Integration points and weights with orthogonal polynomials
We can use Eq.
(14)
to rewrite the above integral as
\begin{equation*}
\int_{-1}^1P_{2N-1}(x)dx=\int_{-1}^1(L_N(x)P_{N-1}(x)+Q_{N-1}(x))dx=\int_{-1}^1Q_{N-1}(x)dx,
\end{equation*}
due to the orthogonality properties of the Legendre polynomials. We see that it suffices
to evaluate the integral over
\int_{-1}^1Q_{N-1}(x)dx in order to evaluate
\int_{-1}^1P_{2N-1}(x)dx . In addition, at the points
x_k where
L_N is zero, we have
\begin{equation*}
P_{2N-1}(x_k)=Q_{N-1}(x_k)\hspace{1cm} k=0,1,\dots, N-1,
\end{equation*}
and we see that through these
N points we can fully define
Q_{N-1}(x) and thereby the
integral. Note that we have chosen to let the numbering of the points run from
0 to
N-1 .
The reason for this choice is that we wish to have the same numbering as the order of a
polynomial of degree
N-1 . This numbering will be useful below when we introduce the matrix
elements which define the integration weights
w_i .