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On computing the best least squares solutions in Hilbers space

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Summary

An iterative method for computing the best least squares solution ofAx=b, for a bounded linear operatorA with closed range, is formulated and studied in Hilbert space. Convergence of the method is characterized in terms ofK U-positive definite operators. A discretization theory for the best least squares problems is presented.

Riassunto

Un metodo iterativo per computare “best least squares solution” diAx=b, doveA è un operatore lineare e limitato con un insieme chiuso di valori, è formulato e studiato nello spazio di Hilbert. La convergenza del metodo è caratterizzata in termini di “K U−positive definite operators”. Una teoria di discretizzazione per il problema di “best least squares” è presentata.

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References

  1. E. Arghiriade,Sur l’inverse generalisee d’un operateur lineaire dans les espaces de Hilbert, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. ser VII, vol. XLVI (1969), 471–477.

    Google Scholar 

  2. A. Ben-Israel,A note on an iterative method for generalized inversion of matrices, Math. Comp. 20 (1966), 439–440.

    Article  MATH  Google Scholar 

  3. A. Berman and R. J. Plemmons,Cones and iterative methods for best least squares solutions of linear systems, Centre de Recherches Mathématiques, Université de Montréal, Montréal (June, 1972).

    Google Scholar 

  4. I. Erdelyi and A. Ben-Israel,Estremal solutions of linear equations and generalized inversion between Hilbert spaces, J. Math. Ann. Appl. 39 (1972), 298–313.

    Article  MATH  MathSciNet  Google Scholar 

  5. G. H. Golub,Numerical methods for solving linear least squares problems, Numer. Math. 7 (1965), 206–216.

    Article  MATH  MathSciNet  Google Scholar 

  6. W. J. Kammerer and M. Z. Nashed,On the convergence of the conjugate gradient method for singular linear operator equations, SIAM J. Numer Anal. 9 (1972), 165–181.

    Article  MathSciNet  Google Scholar 

  7. L. V. Kantorovich and G. P. Akilov,Functional Analysis in Normed Spaces, Translated from Russian by D. E. Brown, Pergamon Press, Oxford, 1964.

    MATH  Google Scholar 

  8. V. J. Katkovnik,The method of least squares in a problem of estimation of the parameters of a nonlinear model, in Cybernetics and Computing Techniques, No. 5: Complex Control Systems (in Russian), pp. 79–91.

  9. A. E. Martyniuk,Certain new applications of the methods of Galerkin type, Mat. Sb. 49 (1959), 85–108 (in Russian).

    MathSciNet  Google Scholar 

  10. M. Z. Nashed,Steepest descent for singular operator equations, SIAM J. Num. Anal. 7 (1970), 358–362.

    Article  MATH  MathSciNet  Google Scholar 

  11. W. V. Petryshyn,On the extrapolated Jacobi or simultaneous displacements method in the solution of matrix and operator equations, Math. Comp. 19 (1965), 37–56.

    Article  MATH  MathSciNet  Google Scholar 

  12. W. V. Petryshyn,On generalized inverses and on the uniform convergence of (1−βK) n with applications to iterative methods, J. Math. Anal. Appl. 18 (1967), 417–439.

    Article  MATH  MathSciNet  Google Scholar 

  13. D. W. Showalter and A. Ben-Israel,Representation and computation of the generalized inverse of a bounded linear operator between Hilbert spaces. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., ser VIII, vol. XLVIII (1970), 120–130.

    Google Scholar 

  14. I. D. Sofronov,On approximate solution of singular integral equations, Dokladi Ak. Nauk. SSSR, 111 (1956), 37–39 (in Russian).

    MATH  MathSciNet  Google Scholar 

  15. J. L. Soulé,Linear Operators in Hilber Space, Gordon and Breach, Notes on mathematics and its applications New York, 1968.

  16. A. E. Taylor,Introduction to Functional Analysis, John Wiley, New York, 1958.

    MATH  Google Scholar 

  17. R. S. Varga,Matrix Iterative Analysis, Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1962.

    Google Scholar 

  18. E. P. Wigner,On weakly positive matrices, Canad. J. Math. 15 (1963), 313–317.

    MATH  MathSciNet  Google Scholar 

  19. J. H. Wilkinson,Rounding Errors in Algebraic Processes, Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1963.

    MATH  Google Scholar 

  20. S. Zlobec,Contributions to Mathematical Programming and Generalized Inversion, Ph. D. Thesis, Northwestern University, Evanston, Illinois, 1970.

    Google Scholar 

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  1. Montreal, Canada

    S. Zlobec

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  1. S. Zlobec
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Introduced by B. Pettineo.

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Zlobec, S. On computing the best least squares solutions in Hilbers space. Rend. Circ. Mat. Palermo 25, 256–270 (1976). https://doi.org/10.1007/BF02849512

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  • DOI: https://doi.org/10.1007/BF02849512

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