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BIT Numerical Mathematics

, Volume 19, Issue 2, pp 214–222 | Cite as

A modification of the secant rule derived from a maximum likelihood principle

  • F. M. Larkin
Article
  • 18 Downloads

Abstract

An estimate of a zero of a complex function, constructed from ordinate information at distinct abscissae, is found from a Maximum Likelihood estimate relative to a normal probability distribution induced by a weak Gaussian distribution on a related Hilbert space. In the case of two ordinate observations this leads to an estimator structurally similar to the Secant Rule, and asymptotically approaching that rule in certain limiting situations. A correspondingly modified version of Newton's method is also derived, and regional and asymptotic convergence results proved.

Keywords

Gaussian Distribution Probability Distribution Hilbert Space Computational Mathematic Maximum Likelihood Estimate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    N. Aronszain,Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404.Google Scholar
  2. 2.
    R. A. Fisher,On the mathematical foundation of theoretical statistics, Phil. Trans. Roy. Soc. 222 (1921), 309.Google Scholar
  3. 3.
    L. Gross,Measurable functions on a Hilbert space, Trans. Amer. Math. Soc. 105 (1962), 372–390.Google Scholar
  4. 4.
    F. M. Larkin,Gaussian measure in Hilbert space, and applications in numerical analysis. Rocky Mt. J. Math. 2 (1972), 397–421.Google Scholar
  5. 5.
    F. M. Larkin,Probabilistic error estimates in spline interpolation and quadrature, in Information Processing 74, 606–609, North-Holland, 1974.Google Scholar
  6. 6.
    F. M. Larkin,Some remarks on the estimation of quadratic functionals, in Theory of Approximation with Applications, 43–63, A. G. Law and B. N. Sahney (eds), Academic Press, New York, 1976.Google Scholar
  7. 7.
    F. M. Larkin,A further optimal property of natural polynomial splines, J. Approx. Th. 22 (1978), 1–8.Google Scholar
  8. 8.
    F. M. Larkin,Probabilistic estimation of poles and zeros of functions, TR-78-66, Computing & Information Science Department, Queen's University, Kingston, Ontario. To appear in J. Approx. Th.Google Scholar
  9. 9.
    H. Meschkowski,Hilbertsche Räume mit Kernfunktion, Springer-Verlag, Berlin, Göttingen and Heidelberg, 1962.Google Scholar
  10. 10.
    J. M. Ortega and W. C. Rheinboldt,Iterative solution of non-linear equations in several variables, Academic Press, New York, 1970.Google Scholar

Copyright information

© BIT Foundations 1979

Authors and Affiliations

  • F. M. Larkin
    • 1
  1. 1.Dept. of Computing and Information ScienceQueen's UniversityKingstonCanada

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