Welcome to our webpage for Efficient Eigensolvers!
This summer, we are working on solving eigenvectors and eigenvalues for a given matrix. Based on the non-existance of the closed formula for the root of an arbitrary polynomial with degree 5 or more, we conclude there is no eigensolver that could solve the eigenvalues for an arbitrary matrix in finite steps even given the perfect accuracy.
Numerically, eigensolvers must be iterative.
For numerical experiments results, we collected page rank scores for subpages under several domains: https://icerm.brown.edu, https://en.wikipedia.com/wiki/, https://cnn.com/, etc. You can find the adjacency matrices and page rank results here.
To see how preprocessing a matrix like applying reduction could save the run time, we measured convergence rates for different eigensolvers with and without Hessenberg Reduction.
Hilbert matrix, a nightmare for generations of scientists working on improving the time complexity of the eigensolver, is revisited by our code. We visualized how bad our eigensolers are when facing this accuracy dilemma caused by Hilbert matrix.
We implemented two major kinds of eigensolvers:
We also introduced other variants involving different techniques, e.g. Rayleigh Quotient Iteration.
A common application for eigensolvers, more specifically, Power Iteration Method, is PageRank. By crawling all the subpages under a given domain, we generated a directed graph with vertices representing subpages and directed edges representing linkages by webcrawler. By manipulating this directed graph’s adjacency matrix and feeding it into the eigensolver, we could converge to the dominant eigenvalue’s eigenvectors, where each value could be regarded as an importance score.
The time complexity for each plain eigensolver is O(n^3), which is problematic for actual applications – when we are crawling 10 million pages and processing a 10 million dimensional matrix – the time consumed is not fun. Hessenberg Reduction comes to help.
Let’s face the tricky Hilbert matrix.
if you also want to know how to …