We provide a unified description of the Fermi operator expansion and recursion methods within the technique of spectral quadrature. Through rigorous error estimates, we prove that this approach is linear-scaling, stable and exponentially convergent. We use this analysis to determine the influence of smearing, band-gap, position of Fermi energy, and spectral width of the Hamiltonian on the convergence rates obtained in practical calculations. Additionally, we establish that super-geometric convergence can be achieved when the erfc function is used for smearing. We validate the spectral quadrature method and the accuracy of our analysis by means of selected examples. (c) 2013 Elsevier B.V. All rights reserved.