Skip to main content
SpringerLink
Log in

Orthogonal Projectors and Systems of Linear Algebraic Equations

  • Published:
Numerical Analysis and Applications Aims and scope Submit manuscript

ABSTRACT

In this paper, an operator iterative procedure for constructing an orthogonal projection of a vector onto a given subspace is proposed. The algorithm is based on Euclidean orthogonalization of power sequences of a special linear transform generated by an initial subspace. A numerical method based on this idea for solving consistent systems of linear algebraic equations is proposed. Numerical results are presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. Voevodin, V.V. and Kuznetsov, Yu.A., Matritsy i vychisleniya (Matrices and Calculations), Moscow: Nauka, 1984.

  2. Aleksandrov, P.S., Lektsii po analiticheskoi geometrii (Lectures on Analytical Geometry), St. Petersburg: Lan, 2008.

  3. Kireev, I.V., An Orthogonal Power Method of Solving the Partial Eigenproblem for a Symmetric Nonnegative Definite Matrix,Vych. Met. Progr., 2016, vol. 17, no. 1, pp. 44–54.

  4. Saad, Y., Iterative Methods for Sparse Linear Systems, 2nd ed., SIAM, 2003.

  5. Galàntai, A., Subspaces, Angles and Pairs of Orthogonal Projections, Lin. Multilin. Alg., 2008, vol. 56, no. 3, pp. 227–260.

  6. Shilov, G.E., Matematicheskii analiz. Konechnomernye lineynye prostranstva (Mathematical Analysis. Finite-Dimensional Linear Spaces), Moscow: Nauka, 1969.

  7. Kireev, I.V., Inexpensive Stopping Criteria in the Conjugate Gradient Method, Vych. Tekhnol., 2015, vol. 20, no. 2, pp. 44–55.

  8. Marcus, M. and Minc, H., A Survey of Matrix Theory and Matrix Inequalities, New York: Dover Publ., 1992.

  9. Kaczmarz, S., Angenäherte auflösung von systemen lineare Gleichungen, Bull. Int. Acad. Polonaise Sci. Lett., 1937, vol. 35, pp. 355–357.

  10. Babenko, V.N., Convergence of the Kaczmarz Projection Algorithm, Zh. Vych. Mat. Mat. Fiz., 1984, vol. 24, no. 10, pp. 1571–1573.

  11. Il’in, V.P., On Kaczmarz’s Iterative Method and Its Generalizations, Sib. Zh. Ind. Mat., 2006, vol. 9, no. 3, pp. 39–49.

  12. Il’in, V.P., Projection Methods in Krylov Subspaces,Zap. Nauch. Sem. POMI, 2018, vol. 472, pp. 103–119.

  13. Liu, J. and Wright, S.J., An Accelerated Randomized Kaczmarz Algorithm, Math. Comp., 2016, vol. 85, pp. 153–178.

  14. Quarteroni, A., Sacco, R., and Saleri, F., Numerical Mathematics. Texts in Applied Mathematics, vol. 37, New York: Springer, 2007.

  15. Liesen, J. and Strakos, Z., Krylov Subspace Methods: Principles and Analysis. Numerical Mathematics and Scientific Computation, Oxford: Oxford Univ. Press, 2013.

  16. Le Dret, H. and Lucquin, B., Partial Differential Equations: Modeling, Analysis and Numerical Approximation. International Series of Numerical Mathematics, Springer, 2016.

  17. Dautov, R.Z. and Karchevskii, M.M., Vvedenie v teoriyu metoda konechnyh elementov. Uchebnoe posobie (An Introduction to the Theory of the Finite Element Method: Textbook), Kazan: Kazan State Univ., 2004.

  18. Quarteroni, A. and Valli, A., Domain Decomposition Methods for Partial Differential Equations. Numerical Mathematics and Scientific Computation, Oxford Sci. Publ., 1999.

Download references

Author information

Authors and Affiliations

  1. Institute of Computational Modeling, Siberian Branch, Russian Academy of Sciences, 660036, Krasnoyarsk, Russia

    I. V. Kireev

  2. Institute of Mathematics and Fundamental Informatics, Siberian Federal University, 660041, Krasnoyarsk, Russia

    I. V. Kireev

Authors
  1. I. V. Kireev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. V. Kireev.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kireev, I.V. Orthogonal Projectors and Systems of Linear Algebraic Equations. Numer. Analys. Appl. 13, 262–270 (2020). https://doi.org/10.1134/S1995423920030064

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1995423920030064

Search

Navigation