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

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Numerical Infinities and Infinitesimals in Optimization

Part of the book series: Emergence, Complexity and Computation ((ECC,volume 43))

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Abstract

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.

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Notes

  1. 1.

    More correctly, we urge to remark that the expressions (33) are obtained neglecting in the quantity \(\alpha _{k+1}\) the infinitesimal terms, i.e. those terms containing negative powers of \(s\textcircled {1}\), that are indeed negligibly small due to the degenerate Step k in CG\(_{\textcircled {1}}\).

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Acknowledgements

The author is thankful to the Editors of the present volume for their great efforts and constant commitment. The author is also grateful for the support he received by both the National Research Council–Marine Technology Research Institute (CNR-INM), and the National Research Group GNCS (Gruppo Nazionale per il Calcolo Scientifico) within IN\(\delta \)AM, Istituto Nazionale di Alta Matematica, Italy.

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  1. Department of Management, University Ca’ Foscari of Venice, Venice, Italy

    Giovanni Fasano

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  1. Giovanni Fasano
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Correspondence to Giovanni Fasano .

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Editors and Affiliations

  1. Dipartimento di Ingegneria Informatica, Modellistica, Elettronica e Sistemistica, University of Calabria, Rende, Italy

    Yaroslav D. Sergeyev

  2. Scuola di Scienze e Tecnologie, Università degli Studi di Camerino, Camerino, Italy

    Renato De Leone

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Fasano, G. (2022). Krylov-Subspace Methods for Quadratic Hypersurfaces: A Grossone–based Perspective. In: Sergeyev, Y.D., De Leone, R. (eds) Numerical Infinities and Infinitesimals in Optimization. Emergence, Complexity and Computation, vol 43. Springer, Cham. https://doi.org/10.1007/978-3-030-93642-6_4

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