Spectral computations for birth and death chains

We consider the spectrum of birth and death chains on an n$n$-path. An iterative scheme is proposed to compute any eigenvalue with exponential convergence rate independent of n$n$. This allows one to determine the whole spectrum in order n²$n^2$ elementary operations. Using the same idea, we also provide a lower bound on the spectral gap, which is of the correct order on some classes of examples.