Title: A Lanczos Procedure for Approximating Eigenvalues of Large Stochastic Matrices
Author: William DeMeo
Advisor: Jonathan Goodman
Institution: Courant Institute of Mathematical Sciences, NYU
Description: A dissertation submitted in partial satisfaction of the requirements for the degree of Master of Science in Mathematics in the Graduate Division of the New York University.
The rate at which a Markov chain converges to a given probability distribution has long been an active area of research. Well known bounds on this rate of convergence involve the subdominant eigenvalue of the chain's underlying transition probability matrix. However, many transition probability matrices are so large that we are unable to store even a vector of the matrix in fast computer memory. Thus, traditional methods for approximating eigenvalues are rendered useless.
In this paper we demonstrate that, if the Markov chain is reversible, and we understand the structure of the chain, we can derive the coefficients of the traditional Lanczos algorithm without storing a single vector. We do this by considering the variational properties of observables on the chain's state space. In the process we present the classical theory which relates the information contained in the Lanczos coefficients to the eigenvalues of the Markov chain.
The main document is msthesis.pdf. It is the only file you need if you want to read the thesis.
The other files in the repository are the LaTeX source code (msthesis.tex) and the computer programs (in the src directory) for running the simulations described in the thesis.
All of the LaTeX source code is contained in a single file, msthesis.tex, which can be compiled with the following command
to produce the msthesis.pdf file, assuming you have a reasonably good installation of the TeXLive package or its equivalent.
The computer program files are in the src directory. I have not tested them recently. If you try them and encounter a problem, please submit an issue.