In this short paper, the authors report a new computational approach in the context of Density Functional Theory (DFT). It is shown how it is possible to speed up the self-consistent cycle (iteration) characterizing one of the most well-known DFT implementations: FLAPW. Generating the Hamiltonian and overlap matrices and solving the associated generalized eigenproblems $Ax = λBx$ constitute the two most time-consuming fractions of each iteration. Two promising directions, implementing the new methodology, are presented that will ultimately improve the performance of the generalized eigensolver and save computational time.