Real-Arithmetic Sketched Eigensolvers for Symmetric Matrices

Abstract

In this work, we develop sketched variants of subspace iteration and TraceMIN, called sSI and sTraceMIN, for real symmetric eigenvalue problems by combining basis whitening with the sketched Rayleigh-Ritz (sRR) procedure. The nonsymmetric projected matrix in sRR may have nonreal eigenpairs, leading to complex arithmetic and additional computational costs. We overcome this difficulty using a real Schur formulation of sRR that keeps the iterations in real arithmetic. We further develop procedures for characteristic subspace selection and convergence monitoring within this formulation. On the theoretical side, we discuss suitable subspace embedding assumptions for sketched eigensolvers that use a fixed sketching matrix throughout the iterations. Under a mild embedding assumption, we establish local convergence results for sSI and sTraceMIN. Numerical experiments demonstrate runtime savings of up to 39% for sSI and 21% for sTraceMIN relative to their standard counterparts on the test problems.