The block grade of a block Krylov space

The aim of the paper is to compile and compare basic theoretical facts on Krylov subspaces and block Krylov subspaces. Many Krylov (sub)space methods for solving a linear system Ax=b have the property that in exact computer arithmetic the true solution is found after ν iterations, where ν is the dimension of the largest Krylov subspace generated by A from r0, the residual of the initial approximation x0. This dimension is called the grade of r0 with respect to A. Though the structure of block Krylov subspaces is more complicated than that of ordinary Krylov subspaces, we introduce here a block grade for which an analogous statement holds when block Krylov space methods are applied to linear systems with multiple, say s, right-hand sides. In this case, the s initial residuals are bundled into a matrix R0 with s columns. The possibility of linear dependence among columns of the block Krylov matrix , which in practical algorithms calls for the deletion (or, deflation) of some columns, requires extra care. Relations between grade and block grade are also established, as well as relations to the corresponding notions of a minimal polynomial and its companion matrix.