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

Abstract

Our problem is to compute an approximation to the largest eigenvalue of an n x n large symmetric positive definite matrix with relative error at most c. We consider only algorithms that use Krylov inforrpation [b, Ab,. .. , Akb] consisting of k matrix-vector multiplications for some unit vector b. 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 we study the power and Lanczos algorithms. For the power algorithm (method) we prove sharp bounds on the average relative error and on the probabilistic relative failure. For the Lanczos algorithm we present only upper bounds. In particular, In(n)Jk characterizes the average relative error of the power algorithm, whereas O((ln(n)jk)l) 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. We show that the probabilistic relative •Supported in part by the National Science Foundation under Grant DCR-86-03674. failure goes to zero roughly as vIn(l-c)k for the power algorithm and at most as ..;n e-(2k-1)y'i for the Lanczos algorithm. These bounds are for a worst case distribution of eigenvalues which may depend on k. vVe also study the behavior in the average and probabilistic cases of the two algorithms for a fixed matrix A 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.

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Optimizing over the Growing Spectrahedron

“…The low rank factorization of X i is obtained in the algorithm via Furthermore, it is known that the function ApproxEV(M, ε) that computes an approximate eigenvector with guarantee ε can be implemented using the Lanczos method that runs in timeÕ N √ L √ ε and returns a valid approximation with high probability, where N is the number of non-zero entries in the matrix M ∈ R n×n and L is a bound on the largest eigenvalue of M , see [11]. Hence, we can conclude with the following corollary.…”

Untitled

“…In the Lanczos method, each iteration can be implemented in one pass over A, whereas our algorithm requires two passes over A in each iteration. Kuczyński and Woźniakowski [27] prove that the Lanczos method, with a randomly chosen starting vector, outputs a vector…”