Deflated and restarted symmetric Lanczos methods for eigenvalues and linear equations with multiple right-hand sides

TL;DR

This paper introduces a deflated restarted symmetric Lanczos algorithm that efficiently computes eigenvalues and solves linear systems with multiple right-hand sides, especially for large matrices from quantum chromodynamics.

Contribution

It presents a novel deflated restarted Lanczos method that improves eigenvector computation and linear system solving with multiple right-hand sides, addressing storage and convergence issues.

Findings

Effective eigenvector computation with restarted Lanczos

Good convergence for linear systems despite restarting

Successful application to large quantum chromodynamics matrices

Abstract

A deflated restarted Lanczos algorithm is given for both solving symmetric linear equations and computing eigenvalues and eigenvectors. The restarting limits the storage so that finding eigenvectors is practical. Meanwhile, the deflating from the presence of the eigenvectors allows the linear equations to generally have good convergence in spite of the restarting. Some reorthogonalization is necessary to control roundoff error, and several approaches are discussed. The eigenvectors generated while solving the linear equations can be used to help solve systems with multiple right-hand sides. Experiments are given with large matrices from quantum chromodynamics that have many right-hand sides.