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On the convergence of Broyden-like methods for nonlinear equations with nondifferentiable terms

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Abstract

In this paper, the author studies a Broyden-like method for solving nonlinear equations with nondifferentiable terms, which uses as updating matrices, approximations for Jacobian matrices of differentiable terms. Local and semilocal convergence theorems are proved. The results generalize those of Broyden, Dennis and Moré.

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References

  • Broyden, C. G. (1965). A class of methods for solving nonlinear simultaneous equations. Math. Comp., 19, 577–593.

    Google Scholar 

  • Broyden, C. G. (1970). The convergence of single-rank quasi-Newton methods, Math. Comp., 24, 365–382.

    Google Scholar 

  • Broyden, C. G., Dennis, J. E.Jr. and Moré, J. J. (1973). On the local and superlinear convergence of quasi-Newton Methods, J. Inst. Math. Appl., 12, 223–245.

    Google Scholar 

  • Chen, X. and Yamamoto, T. (1989). Convergence domains of certain iterative methods for solving nonlinear equations, Numer. Funct. Anal. Optim., 10, 37–48.

    Google Scholar 

  • Dennis, J. E.Jr. (1971). On the convergence of Broyden's methods for nonlinear systems of equations, Math. Comp., 25, 559–567.

    Google Scholar 

  • Dennis, J. E.Jr. and Moré, J. J. (1977). Quasi-Newton methods, motivation and theory, SIAM Rev., 19, 46–89.

    Google Scholar 

  • Gregory, R. T. and Karney, D. L. (1969). A Collection of Matrices for Testing Computational Algorithms, Wiley-Interscience, New York.

    Google Scholar 

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

    Google Scholar 

  • Yamamoto, T. (1987). A note on a posteriori error bound of Zabrejko and Nguen for Zincenko's iteration, Numer. Funct. Anal. Optim., 9, 987–994.

    Google Scholar 

  • Yamamoto, T. and Chen, X. (1990). Ball-convergence theorems and error estimates for certain iterative methods for nonlinear equations, to appear in Japan. J. Appl. Math., 7, 131–143.

    Google Scholar 

  • Zabrejko, P. P. and Nguen, D. F. (1987). The majorant method in the theory of Newton-Kantorovich approximations and the Pták error estimates, Numer, Funct. Anal. Optim., 9, 671–684.

    Google Scholar 

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Chen, X. On the convergence of Broyden-like methods for nonlinear equations with nondifferentiable terms. Ann Inst Stat Math 42, 387–401 (1990). https://doi.org/10.1007/BF00050844

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  • DOI: https://doi.org/10.1007/BF00050844

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