A FAST ITERATIVE ALGORITHM FOR NEAR-DIAGONAL EIGENVALUE PROBLEMS.

We introduce a novel eigenvalue algorithm for near-diagonal matrices inspired by Rayleigh-Schrödinger perturbation theory and termed iterative perturbative theory (IPT). Contrary to standard eigenvalue algorithms, which are either "direct" (to compute all eigenpairs) or "iterative" (to compute just a few), IPT computes any number of eigenpairs with the same basic iterative procedure. Thanks to this perfect parallelism, IPT proves more efficient than classical methods (LAPACK or CUSOLVER for the full-spectrum problem, preconditioned Davidson solvers for extremal eigenvalues). We give sufficient conditions for linear convergence and demonstrate performance on dense and sparse test matrices, including one from quantum chemistry. The code is available at http://github.com/msmerlak/IterativePerturbationTheory.jl. [ABSTRACT FROM AUTHOR]