TITLE: the complexity of Lanczos method
QUESTION [6 upvotes]: Hi, all
I am working on an algorithm which uses Lanczos method to compute K smallest eigenvalue(and their eigenvectos) of a sparse matrix, just want some information or links about the complexity of Lanczos method.
Thank you

REPLY [5 votes]: Complexity analysis of Lanczos seems to be hard to find in the literature. Here are two leads, that might help a bit.
For the largest eigenvalue, you might find the complexity analysis in the following paper to be useful.
Estimating the Largest Eigenvalue by the Power and Lanczos Algorithms with a Random Start by J. Kuczyński and H. Woźniakowski
A very important point raised in the above paper is that even if an eigenvalue is easily estimable, its corresponding eigenvector can be computationally hard to estimate!
Also see Lemma 2 of Fast algorithms for approximate semidefinite programming ... by S. Arora, E. Hazan, and S. Kale, for additional results.

REPLY [2 votes]: Chapter 9 = Lanczos Methods, of "matrix computations", Golub, van Loan, John Hopkins univ press.