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We analyze the the convergence behavior of block GMRES and characterize thephenomenon of stagnation which is then related to the behavior of the block FOMmethod. We generalize the block FOM method to generate well-definedapproximations in the case that block FOM would normally break down, and thesegeneralized solutions are used in our analysis. This behavior is also relatedto the principal angles between the column-space of the previous block GMRESresidual and the current minimum residual constraint space. At iteration $j$,it is shown that the proper generalization of GMRES stagnation to the blocksetting relates to the columnspace of the $j$th block Arnoldi vector. Ouranalysis covers both the cases of normal iterations as well as block Arnoldibreakdown wherein dependent basis vectors are replaced with random ones.Numerical examples are given to illustrate what we have proven, including asmall application problem to demonstrate the validity of the analysis in a lesspathological case.

Stagnation of block GMRES and its relationship to block FOM