Integration points and weights with orthogonal polynomials
Instead of an integration problem we need now to define the coefficient
\alpha_0 .
Since we know the values of
Q_{N-1} at the zeros of
L_N , we may rewrite
Eq.
(15) as
\begin{equation}
Q_{N-1}(x_k)=\sum_{i=0}^{N-1}\alpha_iL_i(x_k)=\sum_{i=0}^{N-1}\alpha_iL_{ik} \hspace{1cm} k=0,1,\dots, N-1.
\tag{16}
\end{equation}
Since the Legendre polynomials are linearly independent of each other, none
of the columns in the matrix
L_{ik} are linear combinations of the others.