Matrix Powers Kernels for Thick-Restart Lanczos with Explicit External Deflation

Abstract

Some scientific and engineering applications need to compute a large number of eigenpairs of a large Hermitian matrix. Though the Lanczos method is effective for computing a few eigenvalues, it can be expensive for computing a large number of eigenpairs (e.g., in terms of computation and communication). To improve the performance of the method, in this paper, we study an s-step variant of thick-restart Lanczos (TRLan) combined with an explicit external deflation (EED). The s-step method generates a set of s basis vectors at a time and reduces the communication costs of generating the basis vectors. We then design a specialized matrix powers kernel (MPK) that further reduces the communication and computational costs by taking advantage of the special properties of the deflation matrix. We conducted numerical experiments of the new TRLan eigensolver using synthetic matrices and matrices from electronic structure calculations. The performance results on the Cori supercomputer at the National Energy Research Scientific Computing Center (NERSC) demonstrate the potential of the specialized MPK to significantly reduce the execution time of the TRLan eigensolver. The speedups of up to 3.1× and 5.3× were obtained in our sequential and parallel runs, respectively.