Skip to main content
SpringerLink
Log in

Generalization of the Cramer Formula

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

We apply the method of parametrized continued fractions to the solution of systems of linear algebraic equations on the basis of their Liouville–Neumann formal power series. We construct an analog of the Cramer formula, which is also applicable to the cases of singular, ill-posed, and rectangular matrices.

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. G. Cramer, Introduction à l'Analyse des Lignes Courbes Geneva (1750).

  2. A. G. Kurosh, A Course in Higher Algebra [in Russian], Nauka, Moscow (1965).

    Google Scholar 

  3. L. A. Zadeh and C. A. Desoer, Linear System Theory. The State Space Approach McGraw-Hill, New York (1963).

    Google Scholar 

  4. W. B. Jones and W. J. Thron, Continued Fractions. Analytic Theory and Applications [Russian translation], Mir, Moscow (1985).

    Google Scholar 

  5. G. A. Baker, Jr., and P. Graves-Morris, Padé Approximants Addison-Wesley, London (1981).

    Google Scholar 

  6. M. S. Syavavko, “J-fractional regularization of linear ill-posed equations,” Ukr. Mat. Zh. 48, No. 8, 1130-1143 (1996).

    Google Scholar 

  7. M. S. Syavavko, Integral Continued Fractions [in Ukrainian], Naukova Dumka, Kiev (1994).

    Google Scholar 

  8. M. S. Syavavko, T. V. Pasechnik, and O. M. Rybyts'ka, “Pseudoinverse operator and rational algorithms for the normal solution of Fredholm integral equations of the first kind,” Élektron. Modelir. 17, No. 1, 10-16 (1995).

    Google Scholar 

  9. F. R. Gantmakher, Theory of Matrices [in Russian], Nauka, Moscow (1967).

  10. M. P. Drasin, “Pseudoinverses in associative rings and semigroups,” Amer. Math. Monthly 65, 506-515 (1958).

    Google Scholar 

  11. A. N. Tikhonov and V. Ya. Arsenin, Methods for Solution of Ill-Posed Problems [in Russian], Nauka, Moscow (1986).

    Google Scholar 

  12. F. Natterer, The Mathematics of Computerized Tomography Wiley, New York (1986).

    Google Scholar 

  13. V. A. Chechkin, Mathematical Informatics [in Russian], Nauka, Moscow (1991).

    Google Scholar 

  14. N. R. Drapper and H. Smith, Applied Regression Analysis Wiley, New York (1981).

    Google Scholar 

  15. E. L. Perchik, “A stable procedure for the solution of integral equations of the first kind without using the concept of finite perturbations,” in: Proceedings of the Mathematical Symposium on the Method of Discrete Singularities [in Russian], Feodosiya (1977).

  16. A. V. Khovanskii, “Regularized Greville method and its applications in transmission computerized tomography,” Mat. Model. 8, No. 11, 109-118 (1996).

    Google Scholar 

  17. M. S. Syavavko and O. M. Rybyts'ka, Mathematical Simulation under Conditions of Indeterminacy [in Ukrainian], Ukains'ki Tekhnolohii, Lviv (2000).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lviv Agricultural University, Dublyany

    M. S. Syavavko

Authors
  1. M. S. Syavavko
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Syavavko, M.S. Generalization of the Cramer Formula. Ukrainian Mathematical Journal 53, 762–784 (2001). https://doi.org/10.1023/A:1012582301200

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1012582301200

Keywords

Search

Navigation