Randomized block Krylov methods for approximating extreme eigenvalues
Randomized block Krylov subspace methods form a powerful class of algorithms for computing the extreme eigenvalues of a symmetric matrix or the extreme singular values of a general matrix. The purpose of this paper is to develop new theoretical bounds on the performance of randomized block Krylov subspace methods for these problems. For matrices with polynomial spectral decay, the randomized block Krylov method can obtain an accurate spectral norm estimate using only a constant number of steps (that depends on the decay rate and the accuracy). Furthermore, the analysis reveals that the behavior of the algorithm depends in a delicate way on the block size. Numerical evidence confirms these predictions.
© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021. Received 30 April 2018; Revised 27 September 2021; Accepted 02 October 2021; Published 03 December 2021. This paper was produced under a subcontract to ONR Award 00014-16-C-2009. The author thanks Shaunak Bopardikar, Petros Drineas, Ilse Ipsen, Youssef Saad, Fadil Santosa, and Rob Webber for helpful discussions and feedback. Mark Embree provided very detailed, thoughtful comments that greatly improved the quality of the paper.
Submitted - 2110.00649.pdf