The computation of a number of the smallest eigenvalues of large and sparse matrices is crucial in various scientific applications, as the Finite Element solution of PDEs, electronic structure calculations or Laplacian of graphs, to mention a few. We propose in this contribution a parallel algorithm which is based on the spectral low-rank modification of a factorized sparse inverse preconditioner (RFSAI) to accelerate Newton-based iterative eigensolvers. Numerical results onto matrices arising from various realistic problems with size up to 5 million unknowns and 2.2×10⁸ nonzero elements account for the efficiency and the scalability of the proposed RFSAI–updated preconditioner.