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Numerical Algorithms

, Volume 55, Issue 4, pp 481–502 | Cite as

Practical Quasi-Newton algorithms for singular nonlinear systems

  • Sandra Buhmiler
  • Nataša Krejić
  • Zorana Lužanin
Original Paper

Abstract

Quasi-Newton methods for solving singular systems of nonlinear equations are considered in this paper. Singular roots cause a number of problems in implementation of iterative methods and in general deteriorate the rate of convergence. We propose two modifications of QN methods based on Newton’s and Shamanski’s method for singular problems. The proposed algorithms belong to the class of two-step iterations. Influence of iterative rule for matrix updates and the choice of parameters that keep iterative sequence within convergence region are empirically analyzed and some conclusions are obtained.

Keywords

Nonlinear system of equations Singular system Quasi-Newton method Local convergence 

Mathematics Subject Classifications (2000)

65H10 47J20 

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References

  1. 1.
    Bouaricha, A., Schnabel, R.B.: Tensor methods for nonlinear least squares problems. SIAM J. Sci. Comput. 21, 1199–1221 (1999)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Broyden, G.: A class of methods for solving nonlinear simultaneous equations. Math. Comput. 19, 577–593 (1965)MathSciNetMATHGoogle Scholar
  3. 3.
    Buhmiler, S., Krejić, N.: A new smoothing quasi-Newton method for nonlinear complementarity problems. J. Comput. Appl. Math. 211, 141–155 (2008)CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Decker, D.W., Kelley, C.T.: Newton’s method at singular point I. SIAM J. Numer. Anal. 17, 66–70 (1980)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Decker, D.W., Kelley, C.T.: Sublinear convergence of the chord method at singular point. Numer. Math. 42, 147–154 (1983)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Decker, D.W., Kelley, C.T.: Broyden’s method for a class of problems having singular Jacobian at the root. SIAM J. Numer. Anal. 22(3), 566–573 (1985)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Feng, D., Frank, P.D., Schnabel, R.B.: Local convergence analysis of tensor methods for nonlinear equations. Math. Program. 62, 427–459 (1993)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Griewank, A.: On solving nonlinear equations with simple singularities or nearly singular solutions. SIAM Rev. 27, 537–563 (1985)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Griewank, A.: The “global” convergence of Broyden-like methods with a suitable line search. J. Aust. Math. Soc. Series B 28, 75–92 (1986)CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Hauser, R., Nedić, J: The continuous Newton–Raphson method can look ahead. SIAM J. Optim. 15, 915–925 (2005)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Hauser, R., Nedić, J.: On the relationship between the convergence rates of iterative and continuous processes. SIAM J. Optim. 18, 52–64 (2007)MathSciNetMATHGoogle Scholar
  12. 12.
    Janovsky, V., Seigre, V.: Qualitative analysis of Newton’s flow. SIAM J. Numer. Anal. 33, 2068–2097 (1996)CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Kelley, C.T., Suresh, R.: A new acceleration method for Newton’s method at singular point. SIAM J. Numer. Anal. 20, 1001–1009 (1983)CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Kelley, C.T.: A Shamanskii-like acceleration scheme for nonlinear equations at singular point. Math. Comput. 47, 609–623 (1986)MathSciNetMATHGoogle Scholar
  15. 15.
    Kelley, C.T., Xue, Z.Q.: Inexact Newton methods for singular problems. Optim. Methods Softw. 2, 249–267 (1993)CrossRefGoogle Scholar
  16. 16.
    Lukšan,L.: Inexact Trust Region Method for Large Sparse Systems of Nonlinear Equations. J Opt. Theory Appl. 81, 569–590 (1994)CrossRefMATHGoogle Scholar
  17. 17.
    Martínez, J.M.: A quasi-Newton method with modification of one column per iteration. Computing 3, 353–362 (1984)CrossRefGoogle Scholar
  18. 18.
    Martínez, J.M.: Practical quasi-Newton methods for solving nonlinear systems. J. Comput. Appl. Math. 124, 97–121 (2000)CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    de Mendonça, L.F., Pérez, R., Lopes, V.L.R.: Inverse q-columns updates methods for solving nonlinear system of equations. J. Comput. Appl. Math. 158, 317–337 (2003)CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    Neuberger, J.W.: The continuous Newton’s method, inverse functions and Nash–Moser. Am. Math. Mon. 114, 432–437 (2007)MathSciNetMATHGoogle Scholar
  21. 21.
    Nie, P.Y.: A null space method for solving system of equations. Appl. Math. Comput. 149, 215–226 (2004)CrossRefMathSciNetMATHGoogle Scholar
  22. 22.
    Reddien, G.W.: On Newton’s method for singular problems. SIAM J. Numer. Anal. 15(5), 993–996 (1978)CrossRefMathSciNetMATHGoogle Scholar
  23. 23.
    Riaza, R.: Attraction domains of degenerate singular equilibria in quasi-linear ODEs. SIAM J. Math. Anal. 36, 678–690 (2004)CrossRefMathSciNetMATHGoogle Scholar
  24. 24.
    Riaza, R.: On the local classification of smooth maps induced by Newton’s method. J. Differ. Equ. 217, 377–392 (2005)CrossRefMathSciNetMATHGoogle Scholar
  25. 25.
    Riaza, R.: Local dynamics of quasilinear ODEs with folded singular equilibria. Nonlinear Anal. 68, 2242–2249 (2008)CrossRefMathSciNetMATHGoogle Scholar
  26. 26.
    Riaza, R., Zufiria, P.J.: Discretization of implicit ODEs for singular root-finding problems. J. Comput. Appl. Math. 140, 695–712 (2002)CrossRefMathSciNetMATHGoogle Scholar
  27. 27.
    Schnabel, R.B., Frank, P. D.: Tensor methods for nonlinear equations. SIAM J. Numer. Anal. 21, 815–843 (1984)CrossRefMathSciNetMATHGoogle Scholar
  28. 28.
    Shen, Y.-Q., Ypma, T.J.: Newton’s method for singular nonlinear equations using approximate left and right nullspaces of the Jacobian. Appl. Numer. Math. 54(2),256–265 (2005)CrossRefMathSciNetMATHGoogle Scholar
  29. 29.
    Spedicato, E., Huang, Z.: Numerical experience with Newton-like methods for nonlinear algebraic systems. Computing 58, 69–89 (1997)CrossRefMathSciNetMATHGoogle Scholar
  30. 30.
    Thomas, S.: Sequential Estimation Techniques For Quasi-Newton Algorithms, Ph.D. thesis, TR 75-227, Cornell University (1975)Google Scholar
  31. 31.
    Yamamoto, T.: Historical developments in convergence analysis for Newton’s and Newton-like methods. J. Comput. Appl. Math. 124, 1–23 (2000)CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2010

Authors and Affiliations

  • Sandra Buhmiler
    • 1
  • Nataša Krejić
    • 2
  • Zorana Lužanin
    • 2
  1. 1.Department of Mathematics, Faculty of EngineeringUniversity of Novi SadNovi SadSerbia
  2. 2.Department of Mathematics and Informatics, Faculty of ScienceUniversity of Novi SadNovi SadSerbia

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