Abstract
Krylov subspace methods are popular iterative methods to solve large sparse linear systems in the real-world computations due to their cheap memory requirement and computational cost. In this paper, we discuss the solution of singular systems. We will show that the consistency of a singular linear system is not a sufficient condition for a Krylov subspace method to successfully find a solution to the system. The choice of initial guess is a crucial step. If the initial guess is properly chosen, a Krylov method almost surely converges to find a solution from the point of view of probability, otherwise a Krylov subspace method surely diverges. Moreover, our algorithm applied to parallel calculation is discussed in the paper.
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