Fast and Accurate Randomized Algorithms for Linear Systems and Eigenvalue Problems
Yuji Nakatsukasa, Joel A. Tropp
Abstract
This paper develops a class of algorithms for general linear systems and eigenvalue problems. These algorithms apply fast randomized dimension reduction (``sketching") to accelerate standard subspace projection methods, such as GMRES and Rayleigh--Ritz. This modification makes it possible to incorporate nontraditional bases for the approximation subspace that are easier to construct. When the basis is numerically full rank, the new algorithms have accuracy similar to classic methods but run faster and may use less storage. For model problems, numerical experiments show large advantages over the optimized MATLAB routines, including a 70\\times speedup over gmres and a 10\\times speedup over eigs.
Topics & Concepts
MathematicsEigenvalues and eigenvectorsAlgorithmLinear systemDivide-and-conquer eigenvalue algorithmApplied mathematicsAlgebra over a fieldCalculus (dental)Mathematical analysisPure mathematicsDentistryQuantum mechanicsMedicinePhysicsStochastic Gradient Optimization TechniquesSparse and Compressive Sensing TechniquesComplexity and Algorithms in Graphs