Krylov-Subspace Methods for Quadratic Hypersurfaces: A Grossone–based Perspective

We study the role of the recently introduced infinite number grossone, to deal with two renowned Krylov-subspace methods for symmetric (possibly indefinite) linear systems. We preliminarily explore the relationship between the Conjugate Gradient (CG) method and the Lanczos process, along with their specific role of yielding tridiagonal matrices which retain large information on the original linear system matrix. Then, we show that on one hand there is not immediate evidence of an advantage from embedding grossone within the Lanczos process. On the other hand, coupling the CG with grossone shows clear theoretical improvements. Furthermore, reformulating the CG iteration through a grossone-based framework allows to encompass also a certain number of Krylov-subspace methods relying on conjugacy among vectors. The last generalization remarkably justifies the use of a grossone-based reformulation of the CG to solve also indefinite linear systems. Finally, pairing the CG with the algebra of grossone easily provides relevant geometric properties of quadratic hypersurfaces.