On stagnation of the DGMRES method

Let a be an n-by-n matrix with index α > 0 and b ∈ cn. in this paper,the problem of stagnation of the dgmres method for the singular linearsystem ax = b is considered. we show that dgmres(a, b, α) has partialstagnation of order at least k if and only if (0, . . . , 0) belongs to the the jointnumerical range of matrices {b^α+1, . . . ,b^α+k}, where b is a compressionof a to the range of a^α. also, we characterize the nonsingular part of amatrices a such that dgmres(a, b, α) does not stagnate for all b ∈ cn.moreover, a sufficient condition for non-existence of real stagnation vectorsb ∈ r(a^α) for the dgmres method is presented, and the dgmresstagnation of special matrices are studied.