Estimating the Largest Eigenvalue by the Power and Lanczos Algorithms with a Random Start

Abstract

This paper addresses the problem of computing an approximation to the largest eigenvalue of an $n \times n$ large symmetric positive definite matrix with relative error at most $\varepsilon $. Only algorithms that use Krylov information $[b,Ab, \cdots ,A^k b]$ consisting of k matrix-vector multiplications for some unit vector b are considered. If the vector b is chosen deterministically, then the problem cannot be solved no matter how many matrix-vector multiplications are performed and what algorithm is used. If, however, the vector b is chosen randomly with respect to the uniform distribution over the unit sphere, then the problem can be solved on the average and probabilistically. More precisely, for a randomly chosen vector b, the power and Lanczos algorithms are studied. For the power algorithm (method), sharp bounds on the average relative error and on the probabilistic relative failure are proven. For the Lanczos algorithm only upper bounds are presented. In particular, $\ln ( n )/( n ) k $ characterizes the average relative error of the power algorithm, whereas $O( ( \ln ( n )/ k )^2 )$ is an upper bound on the average relative error of the Lanczos algorithm. In the probabilistic case, the algorithm is characterized by its probabilistic relative failure, which is defined as the measure of the set of vectors b for which the algorithm fails. It is shown that the probabilistic relative failure goes to zero roughly as $\sqrt n (1 - \varepsilon )^k $ for the power algorithm and at most as $\sqrt n e^{ - (2k - 1)\sqrt \varepsilon } $ for the Lanczos algorithm. These bounds are for a worst case distribution of eigenvalues which may depend on k. The behavior in the average and probabilistic cases of the two algorithms for a fixed matrix A is also studied as the number of matrix-vector multiplications k increases. The bounds for the power algorithm depend then on the ratio of the two largest eigenvalues and their multiplicities. The bounds for the Lanczos algorithm depend on the ratio between the difference of the two largest eigenvalues and the difference of the largest and the smallest eigenvalues.