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Lanczos Conjugate-Gradient Method and Pseudoinverse Computation on Indefinite and Singular Systems

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  • G. Fasano

    (CNR
    Istituto Nazionale per Studi ed Esperienze di Architettura Navale (INSEAN))

Abstract

This paper extends some theoretical properties of the conjugate gradient-type method FLR (Ref. 1) for iteratively solving indefinite linear systems of equations. The latter algorithm is a generalization of the conjugate gradient method by Hestenes and Stiefel (CG, Ref. 2). We develop a complete relationship between the FLR algorithm and the Lanczos process, in the case of indefinite and possibly singular matrices. Then, we develop simple theoretical results for the FLR algorithm in order to construct an approximation of the Moore-Penrose pseudoinverse of an indefinite matrix. Our approach supplies the theoretical framework for applications within unconstrained optimization.

Suggested Citation

  • G. Fasano, 2007. "Lanczos Conjugate-Gradient Method and Pseudoinverse Computation on Indefinite and Singular Systems," Journal of Optimization Theory and Applications, Springer, vol. 132(2), pages 267-285, February.
    • Handle: RePEc:spr:joptap:v:132:y:2007:i:2:d:10.1007_s10957-006-9119-3
    DOI: 10.1007/s10957-006-9119-3
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    References listed on IDEAS

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    1. G. Fasano, 2005. "Planar Conjugate Gradient Algorithm for Large-Scale Unconstrained Optimization, Part 2: Application," Journal of Optimization Theory and Applications, Springer, vol. 125(3), pages 543-558, June.
    2. F. Lampariello & M. Sciandrone, 2003. "Use of the Minimum-Norm Search Direction in a Nonmonotone Version of the Gauss-Newton Method," Journal of Optimization Theory and Applications, Springer, vol. 119(1), pages 65-82, October.
    3. G. Fasano, 2005. "Planar Conjugate Gradient Algorithm for Large-Scale Unconstrained Optimization, Part 1: Theory," Journal of Optimization Theory and Applications, Springer, vol. 125(3), pages 523-541, June.
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    Citations

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    Cited by:

    1. Andrea Caliciotti & Giovanni Fasano & Florian Potra & Massimo Roma, 2020. "Issues on the use of a modified Bunch and Kaufman decomposition for large scale Newton’s equation," Computational Optimization and Applications, Springer, vol. 77(3), pages 627-651, December.
    2. Renato De Leone & Giovanni Fasano & Yaroslav D. Sergeyev, 2018. "Planar methods and grossone for the Conjugate Gradient breakdown in nonlinear programming," Computational Optimization and Applications, Springer, vol. 71(1), pages 73-93, September.
    3. Renato Leone & Giovanni Fasano & Massimo Roma & Yaroslav D. Sergeyev, 2020. "Iterative Grossone-Based Computation of Negative Curvature Directions in Large-Scale Optimization," Journal of Optimization Theory and Applications, Springer, vol. 186(2), pages 554-589, August.
    4. Giovanni Fasano, 2015. "A Framework of Conjugate Direction Methods for Symmetric Linear Systems in Optimization," Journal of Optimization Theory and Applications, Springer, vol. 164(3), pages 883-914, March.
    5. Giovanni Fasano & Massimo Roma, 2013. "Preconditioning Newton–Krylov methods in nonconvex large scale optimization," Computational Optimization and Applications, Springer, vol. 56(2), pages 253-290, October.
    6. Giovanni Fasano, 2008. "Notes on a 3-term Conjugacy Recurrence for the Iterative Solution of Symmetric Linear Systems," Working Papers 179, Department of Applied Mathematics, Università Ca' Foscari Venezia.

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