Convergence properties of block GMRES and matrix polynomials

This paper studies convergence properties of the block GMRES algorithm when applied to nonsymmetric systems with multiple right-hand sides. A convergence theory is developed based on a representation of the method using matrix-valued polynomials. Relations between the roots of the residual polynomial for block GMRES and the matrix epsilon-pseudospectrum are derived, and illustrated with numerical experiments. The role of invariant subspaces in the effectiveness of block methods is also discussed.