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A survey of homotopy methods for smooth mappings

  • E. L. Allgower
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 878)

Keywords

Bifurcation Point Regular Point Continuation Method Nonlinear Eigenvalue Problem Bifurcation Problem 
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References

  1. [1]
    R. Abraham and J. Robbin. Transversal Mappings and Flows, Benjamin, New York-Amsterdam, 1967.zbMATHGoogle Scholar
  2. [2]
    J.C. Alexander. The topological theory of an embedding method, Continuation methods, H. Wacker, ed., Academic Press, New York, 1978.Google Scholar
  3. [3]
    J.C. Alexander and J.A. Yorke. The homotopy continuation method: Numerically implementable topological procedures, Trans.Amer.Math.Soc. 242 (1978), 271–284.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    E.L. Allgower. On a discretization of y″ + λyk = 0, Topics in Numerical Analysis II, J.J.H. Miller, ed., Academic Press, New York, pp. 1–15, 1975.CrossRefGoogle Scholar
  5. [5]
    E.L. 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
  6. [6]
    E.L.Allgower and K.Georg. Homotopy methods for approximating several solutions to nonlinear systems of equations, in Numerical Solution of Highly Nonlinear Problems, W. Förster, ed., North-Holland, 1980.Google Scholar
  7. [7]
    E.L.Allgower, K.Böhmer and S.F.McCormick. Discrete correction methods for operator equations, these proceedings.Google Scholar
  8. [8]
    P. Anselone and R. Moore. An extension of the Newton-Kantorovich method for solving nonlinear equations with an application to elasticity, J. Math.Anal.Appl., 13 (1966), 476–501.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    E. Bohl. Chord techniques and Newton's method for discrete bifurcation problems, Numer.Math., 34 (1980), 111–124.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    E. Bohl. On the bifurcation diagram of discrete analogues for ordinary bifurcation problems, Math.Meth. in the Appl.Sci., 1 (1979), 566–571.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    F.J. Branin, Jr. Widely convergent method for finding multiple solutions of simultaneous nonlinear equations, IBM J.Res.Develop. 16 (1972), 504–522.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    F.J.Branin,Jr. and K.S.Hoo. A method for finding multiple extrema of a function of n variables, in: Numerical Methods for Nonlinear Optimization, F.Lootsma, ed., Academic Press, pp. 231–327, 1972).Google Scholar
  13. [13]
    S.N. Chow, J. Mallet-Paret and J.A. Yorke. Finding zeros of maps: Homotopy methods that are constructive with probability one, Math.Comput., 32 (1978), 887–899.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    L.O. Chua and A. Ushida. A switching-parameter algorithm for finding multiple solutions of nonlinear resistive circuits, IEEE Trans.Circuit Theory and Applications, 4 (1976), 215–239.CrossRefzbMATHGoogle Scholar
  15. [15]
    M.G. Crandall and P.H. Rabinowitz. Bifurcation from simple eigenvalues, J.Func.Anal., 8 (1971), 321–340.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    D. Davidenko. On a new method of numerically integrating a system of nonlinear equations, Dokl.Akad.Nauk SSSR, 88 (1953), 601–604. (In Russian)MathSciNetGoogle Scholar
  17. [17]
    P. Deufelhard. A modified continuation method for the numerical solution of nonlinear two-point boundary value problems by shooting techniques, Numer.Math. 26 (1976), 327–343.MathSciNetCrossRefGoogle Scholar
  18. [18]
    P. Deufelhard. A step size control for continuation methods and its special application to multiple shooting techniques, Numer.Math. 33 (1979), 115–146.MathSciNetCrossRefGoogle Scholar
  19. [19]
    F.-J. Drexler. Eine Methode zur Berechnung sämtlicher Lösungen von Polynomgleichungssystemen, Numer.Math., 29 (1977), 45–58.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    F.-J. Drexler. A homotopy method for the calculation of all zeros of zero-dimensional polynomial ideals, Continuation methods, H. Wacker, ed., Academic Press, New York, pp. 69–94, 1978.Google Scholar
  21. [21]
    R.E. Gaines. Difference equations associated with boundary value problems for second-order nonlinear ordinary differential equations, SIAM J.Num.Anal., 11 (1974), 411–434.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    C.B.Garcia and W.I.Zangwill. Global continuation methods for finding all solutions to polynomial systems of equations in n variables, Int'l. Symp. on External Methods and Sys.Anal., Austin, TX, Univ. of Chicago, Dept. of Economics and Graduate School of Business, Report 7755, 1977.Google Scholar
  23. [23]
    K.Georg. On tracing an implicitly defined curve by quasi-Newton steps and calculating bifurcations by local perturbations, to appear in SIAM J.Sci.Stat.Computing.Google Scholar
  24. [24]
    K.Georg. Numerical integration of the Davidenko equation, these proceedings.Google Scholar
  25. [25]
