Randomized Subspace Iteration Method for Eigenvalue Problems

Abstract

This thesis presents the randomized subspace iteration method for eigenvalue problems. In our analysis, we have considered symmetric positive definite eigenvalue problem. We present deterministic and probabilistic bounds for three quantities. First, we present the deterministic and probabilistic bounds for the canonical angles between the exact and the approximate eigenvector subspaces. Second, we give deterministic and probabilistic bounds for the sine of angle between the eigenvectors of the exact eigenvector subspace and the approximated eigenvector subspace. Third, we also present deterministic and probabilistic bounds for the accuracy of eigenvalues using the randomized subspace iteration. The probabilistic bounds are provided when a Gaussian random matrix is used as the initial subspace. Finally, we illustrate our theoretical results numerically using several different test matrices.