Skip to main content
SpringerLink
Log in

Recent and Past Developments in the Simplicial Approximation Approach to Solving Nonlinear Equations—A Subjective View

  • Conference paper
Extremal Methods and Systems Analysis

Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 174))

Abstract

Several contributions to solving nonlinear equations are discussed, along with recent results and some possible future directions.

The work of this author is supported in part by ONR Grant No. N00014-75-C-0495 and NSF Grant No. ENG 76-81058.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Avriel, M., AND J. P. Dauer: A Homotopy Based Approach to Unconstrained Optimization. Department of Operations Research, Technical Report 76–14, Stanford University, July, 1976.

    Google Scholar 

  2. Branin, F. H., AND S. K. Hoo: A Method for Finding Multiple Extrema of a Function of n Variables. Proceedings of the Conference on Numerical Methods for Nonlinear Optimization, University of Dundee, Scotland, June 28-July 1, 1971. Numerical Methods of Nonlinear Optimisation, Academic Press, London, August, 1972.

    Google Scholar 

  3. Broyden, C. G.: A New Method of Solving Nonlinear Simultaneous Equations. Computer Journal 12.1 (1969), 95–100.

    Google Scholar 

  4. Charnes, A.., C. B. Garcia, AND C. E. Lemke: Constructive Proofs of Theorems Relating to F(x) = y, with Applications. To appear in Mathematical Program-ming.

    Google Scholar 

  5. Davidenko, D.: On the Approximate Solution of a System of Nonlinear Equations. Ukrain. Mat. z., 5 (1953), l96–206.

    Google Scholar 

  6. Deist, F. H., AND L. Sefor: Solution of Systems of Nonlinear Equations by Parameter Variation. Computer Journal 10 (1967), 78–82.

    Article  Google Scholar 

  7. Eaves, B. C.: Homotopies for Computation of Fixed Points. Mathematical Programming 3, 1 (1972) 1–22.

    Google Scholar 

  8. Eaves, B. C., AND H. Scarf: The Solution of Systems of Piecewise Linear Equations. Mathematics of Operations Research 1, 1 (1976), 1–27.

    Article  Google Scholar 

  9. Fisher, M. L., AND F. J. Gould: A Simplicial Algorithm for the Nonlinear Complementary Problem. A talk given at the 8th Mathematical Programm ing Symposium, Stanford University, August, 1973. Mathematical Programming 6 (1974), 281–300.

    Article  Google Scholar 

  10. Fisher, M. L., AND F. J. Gould, AND J. W. Tolle: A Simplicial Approximation Algorithm for Solving Systems of Nonlinear Equations. Center for Mathematical Studies in Business and Economics Report No. 7421, University of Chicago, May, 1974 and Instituto Nazionale Di Alta Matematica: Symposia Mathematica 19 (1976), 73–90.

    Google Scholar 

  11. Freudenstein, F., AND B. Roth: Numerical Solutions of Systems of Nonlinear Equations. J. Assoc. Comp. Mach., 10 (1963), 550–556

    Article  Google Scholar 

  12. ]Garcia, C. B., AND F. J. Gould: An Algorithm Based on the Equivalence of Vector and Scalar Labels in Simplicial Approximation. Center for Mathematical Studies in Business and Economics Report No. 7626, University of Chicago, July, 1976.

    Google Scholar 

  13. Garcia, C. B., AND F. J. Gould: Scalar Labelings for Homotopy Paths. To appear in Mathematical Programming.

    Google Scholar 

  14. Garcia, C. B., AND F. J. Gould: A Theorem on Homotopy Paths. Center for Mathematical Studies in Business and Economics Report No. 7734, University of Chicago, June 1977, to appear in Mathematics of Operations Research.

    Google Scholar 

  15. Garcia, C. B., AND F. J. Gould: Relations Between Complementary Pivoting Algorithms and Local and Global Newton Methods. Center for Mathematical Studies in Business and Economics Report No. 7735, University of Chicago, June 1977, to appear in SIAM Review.

    Google Scholar 

  16. Garcia. C.B.,AND W. I. Zangwill: Determining All Solutions to Certain Systems of Nonlinear Equations. Center for Mathematical Studies in Business and Economics Report No. 7712, University of Chicago, January 1977, to appear in Mathematics of Operations Research.

