Accuracy of numerical methods in finding eigenvalues I have a problem with assesing the accuracy of my numerical calculation. I have a 2nd order ODE. It is an eigenvalue problem of the form:
$ y'' + ay' + \lambda^2y = 0 $
and the boundary condiations are:
$ y(0) = y(1) = 0 $
This equation describes a vibrating string, clamped at x=0 and x=1, with a certain mass distribution. I want to be able to calculate the eigenvalues of this equation numerically. I do this by using the Runge-Kutta method to find solutions to the equation with initial conditions:
$ y(0) = 0, y'(0) = 1 $
with different values of $\lambda$ and then looking for the ones that are zero at x=1.
I terminate my search for eigenvalues when I find a function that has
$y(1) < \delta$
where $\delta$ is a predefined constant. 
Now the problem is that I'd like to know how accurate my calculations are, i.e. how do I choose the values of $\delta$ and the stepsize used in computing the values of the potential eigenfunction, so that the error in the eigenvalue is less than, say, $ 10^{-3}$? 

Rewrite the problem as
$$
\text{find }y,\rho\text{ such that }Ly=\rho y\text{ and }y\left(0\right)=y\left(1\right)=0.
$$
It is implicit in the notation that $y$ is smooth enough. You impose
$y^{\prime}\left(0\right)=1$, though this seems arbitrary; I will
ignore this.
You are interested in numerical solutions, so you will eventually
"discretize" your operator $L$. Let's call this discretization
$L_{N}\in\mathbb{R}^{N\times N}$. Then, your problem becomes
$$
\text{find }\mathbf{y},\rho\text{ such that }L_{N}\mathbf{y}=\rho\mathbf{y}.
$$
The first and last rows of $L_{N}$ should incorporate the boundary
conditions. In this form, the problem is simply one of finding eigenvalues
and eigenvectors of $L_{N}$, for which there exist good numerical
methods. This might also make your analysis regarding the error easier
to handle.
