because the difference of two such polynomials would have $n+1$ distinct roots while having degree at most $n$.

For fixed interpolation nodes $c_0,\dots ,c_{s-1}$, interpolate any $p\in {\mathbb{P}}_{s-1}[x]$ asIntegrating shows that the unique weights making the quadrature rule exact through degree $s-1$ are

Now let the nodes be the zeros of the degree-$s$ orthogonal polynomial $q_s$. Given any $p\in {\mathbb{P}}_{2s-1}[x]$, polynomial division givesOrthogonality gives ${\int }_a^b{q}_{s}h=0$, while $q_s(c_k)=0$ makes the quadrature sum for $q_{s}h$ vanish. The rule is already exact on $r$, so it is exact on $p$. This is the defining precision property of Gaussian quadrature.

The displayed Rodrigues formula gives the Legendre polynomialA four-node Gaussian rule is exact through degree $2\cdot 4-1=7$. Solving $q_4(x)=0$ first for $x^2$ gives the four nodesThe rule is not exact for every polynomial of degree eight: its quadrature value on $q_4^2$ is zero because every node is a root of $q_4$, whereas