    Google Scholar 

  17. Garcia. C.B.,AND W. I. Zangwill: Finding All Solutions to Polynomial Systems and Other Systems of Equations. Center for Mathematical Studies in Business and Economics Report No. 7738, University of Chicago, July 1977.

    Google Scholar 

  18. Gould, F. J., AND J. W. Tolle: A Unified Approach to Complementarity in Optimization. Discrete Mathematics 7 (1974), 225–271.

    Article  Google Scholar 

  19. Gould, F. J., AND J. W. Tolle: An Existence Theorem for Solutions to f(x) = 0. Mathematical Programming 11 (1976), 252–262.

    Article  Google Scholar 

  20. JACOVLEV, M. N.: On the Solution of Systems of Nonlinear Equations by Differentiation with Respect to a Parameter. U.S.S.R. Comput. Math. and Math. Phys., 4 (1964), 146–149.

    Google Scholar 

  21. KIZNER, W.: A Numerical Method for Finding Solutions of Nonlinear Equations. J. Soc. Indust. Appl. Math., 12 (1964) 424–428.

    Article  Google Scholar 

  22. Kuhn, H. W.: Simplicial Approximation of Fixed Points. Proc. Natl. Acad. Sci. 61 (1968).

    Google Scholar 

  23. Kuhn, H. W.: Some Combinatorial Lemmas In Topology. IBM J. Res. Develop. 4 (1960), 518–524.

    Google Scholar 

  24. Kuhn, H. W.: The Sandwich Method. A talk given at the 8th Mathematical Programming Symposium, Stanford University, August 1973.

    Google Scholar 

  25. Kuhn, H. W.: AND J. G. Mackinnon: Sandwich Method for Finding Fixed Points. J. Opt. Theory and Appl. 17 (1975), 189–204.

    Google Scholar 

  26. Lemke, C. E.: Bimatrix Equilibrium Points and Mathematical Programming Management Science 11 (1965), 681–689.

    Google Scholar 

  27. Levenberg, K.: A Method for the Solution of Certain Nonlinear Problems in Least Squares. Quart. Appl. Math., 2, (1944), 164–168.

    Google Scholar 

  28. Marquardt, D. W.: An Algorithm for Least Squares Estimation of Nonlinear Para-meters. J. Soc. Indust. Appl. Math., 11, (1963), 431–441.

    Article  Google Scholar 

  29. Merrill, 0. H.: Applications and Extensions of an Algorithm that Computes Fixed Points of Certain Upper Semi-Continuous Point-to-Set Mappings. University of Michigan: Ph.D. dissertation, 1972.

    Google Scholar 

  30. Meyer, G. H.: On Solving Nonlinear Equations with a One-Parameter Operator Imbedding. SIAM J. Numer. Anal., 5, (1967), 739–752.

    Article  Google Scholar 

  31. Milnor, J.: Topology from the Differentiable Viewpoint. University Press of Virginia, 1965.

    Google Scholar 

  32. Ortega, J. M., AND W. C. Rheinboldt: Iterative Solutions of Nonlinear Equations in Several Variables. Academic Press, 1970, 234.

    Google Scholar 

  33. Scarf, H.: The Approximation of Fixed Points of a Continuous Mapping. SIAM Journal of Applied Mathematics 15 (1967), 1328–1342.

    Article  Google Scholar 

  34. Smale, S.: Convergent Process of Price Adjustment and Global Newton Methods. Cowles Foundation Report, Yale University 1975.

    Google Scholar 

  35. Sternberg, S.: Lectures on Differential Geometry. Prentice-Hall, 1964.

    Google Scholar 

Download references

Authors
  1. F. J. Gould
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Operations Research Institute for Management Science and Engineering, The Washington University, 20052, Washington, DC, USA

    Anthony V. Fiacco

  2. Department of Mathematics, Carnegie-Mellon University, 15213, Pittsburgh, PA, USA

    Kenneth O. Kortanek

Rights and permissions

Reprints and permissions

Copyright information

© 1980 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Gould, F.J. (1980). Recent and Past Developments in the Simplicial Approximation Approach to Solving Nonlinear Equations—A Subjective View. In: Fiacco, A.V., Kortanek, K.O. (eds) Extremal Methods and Systems Analysis. Lecture Notes in Economics and Mathematical Systems, vol 174. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-46414-0_24

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-46414-0_24

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-09730-3

  • Online ISBN: 978-3-642-46414-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation