Abstract

This thesis explores the connection between Crouzeix's conjecture and the convergence of the GMRES algorithm. GMRES is a popular iterative method for solving linear systems and is one of the many Krylov methods. Despite its popularity, the convergence of GMRES is not completely understood. While the spectrum can in some cases be a good indicator of convergence, it has been shown that in general, the spectrum does not provide sufficient information to fully explain the behavior of GMRES iterations. Other sets associated with a matrix that can also help predict convergence are the pseudospectrum and the numerical range. This work focuses on convergence bounds obtained by considering the latter. In particular, it focuses on the application of Crouzeix's conjecture, which relates the norm of a matrix polynomial to the size of that polynomial over the numerical range, to describing GMRES convergence.

Degree

MS

College and Department

Physical and Mathematical Sciences; Mathematics

Rights

http://lib.byu.edu/about/copyright/

Date Submitted

2011-07-13

Document Type

Thesis

Handle

http://hdl.lib.byu.edu/1877/etd4678

Keywords

GMRES, Michel Crouzeix, Faber Polynomials, Complex Approximation, Krylov Subspace, Convergence, Iterative Methods, Linear Systems

Included in

Mathematics Commons

Share

COinS