Past exam of the numerical analysis course of the University of Oxford / 2021 / 1 / c / i

Let $L_{n+1}$ denote the conventional Legendre polynomial, normalized by $L_{n+1}(1)=1$. The $n+1$ quadrature nodes are its distinct polynomial roots in $(-1,1)$. For these nodes form the Lagrange cardinal polynomialsChoose the quadrature weights by integrating these polynomials:An equivalent standard formula isThe reason for selecting these quadrature nodes is the resulting degree of precision of a quadrature rule. For any polynomial $p$ of polynomial degree at most $2n+1$, polynomial division givesThe orthogonality of the Legendre polynomial makes ${\int }_{-1}^1{L}_{n+1}s=0$, while $p(x_i)=r(x_i)$ at each quadrature node. The definition of the quadrature weights integrates $r$ exactly, soIn particular, ${\sum }_i{w}_i=2$. Applying this exactness to ${\ell }_i^2$ also proves $w_i={\int }_{-1}^1{\ell }_i(x{)}^{2}dx>0$.