Inexact trust region method for large sparse systems of nonlinear equations

  • L. Lukšan
Contributed Papers

Abstract

The main purpose of this paper is to prove the global convergence of the new trust region method based on the smoothed CGS algorithm. This method is surprisingly convenient for the numerical solution of large sparse systems of nonlinear equations, as is demonstrated by numerical experiments. A modification of the proposed trust region method does not use matrices, so it can be used for large dense systems of nonlinear equations.

Key Words

Nonlinear equations sparse systems trust region methods 

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References

  1. 1.
    Powell, M. J. D.,On the Global Convergence of Trust Region Algorithms for Unconstrained Minimization, Mathematical Programming, Vol. 29, pp. 297–303, 1984.Google Scholar
  2. 2.
    Shultz, G. A., Schnabel, R. B., andByrd, R. H.,A Family of Trust-Region-Based Algorithms for Unconstrained Minimization with Strong Global Convergence Properties, SIAM Journal on Numerical Analysis, Vol. 22, pp. 47–67, 1985.Google Scholar
  3. 3.
    Steihaug, T.,The Conjugate Gradient Method and Trust Regions in Large-Scale Optimization, SIAM Journal on Numerical Analysis, Vol. 20, pp. 626–637, 1983.Google Scholar
  4. 4.
    Sonneveld, P.,CGS, a Fast Lanczos-Type Solver for Nonsymmetric Linear Systems, SIAM Journal on Scientific and Statistical Computations, Vol. 10, pp. 36–52, 1989Google Scholar
  5. 5.
    Tong, C. H.,A Comparative Study of Preconditioned Lanczos Methods for Nonsymmetric Linear Systems, Report No. SAND91-8240B, Sandia National Laboratories, Livermore, California, 1992.Google Scholar
  6. 6.
    Gill, P. E., andMurray, W.,Newton Type Methods for Unconstrained and Linearly Constrained Optimization, Mathematical Programming, Vol. 7, pp. 311–350, 1974.Google Scholar
  7. 7.
    Lukšan, L.,Inexact Trust Region Method for Large Sparse Nonlinear Least Squares, Kybernetika, Vol. 29, pp. 305–324, 1993.Google Scholar
  8. 8.
    Paige, C. C., andSaunders, M. A.,LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares, ACM Transactions on Mathematical Software, Vol. 8, pp. 43–71, 1982.Google Scholar
  9. 9.
    Bogle, I. D. L., andPerkins, J. D.,A New Sparsity-Preserving Quasi-Newton Update for Solving Nonlinear Equations, SIAM Journal on Scientific and Statistical Computations, Vol. 11, pp. 621–630, 1990.Google Scholar
  10. 10.
    Toint, P. L.,Numerical Solution of Large Sets of Algebraic Equations, Mathematics of Computation, Vol. 46, pp. 175–189, 1986.Google Scholar
  11. 11.
    Gomez-Ruggiero, M. A., Martinez, J. M., andMoretti, A. C.,Comparing Algorithms for Solving Sparse Nonlinear Systems of Equations. SIAM Journal on Scientific and Statistical Computations, Vol. 13, pp. 459–483, 1992.Google Scholar
  12. 12.
    Li, G.,Successive Column Correction Algorithms for Solving Sparse Nonlinear Systems of Equations, Mathematical Programming, Vol. 43, pp. 187–207, 1989.Google Scholar
  13. 13.
    Moré, J. J., Garbow, B. S., andHillström, K. E.,Testing Unconstrained Optimization Software, ACM Transactions on Mathematical Software, Vol. 7, pp. 17–41, 1981.Google Scholar

Copyright information

© Plenum Publishing Corporation 1994

Authors and Affiliations

  • L. Lukšan
    • 1
  1. 1.Institute of Computer ScienceAcademy of Sciences of the Czech RepublicPragueCzech Republic

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