Computable Convergence Bounds for GMRES

Computable Convergence Bounds for GMRES

The purpose of this paper is to derive new computable convergence bounds for GMRES. The new bounds depend on the initial guess and are thus conceptually different from standard “worst-case” bounds. Most importantly, approximations to the new bounds can be computed from information generated during the run of a certain GMRES implementation. The approximations allow predictions of how the algorit...

How Descriptive Are Gmres Convergence Bounds?

Eigenvalues with the eigenvector condition number, the eld of values, and pseu-dospectra have all been suggested as the basis for convergence bounds for minimum residual Krylov subspace methods applied to non-normal coeecient matrices. This paper analyzes and compares these bounds, illustrating with six examples the success and failure of each one. Reened bounds based on eigenvalues and the eld...

Computable Bounds for Polynomial Ergodicity

Convergence analysis of the global FOM and GMRES methods for solving matrix equations $AXB=C$ with SPD coefficients

In this paper‎, ‎we study convergence behavior of the global FOM (Gl-FOM) and global GMRES (Gl-GMRES) methods for solving the matrix equation $AXB=C$ where $A$ and $B$ are symmetric positive definite (SPD)‎. ‎We present some new theoretical results of these methods such as computable exact expressions and upper bounds for the norm of the error and residual‎. ‎In particular‎, ‎the obtained upper...

On Mesh Independence of Convergence Bounds for Additive Schwarz Preconditioned Gmres

Additive Schwarz preconditioners, when including a coarse grid correction, are said to be optimal for certain discretized partial differential equations, in the sense that bounds on the convergence of iterative methods are independent of the mesh size h. Cai and Zou [Numer. Linear Algebra Appl., 9:379–397, 2002] showed with a one-dimensional example that in the absence of a coarse grid correcti...