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, Volume 21, Issue 2, pp 113–125 | Cite as

A unified derivation of quasi-Newton methods for solving non-sparse and sparse nonlinear equations

  • R. P. Tewarson
Article

Abstract

An augmented quasi-Newton equation is solved by using theW-V generalized inverse to give a unified derivation of the known quasi-Newton methods for solving systems of nonlinear algebraic equations. This approach makes it possible to get new formulas for sparse and non-sparse systems, and also to determine what norms of the update matrices are minimized when several useful quasi-Newton update formulas are derived.

Keywords

Computational Mathematic Algebraic Equation Nonlinear Equation Nonlinear Algebraic Equation Unify Derivation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Eine einheitliche Ableitung von Quasi-Newtonschen Methoden zur Lösung von nicht dünnbesetzten und dünnbesetzten nichtlinearen Gleichungen

Zusammenfassung

Eine erweiterte Quasi-Newtonsche Gleichung wird gelöst unter Verwendung derW-V verallgemeinerten Inverse und ergibt eine einheitliche Ableitung der bekannten Quasi-Newtonschen Methode für die Lösung von nichtlinearen algebraischen Gleichungssystemen. Dieser Zugang ermöglicht es uns, neue Formeln für dünnbesetzte und nicht dünnbesetzte Systeme zu erhalten und auch zu bestimmen, welche Normen der Update-Matrizen bei verschiedenen brauchbaren Quasi-Newtonschen Update-Formeln minimisiert werden.

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References

  1. [1]
    Dennis, J. E., Jr., Moré, J. J.: Quasi-Newton methods, motivation and theory. SIAM Rev.19, 46–89 (1977).CrossRefGoogle Scholar
  2. [2]
    Schubert, L. K.: Modification of quasi-Newton method for nonlinear equations with a sparse Jacobian. Math. Comp.24, 27–30 (1970).Google Scholar
  3. [3]
    Broyden, C. G.: The convergence of an algorithm for solving sparse nonlinear systems. Math. Comp.25, 285–294 (1971).Google Scholar
  4. [4]
    Broyden, C. G.: A class of methods for solving nonlinear simultaneous equations. Math. Comp.19, 577–593 (1965).Google Scholar
  5. [5]
    Rheinboldt, W. C., Vandergraft, J. S.: On the local convergence of update methods. SIAM J. on Numer. Anal.11, 1069–1085 (1974).CrossRefGoogle Scholar
  6. [6]
    Broyden, C. G., Dennis, J. E., Moré, J. J.: On the local and superlinear convergence of quasi-Newton methods. J. Inst. Math. Appl.12, 223–246 (1973).Google Scholar
  7. [7]
    Herring, G. P.: A note on generalized interpolation and the pseudoinverse. SIAM J. on Numer. Anal.4, 548–556 (1967).CrossRefGoogle Scholar
  8. [8]
    Tewarson, R. P.: Use of smoothing and damping techniques in the solution of nonlinear equations. SIAM Rev.19, 35–45 (1977).CrossRefGoogle Scholar
  9. [9]
    Albert, A.: Regression and the Moore-Penrose Pseudoinverse. New York: Academic Press 1972.Google Scholar
  10. [10]
    Goldfarb, D.: A family of variable-metric methods derived by variational means. Math. Comp.24, 23–26 (1970).Google Scholar
  11. [11]
    Greenstadt, J.: Variations on variable-metric methods. Math. Comp.24, 1–18 (1970).Google Scholar
  12. [12]
    Huang, H. Y.: Unified approach to quadratically convergent algorithms for function minimization. J. Optimization Theory Appl.5, 405–423 (1970).CrossRefGoogle Scholar
  13. [13]
    Pearson, J. D.: Variable metric methods of minimization. Comput. J.12, 171–178 (1969).CrossRefGoogle Scholar
  14. [14]
    Tewarson, R. P.: On the use of generalized inverses in function minimization. Computing6, 241–248 (1970).CrossRefGoogle Scholar
  15. [15]
    Adachi, A.: On variable metric algorithms. J. Opt. Th. Appl.7, 391–410 (1971).CrossRefGoogle Scholar
  16. [16]
    Toint, Ph. L.: On sparse and symmetric matrix updating subject to a linear equation. Math. Comp.31, 954–961 (1977).Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • R. P. Tewarson
    • 1
  1. 1.Department of Applied Mathematics and StatisticsState University of New YorkStony BrookUSA

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