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Inexact Methods: Forcing Terms and Conditioning

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Abstract

In this paper, we consider inexact Newton and Newton-like methods andprovide new convergence conditions relating the forcing terms to theconditioning of the iteration matrices. These results can be exploited wheninexact methods with iterative linear solvers are used. In this framework,preconditioning techniques can be used to improve the performance ofiterative linear solvers and to avoid the need of excessively small forcingterms. Numerical experiments validating the theoretical results arediscussed.

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References

  1. Brown, P. N., and Saad, Y., Convergence Theory ofNonlinear Newton–Krylov Algorithms, SIAM Journal on Optimization, Vol. 4, pp. 297–330, 1994.

    Google Scholar 

  2. Dembo, R. S., Eisenstat, S. C., and Steihaug, T., Inexact Newton Methods, SIAM Journal on Numerical Analysis, Vol. 19, pp. 400–408, 1982.

    Google Scholar 

  3. Eisenstat, S. C., and Walker, H. F., Choosing the Forcing Terms in an Inexact Newton Method, SIAM Journal on Scienti.c Computation, Vol. 17, pp. 16–32, 1996.

    Google Scholar 

  4. Kelley, C. T., Iterative Methods for Linear and Nonlinear Equations, Frontiers in Applied Mathematics, SIAM, Philadelphia, Pennsylvania, Vol. 16, 1995.

    Google Scholar 

  5. Luksan, L., Inexact Trust–Region Method for Large Sparse Systems of Nonlinear Equations, Journal of Optimization Theory and Applications, Vol. 81, pp. 569–591, 1994.

    Google Scholar 

  6. Luksan, L., and Vlceck, J., Computational Experience with Globally Convergent Descent Methods for Large Sparse Systems of Nonlinear Equations, Optimization Methods and Software, Vol. 8, pp. 201–223, 1998.

    Google Scholar 

  7. Morini, B., Convergence Behavior ofInexact Newton Methods, Mathematics of Computation, Vol. 68, pp. 1605–1613, 1999.

    Google Scholar 

  8. Ypma, T. J., Local Convergence ofInexact Newton Methods, SIAM Journal on Numerical Analysis, Vol. 21, pp. 583–590, 1984.

    Google Scholar 

  9. Brown, P. N., and Hindmarsh, A. C., Matrix-Free Methods for Stiff Systems of ODEs, SIAM Journal on Numerical Analysis, Vol. 23, pp. 610–638, 1986.

    Google Scholar 

  10. Jackson, K. R., and Seward, K. R., Adaptive Linear Equation Solvers in Codes for Large Stiff Systems of ODEs, SIAM Journal on Scienti.c and Statistic Computation, Vol. 14, pp. 800–823, 1993.

    Google Scholar 

  11. Morini, B., and Macconi, M., Inexact Methods in the Numerical Solution of Stiff Initial-Value Problems, Computing, Vol. 63, pp. 265–281, 1999.

    Google Scholar 

  12. Pernice, M., and Walker, H. F., NITSOL: A Newton Iterative Solver, SIAM Journal on Scientific Computation, Vol. 19, pp. 302–318, 1998.

    Google Scholar 

  13. Rheinboldt, W. C., On Measures ofIll-Conditioning for Nonlinear Equations, Mathematics of Computation, Vol. 30, pp. 104–111, 1976.

    Google Scholar 

  14. Ortega, J. M., and Rheinboldt, W. C., Iterative Solution ofNonlinear Equations in Several Variables, Academic Press, New York, NY, 1970.

    Google Scholar 

  15. Dennis, J. E., and Schnabel, R. B., Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice Hall, Englewood Cliff, New Jersey, 1983.

    Google Scholar 

  16. Axelsson, O., Iterative Solution Methods, Cambridge University Press, Cambridge, England, 1994.

    Google Scholar 

  17. Greenbaum, A., Iterative Methods for Solving Linear Systems, Frontiers in Applied Mathematics, SIAM, Philadephia, Pennsylvania, Vol. 17, 1997.

    Google Scholar 

  18. Broyden, C.G., The Convergence ofan Algorithm for Solving Sparse Nonlinear Systems, Mathematics of Computation, Vol. 25, pp. 285–294, 1971.

    Google Scholar 

  19. Schubert, L. K., Modification of a Quasi-Newton Method for Nonlinear Equations with Sparse Jacobian, Mathematics of Computation, Vol. 24, pp. 17–30, 1970.

    Google Scholar 

  20. Saad, Y., and Schultz, M. H., GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems, SIAM Journal on Scientific and Statistic Computation, Vol. 7, pp. 856–869, 1986.

    Google Scholar 

  21. Kelley, C. T., Solution ofthe Chandrasekhar H-Equation by Newton's Method, Journal of Mathematical Physics, Vol. 21, pp. 1625–1628, 1980.

    Google Scholar 

  22. Golub, H., and Van Loan, C. F., Matrix Computations, 2nd Edition, Johns Hopkins University Press, Baltimore, Maryland, 1989.

    Google Scholar 

  23. Bischof, C. H., Incremental Condition Estimation, SIAM Journal on Matrix Analysis and Applications, Vol. 11, pp. 335–381, 1989.

    Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Energetica, ``Sergio Stecco”, Firenze, Italia

    M. G. Gasparo (Associate Professor)

  2. Dipartimento di Energetica, ``Sergio Stecco”, Firenze, Italia

    B. Morini (Researcher)

Authors
  1. M. G. Gasparo
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  2. B. Morini
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Gasparo, M.G., Morini, B. Inexact Methods: Forcing Terms and Conditioning. Journal of Optimization Theory and Applications 107, 573–589 (2000). https://doi.org/10.1023/A:1026499216100

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