Krylov Method Revisited with an Application to the Localization of Eigenvalues

Our aim is to localize matrix eigenvalues in the sense that we build a sufficiently small neighborhood for each of them (or for a cluster), through not prohibitively expensive computations. Our results enter the framework started with Gerschgorin disks and deals at the present time with pseudospectra. The set of theoretical tools we have chosen to use does not avoid the notion of the characteristic polynomial. Certainly, when some computations are performed on it, the well-known ill-conditioning of its coefficients with respect to the matrix entries is properly and carefully handled.