  Polynomially filtered exact diagonalization method (POLFED) for large sparse
matrices is introduced. The algorithm finds an optimal basis of a subspace
spanned by eigenvectors with eigenvalues close to a specified energy target by
a spectral transformation using a high order polynomial of the matrix. The
memory requirements scale better with system size than in the state-of-the-art
shift-invert approach. The potential of POLFED is demonstrated examining
many-body localization transition in 1D interacting quantum spin-1/2 chains. We
investigate the disorder strength and system size scaling of Thouless time.
System size dependence of bipartite entanglement entropy and of the gap ratio
highlights the importance of finite-size effects in the system. We discuss
possible scenarios regarding the many-body localization transition obtaining
estimates for the critical disorder strength.
