In this conference we shall discuss the use of the singularity subtraction technique as a method to approximate eigenvectors and eigenvalues of compact integral operators defined through a weakly singular convolution kernel. We shall give a precise definition of weak singularity and explain why the numerical procedure gives reasonably good spectral approximations. In order to improve their quality, two commonly used strategies will be tempted: iterative refinement and acceleration. We will explain the reason why one of these works and not the other. The underlying research is a joint work with Alain Largillier from the same team, and Balmohan V. Limaye from the Department of Mathematics at the Indian Institute of Technology Bombay.