I am solving an eigenvalue Problem, the paper I'm reading says to solve
$$\int_{0}^N A(i,j) \phi(j) dr(j) = \lambda \phi(j)$$
and to approximate the above by
$$\sum_{i=1}^N A(i,j) \phi(j) H_j = \lambda \phi(j)$$
- where $H_j = \frac{\Delta r}{3} k_i$ and $\Delta r$ is the gridspace.
- and $k_i$ is the weighting function.
Which weighting function should I use? Based on the $H_j$ they give, is it somehow implied which weighting factor should be used, such as Simpson's rule? (they don't specify).
Edit: I decided this is just simpson's quadrature they are applying. So weigh each row the of the matrix by the usual, eg the first row is $k_i = (A(1,1)+ 4A(1,2)+2A(1,3)+\cdots + 2A(1,n-2)+4A(1,n-1)+A(1,n))$
To be clear I am not sure this weight is necessary since I seem to be getting the same eigenfunctions