Estimating a Largest Eigenvector by Polynomial Algorithms with a Random Start

In 7] and 8], the power and Lanczos algorithms with random start for estimating the largest eigenvalue of an n n large symmetric positive deenite matrix were analyzed. In this paper we continue this study by estimating an eigenvector corresponding to the largest eigenvalue. We analyze polynomial algorithms using Krylov information for two error criteria: the randomized error and the randomized residual error. For the randomized error, in contrast to 7] and 8], we prove that it is not possible to get distribution-free bounds. In fact, there exists a matrix for which all polynomial algorithms fail for approximating an eigenvector corresponding to the largest eigenvalue as long as the number of steps is less than n. This shows that the problem of estimating such an eigenvector is much harder than the problem of estimating the largest eigenvalue, and that even the random 1 start does not help. For the randomized error, the bounds must depend on the distribution of eigenvalues of the matrix. For the power algorithm, bounds depending on the ratio of the two largest eigenvalues are presented in 2]. We supply such bounds for the Lanczos algorithm. For the randomized residual error, distribution-free bounds exist and we supply such bounds for the power and Lanczos algorithms. Asymptotic bounds, as well as bounds depending on the ratio of the two largest eigenvalues, are also presented.