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The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics pp 197–208Cite as

Error Bounds for Newton’s Method Under the Kantorovich Assumptions

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Abstract

A principle which finds sharp a posteriori error bounds for Newton’s method is given under the assumptions of Kantorovich’s theorem. On the basis of this principle, new error bounds are derived and comparison is made among the known bounds of Dennis [1], Gragg-Tapia [4], Kantorovich [5], [6], Kornstaedt [8], Lancaster [9], Miel [11], Moret [13], Ostrowski [16], [17], Potra [18], and Potra-Pták [19].

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References

  1. R. G. Bartle, “Newton’s method in Banach spaces,” Proc. Amer. Math. Soc., 6 (1955), 827–831.

    MathSciNet  MATH  Google Scholar 

  2. J. E. Dennis, Jr., “On the Kantorovich hypothesis for Newton’s method,” SIAM J. Numer. Anal., 6 (1969), 493–507.

    Article  MathSciNet  MATH  Google Scholar 

  3. B. Döring, “Uber das Newtonsche Näherungsverfahren,” Math. Phys. Sern.-Ber., 16 (1969), 27–40.

    MATH  Google Scholar 

  4. W. B. Gragg and R. A. Tapia, “Optimal error bounds for the Newton-Kantoro- vich theorem,” SIAM J. Numer. Anal, 11 (1974), 10–13.

    Article  MathSciNet  MATH  Google Scholar 

  5. L. V. Kantorovich, “On Newton’s method for functional equations,” Dokl. Akad. Nauk. SSSR, 59 (1948), 1237–1240.

    Google Scholar 

  6. L. V. Kantorovich, “The majorant principle and Newton’s method,” Dokl. Akad. Nauk. SSSR, 76 (1951), 17–20.

    Google Scholar 

  7. L. V. Kantorovich and G. P. Akilov, Functional Analysis in Normed Spaces, Pergamon Press, Oxford, 1964.

    MATH  Google Scholar 

  8. H. J. Kornstaedt, “Funktionalungleichungen und Iterations verfahren,” Aequat. Math., 13 (1975), 21–45.

    Article  MathSciNet  MATH  Google Scholar 

  9. P. Lancaster, “Error analysis for the Newton-Raphson method,” Numer. Math., 9 (1966), 55 - 68.

    Article  MathSciNet  MATH  Google Scholar 

  10. G. J. Miel, “The Kantorovich theorem with optimal error bounds,” Amer. Math. Monthly, 86 (1979), 212–215.

    Article  MathSciNet  MATH  Google Scholar 

  11. G.J. Miel, “Majorizing sequences and error bounds for iterative methods,” Math. Comp., 34 (1980), 185–202.

    Article  MathSciNet  MATH  Google Scholar 

  12. G. J. Miel, “An updated version of the Kantorovich theorem for Newton’s method,” Computing, 27 (1981), 237–244.

    Article  MathSciNet  MATH  Google Scholar 

  13. I. Moret, “A note on Newton type iterative methods,” Computing, 33 (1984), 65–73.

    Article  MathSciNet  MATH  Google Scholar 

  14. J. M. Ortega, “The Newton-Kantorovich theorem,” Amer. Math. Monthly, 75 (1968), 658–660.

    Article  MathSciNet  MATH  Google Scholar 

  15. J. M. Ortega and W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.

    MATH  Google Scholar 

  16. A. M. Ostrowski, “La method de Newton dans les espaces de Banach,” C. R. Acad. Sei. Paris, 27 (A) (1971), 1251–1253.

    Google Scholar 

  17. A. M. Ostrowski, Solution of Equations in Euclidean and Banach Spaces, Academic Press, New York, 1973.

    MATH  Google Scholar 

  18. F. A. Potra, “On the a posteriori error estimates for Newton’s method,” Beiträge zur Numerische Mathematik, 12 (1984), 125–138.

    MathSciNet  Google Scholar 

  19. F. A. Potra and V. Ptak, “Sharp error bounds for Newton’s process,” Numer. Math., 34 (1980), 63–72.

    Article  MathSciNet  MATH  Google Scholar 

  20. L. B. Rail, “A note on the convergence of Newton’s method,” SIAM J. Numer. Anal., 11 (1974), 34–36.

    Article  MathSciNet  Google Scholar 

  21. L. B. Rail, Computational Solution of Nonlinear Operator Equations, Krieger, Huntington, New York, 1979.

    Google Scholar 

  22. L. B. Rail and R. A. Tapia, “The Kantorovich theorem and error estimates for Newton’s method,” MRC Technical Summary Report No. 1043, University of Wisconsin, 1970.

    Google Scholar 

  23. W. C. Rheinboldt, “A unified convergence theory for a class of iterative process,” SIAM J. Numer. Anal., 5 (1968), 42–63.

    Article  MathSciNet  MATH  Google Scholar 

  24. J. W. Schmidt, “Regular-falsi Verfahren mit konsistenter Steigung und Majo- rantenprinzip,” Periodica Math. Hungarica, 5 (1974), 187–193.

    Article  MATH  Google Scholar 

  25. J. W. Schmidt, “Unter Fehlerschranken fur Regular-falsi-Verfahren,” Periodica Math. Hungarica, 5 (1978), 241–247.

    Article  Google Scholar 

  26. R. A. Tapia, “The Kantorovich theorem for Newton’s method,” Amer. Math. Monthly, 78 (1971), 389–392.

    Article  MathSciNet  MATH  Google Scholar 

  27. T. Yamamoto, “Error bounds for Newton’s process derived from the Kantorovich theorem,” Japan J. Appl. Math, 2 (1985), 285–292.

    Article  MathSciNet  MATH  Google Scholar 

  28. T. Yamamoto, “Error bounds for Newton’s iterates derived from the Kantorovich theorem,” MRC Technical Summary Report No. 2843, University of Wisconsin, 1985; Numer. Math., 48 (1986), 91–98.

    Article  MathSciNet  MATH  Google Scholar 

  29. T. Yamamoto, “A unified derivation of several error bounds for Newton’s process,” J. Comp. Appl. Math., 12& 13 (1985), 179–191.

    Article  Google Scholar 

  30. T. Yamamoto, “Error bounds for Newton-like methods under Kantorovich type assumptions,” MRC Technical Summary Report No. 2846, University of Wisconsin, 1985.

    Google Scholar 

  31. T. Yamamoto, “A convergence theorem for Newton’s method in Banach spaces,” MRC Technical Summary Report No. 2879, University of Wisconsin, 1985.

    Google Scholar 

  32. T. Yamamoto, “A method for finding sharp error bounds for Newton’s method under the Kantorovich assumptions,” to appear in Numer. Math.

    Google Scholar 

  33. T. J. Ypma, “Affine invariant convergence results for Newton’s method,” BIT, 22 (1982), 108–118.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Ehime University, Matsuyama, 790, Japan

    Tetsuro Yamamoto (Faculty of Science)

Authors
  1. Tetsuro Yamamoto
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Wyoming, Laramie, WY, 82071, USA

    Richard E. Ewing  & Kenneth I. Gross  & 

  2. Department of Mathematics, Texas Tech University, Lubbock, TX, 79409, USA

    Clyde F. Martin

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© 1986 Springer-Verlag New York Inc.

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Yamamoto, T. (1986). Error Bounds for Newton’s Method Under the Kantorovich Assumptions. In: Ewing, R.E., Gross, K.I., Martin, C.F. (eds) The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4984-9_14

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  • DOI: https://doi.org/10.1007/978-1-4612-4984-9_14

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-9385-9

  • Online ISBN: 978-1-4612-4984-9

  • eBook Packages: Springer Book Archive

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