Cybernetics and Systems Analysis

, Volume 44, Issue 6, pp 863–874 | Cite as

Reliability analysis of computer solutions of systems of linear algebraic equations with approximate initial data

  • A. N. Khimich
  • E. A. Nikolaevskaya
Systems Analysis

A weighted least squares problem {ie863-01} with positive definite weights M and N is considered, where A ∈ Rm×n is a rank-deficient matrix, b ∈ Rm. The hereditary, computational, and global errors of a weighted normal pseudosolution are estimated for perturbed initial data, including the case where the rank of the perturbed matrix varies.


weighted pseudoinverse matrix weighted normal pseudosolution weighted least squares problem global error 


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  1. 1.
    A. Ben Israel and T. N. E. Greville, Generalized Inverse: Theory and Applications, Springer Verlag, New York (2003).MATHGoogle Scholar
  2. 2.
    A. Albert, Regression, Pseudoinversion, and Recurrent Estimation [Russian translation], Nauka, Moscow (1977).Google Scholar
  3. 3.
    N. F. Kirichenko, “Analytical representation of perturbations of pseudoinverse matrices,” Cybern. Syst. Analysis, 33, No. 2, 230–238 (1997).MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    I. V. Sergienko, E. F. Galba, and V. S. Deineka, “Limiting representations of weighted pseudoinverse matrices with positive definite weights. Problem regularization,” Cybern. Syst. Analysis, 39, No. 6, 816–830 (2003).MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Kh. D. Ikramov and M. Matin Far, “Updating the minimum-norm solutions to the recursive least squares problem with linear equality constraints,” Comp. Math. Math. Phys., 44, No. 10, 1640–1648 (2004).MathSciNetGoogle Scholar
  6. 6.
    G. H. Golub and C. F. Van Loan, Matrix Computations, The Johns Hopkins University Press, Baltimore (1996).MATHGoogle Scholar
  7. 7.
    S. K. Godunov, A. G. Antonov, O. P. Kirilyuk, and V. N. Kostin, Guaranteed Accuracy of the Solution of Systems of Linear Algebraic Equations in Euclidean Spaces [in Russian], Nauka, Novosibirsk (1992).Google Scholar
  8. 8.
    L. Elden, “Perturbation theory for the least squares problem with linear equality constraints,” SIAM J. Numer. Anal., 17, 338–350 (1980).MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    A. Bjork, Numerical Methods for Least Squares Problems, SIAM, Philadelphia, PA (1996).Google Scholar
  10. 10.
    A. N. Malyshev, An Introduction to Computational Linear Algebra [in Russian], Nauka, Novosibirsk (1991).Google Scholar
  11. 11.
    V. V. Voevodin and Yu. A. Kuznetsov, Matrices and Computations [in Russian], Nauka, Moscow (1989).Google Scholar
  12. 12.
    A. N. Khimich, “Perturbation bounds for the least squares problem,” Cybern. Syst. Analysis, 32, No. 3, 434–436 (1996).MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    A. N. Khimich, S. A. Voitsekhovskii, and R. N. Brusnikin, “Reliability of solutions for linear mathematical models with approximate initial data,” Matem. Mashiny i Sistemy, No. 3, 3–17 (2004).Google Scholar
  14. 14.
    L. Elden, “A weighted pseudoinverse, generalized singular values and constrained least squares problems,” BIT. 22, 487–502 (1982).MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Y. Wei and H. Wu, “Expression for the perturbation of the weighted Moore-Penrose inverse,” Comput. and Math. with Appl., 39, 13–18 (2000).MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    M. Wei, Supremum and Stability of Weighted Pseudoinverses and Weighted Least Squares Problems: Analysis and Computations, Huntington, New York (2001).Google Scholar
  17. 17.
    D. Wang, “Some topics on weighted Moore-Penrose inverse, weighted least squares and weighted regularized Tikhonov problems,” Appl. Math. and Comput., 157, 243–267 (2004).MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Y. Wei and D. Wang, “Condition numbers and perturbation of weighted Moore-Penrose inverse and weighted least squares problem,” Appl. Math. and Comput., 145, 45–58 (2003).MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    C. F. Van Loan, “Generalizing the singular value decomposition,” SIAM J. Numer. Anal., 13, 76–83 (1976).MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    I. N. Molchanov and E. F. Galba, “A weighted pseudoinverse for complex matrices,” Ukr. Math. J., 35, No. 1, 46–50 (1983).MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    C. L. Lawson and R. J. Henson, Solving Least Squares Problems, Prentice-Hall, Englewood Cliffs, NJ (1986).Google Scholar
  22. 22.
    A. N. Khimich and E. A. Nikolaevskaya, “Estimating the error of the solution of the weighted lest squares problem,” Komp. Matem., No. 3, 36–45 (2006).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.V. M. Glushkov Institute of CyberneticsNational Academy of Sciences of UkraineKyivUkraine

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