Loading [MathJax]/extensions/TeX/boldsymbol.js

 

 

 

Application to the case N=2

Obviously, there is no problem in changing the numbering of the matrix elements i,k=0,1,2,\dots,N-1 to i,k=1,2,\dots,N . We have chosen to start from zero, since we deal with polynomials of degree N-1 .

Summarizing, for Legendre polynomials with N=2 we have weights \begin{equation*} \omega : \left\{1,1\right\}, \end{equation*} and mesh points \begin{equation*} x : \left\{-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\right\}. \end{equation*}