We discuss a main computational problem confronted by physicists and chemists in the ab initio (from quantum mechanical first principles) calculation of materials properties. Ab initio calculations typically require the self-consistent solution of Schroedinger and Poisson equations, which in turn requires the repeated solution of large eigen- and linear systems with increased accuracy, and better initial guesses available, in successive iterations. Our approach employs a finite-element basis set, and so produces sparse (generalized) eigenproblems and linear systems. We discuss the effectiveness of steepest descent and conjugate gradient based approaches with various levels of preconditioning in this context.