Advertisement

An introduction to variable dimension algorithms for solving systems of equations

  • M. Kojima
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 878)

Keywords

Complementarity Problem Continuation Method Bounded Open Subset Simplicial Subdivision Homotopy Continuation Method 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    J. C. Allexander, "The topological theory of an embedding method", in: H. Wacker, ed., Continuation Methods (Academic Press, New York, 1978) pp.36–67.Google Scholar
  2. [2]
    J. C. Allexander and J. A. Yorke, "The homotopy continuation methods, Numerically implementable topological procedures", Trans. Amer. Math. Soc. 242 (1978) 271–284.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    E. Allgower and K. Georg, "Simplicial and continuation methods for approximating fixed points and solutions to systems of equations", SIAM Review 22 (1980) 28–85.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    S. N. Chow, J. Mallet-Paret and J. A. Yorke, "Finding zeros of maps: Homotopy methods that are constructive with probability one", Math. Comp. 32 (1978) 887–899.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    B. C. Eaves, "A short course in solving equations with PL homotopies", SIAM-AMS Proc. (1976) 73–143.Google Scholar
  6. [6]
    R. M. Freund, "Variable-dimension complexes with applications", Tech. Rept. SOL 80-11, Dept. of Operations Research, Stanford University, Stanford, California, June 1980.Google Scholar
  7. [7]
    K. Georg, "On tracing an implicitly defined curve by quasi-Newton steps and calculating bifurcation by local perturbations", University of Bonn, Bonn, Jan. 1980.Google Scholar
  8. [8]
    R. B. Kearfott, "A derivative-free arc continuation method and a bifurcation technique", University of South Louisiana, 1980.Google Scholar
  9. [9]
    R. B. Kellogg, T. Y. Li and J. A. Yorke, "A constructive proof of the Brouwer fixed point theorem and computational results", SIAM J. Numer. Anal. 4 (1976) 473–483.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    M. Kojima, "A note on ‘A new algorithm for computing fixed points’ by van der Laan and Talman", in: Forster, ed., Numerical Solution of Highly Nonlinear Problems, Fixed Point Algorithms and Complementarity Problems (North-Holland, New York, 1980) pp.37–42.Google Scholar
  11. [11]
    M. Kojima and Y. Yamamoto, "Variable dimension algorithms, Part I: Basic theory", Res. Rept. B-77, Dept. of Information Sciences, Tokyo Institute of Technology, Tokyo, Dec. 1979.Google Scholar
  12. [12]
    M. Kojima and Y. Yamamoto, "Variable dimension algorithms, Part II: Some new algorithms and triangulations with continuous refinement of mesh size", Res. Rept. B-82, Dept. of Information Sciences, Tokyo Institute of Technology, Tokyo, May 1980.Google Scholar
  13. [13]
    G. van der Laan, "Simplicial fixed point algorithms", Ph. D. Dissertation, Free University Amsterdam, 1980.Google Scholar
  14. [14]
    G. van der Laan and A. J. J. Talman, "A restart algorithm for computing fixed points without extra dimension", Math. Programming 17 (1979) 74–84.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    G. van der Laan and A. J. J. Talman, "A restart algorithm without an artificial level for computing fixed points on unbounded regions", in: H. O. Peitgen and H. O. Walther, ed., Functional Differential Equations and Approximation of Fixed Points, Lecture Notes in Mathematics 730 (Springer, Berlin, 1979) pp.247–256.CrossRefGoogle Scholar
  16. [16]
    G. van der Laan and A. J. J. Talman, "Convergence and properties of recent variable dimension algorithms", in: W. Forster, ed., Numerical Solution of Highly Nonlinear Problems, Fixed Point Algorithms and Complementarity Problems (North-Holland, New York, 1980) pp.3–36.Google Scholar
  17. [17]
    G. van der Laan and A. J. J. Talman, "On the computation of of fixed points in the product space of the unit simplices and an application to non cooperative n-person games", Free University, Amsterdam, Oct. 1978.Google Scholar
  18. [18]
    G. van der Laan and A. J. J. Talman, "A class of simplicial subdivisions for restart fixed point algorithms without an extra dimension", Free University, Amsterdam, Dec. 1980.zbMATHGoogle Scholar
  19. [19]
    O. H. Merrill, "Applications and extentions of an algorithm that computes fixed points of a certain upper semi-continuous point to set mapping", Ph. D. Dissertation, Dept. of Industrial Engineering, University of Michigan, 1972.Google Scholar
  20. [20]
    J. M. Ortega and W. C. Rheinboldt, Iterative Solutions of Nonlinear Equations in Several Variables (Academic Press, New York, 1970).zbMATHGoogle Scholar
  21. [21]
    R. M. Reiser, "A modified integer labelling for complementarity algorithms", Institut für Operations Research der Universtät Zürich, June 1978.Google Scholar
  22. [22]
    D. Saupe, "Predictor-corrector methods and simplicial continuation algorithms", presented at the conference on Numerical Solutions of Nonlinear Equations, Simplicial & Classical Methods, University of Bremen, July 1980.Google Scholar
  23. [23]
    A. J. J. Talman, "Variable dimension fixed point algorithms and triangulations", Ph. D. Dissertation, Free University, Amsterdam, 1980.zbMATHGoogle Scholar
  24. [24]
    M. J. Todd, "Union Jack triangulations", in S. Karamardian, ed., Fixed Points: Algorithms and Applications (Academic Press, New York, 1977).Google Scholar
  25. [25]
    M. J. Todd, "Fixed-point algorithms that allow restarting without an extra dimension", Tech. Rept. No.379, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York, Sept. 1978.Google Scholar
  26. [26]
    M. J. Todd, "Global and local convergence and monotonicity results for a recent variable dimension simplicial algorithm", in: W. Forster, ed., Numerical Solution of Highly Nonlinear Problems, Fixed Point Algorithms and Complementarity Problems (North-Holland, New York, 1980) pp. 43–69.Google Scholar
  27. [27]
    M. J. Todd and A. H. Wright, "A variable-dimension simplicial algorithm for antipodal fixed-point theorems", Tech. Rept. No.417, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York, April 1979.zbMATHGoogle Scholar
  28. [28]
    A. H. Wright, "The octahedral algorithm, a new simplicial fixed point algorithm", Mathematics rept. No.61, Western Michigan University, Oct. 1979.Google Scholar

Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • M. Kojima
    • 1
  1. 1.Department of Information SciencesTokyo Institute of TechnologyMeguro, TokyoJapan

Personalised recommendations