    H.Hackl, H.Wacker and W.Zulehner. Aufwandsoptimale Schrittweitensteuerung by Einbettungsmethoden, in Constructive Methods for Nonlinear Boundary Value Problems and Nonlinear Oscillations, Birkhauser, Basel, ISNM 48 (1979), eds. J.Albrecht, L.Collatz and K.Kirchgassner, pp. 48–67.Google Scholar
  26. [26]
    C. Haselgrove. Solution of nonlinear equations and of differential equations with two-point boundary conditions, Comput.J. 4 (1961), 255–259.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    M. Hirsch and S. Smale. On algorithms for solving f(x) = 0, Comm.Pure Appl.Math., 32 (1979), 281–312.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    H.Jurgens, H.-O. Peitgen and D.Saupe. Topological perturbations in the numerical study of nonlinear eigenvalue and bifurcation problems, in Analysis and Computation of Fixed Points, ed. S.Robinson, Academic Press.Google Scholar
  29. [29]
    R.B.Kearfott. A derivative-free arc continuation method and a bifurcation technique, preprint (also see these proceedings).Google Scholar
  30. [30]
    J.P. Keener and H.B. Keller. Perturbed bifurcation theory, Arch.Rat.Mech.Anal. 50 (1973), 159–175.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    H.B. Keller. Numerical solution of bifurcation and nonlinear eigenvalue problems, in Applications of Bifurcation Theory, ed.: P.H. Rabinowitz, Academic Press, New York, pp. 359–384, 1977.Google Scholar
  32. [32]
    H.B. Keller. Global homotopies and Newton methods, in Recent Advances in Numerical Analysis, eds: C. deBoor and G.H. Golub, Academic Press, New York, pp. 73–94, 1978.CrossRefGoogle Scholar
  33. [33]
    R.B. Kellogg, T.Y. Li and J. Yorke. A constructive proof of the Brouwer fixed point theorem and computational results, SIAM J.Numer.Anal., 4 (1976), 473–483.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    E. Lahaye. Une méthode de résolution d'une catégorie d'équations transcendantes, C.R.Acad.Sci., Paris, 198 (1934), 1840–1842.zbMATHGoogle Scholar
  35. [35]
    W.F. Langford. Numerical solution of bifurcation problems for ordinary differential equations, preprint, McGill University, Montreal (1976).zbMATHGoogle Scholar
  36. [36]
    T.Y. Li. Numerical aspects of the continuation method-flow charts bf a simple algorithm, Proc. of Symp. on analysis and computation of fixed points, Madison, WI, ed. S.M. Robinson, Academic Press, New York, 1979.Google Scholar
  37. [37]
    R. Menzel and H. Schwetlick. Über einen Ordnungsbegriff bei Einbettungsalgorithm zur Lösung nichtlinearer Gleichungen, Computing 16 (1976), 187–199.MathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    R. Menzel and H. Schwetlick. Zur Lösung parameterabhängiger nichtlinearer Gleichungen mit singulären Jacobi-Matrizen, Numer.Math., 30 (1978), 65–79.MathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    R. Menzel. Ein implementierbarer Algorithmus zur Lösung nichtlinearer Gleichungssysteme bei schwach singulärer Einbettung, Beiträge zur Numerischen Mathematik, 8 (1980), 99–111.MathSciNetzbMATHGoogle Scholar
  40. [40]
    G. Meyer. On solving nonlinear equations with a one-parameter operator imbedding, SIAM J.Numer.Anal., 5 (1968), 739–752.MathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    J.W. Milnor. Topology from the Differentiable Viewpoint, University Press of Virginia, Charlottesville, VA, 1969.zbMATHGoogle Scholar
  42. [42]
    H.D.Mittelman and H.Weber. Numerical treatment of bifurcation problems, University of Dortmund, preprint, 1979.Google Scholar
  43. [43]
    Paul Nelson, Jr. Subcriticality for submultiplying steady-state neutron diffusion, in Nonlinear diffusion, ed. John Nohel, Research Notes in Math. 14, Pitman, London.Google Scholar
  44. [44]
    J.M. Ortega and W.C. Rheinboldt. Iterative Solutions of Nonlinear Equations in Several Variables, Academic Press, New York-London, 1970.zbMATHGoogle Scholar
  45. [45]
    H.O. Peitgen and H.O. Walther, eds., Functional Differential Equations and Approximation of Fixed Points, Springer L.N.730Google Scholar
  46. [46]
    H.O. Peitgen and M. Prüfer. The Leray Schauder continuation method is a constructive element in the numerical study of nonlinear eigenvalue and bifurcation problems, in [45], pp. 326–409.Google Scholar
  47. [47]
    H.O. Peitgen, D. Saupe and K. Schmitt. Nonlinear elliptic boundary value problems versus their finite difference approximations …, J. reine angew. Mathematik 322 (1981), 74–117.MathSciNetzbMATHGoogle Scholar
  48. [48]
    P.H. Rabinowitz. Some global results for nonlinear eigenvalue problems, J.Func.Anal., 7 (1971), 487–513.MathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    W.C. Rheinboldt. Numerical methods for a class of finite-dimensional bifurcation problems, SIAM J.Numer.Anal., 15 (1978), 1–11.MathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    W.C. Rheinboldt. Solution field of nonlinear equations and continuation methods, SIAM J.Numer.Anal., 17 (1980), 221–237.MathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    E. Riks. The application of Newton's Method to the problem of elastic stability, J.Appl.Mech.Techn.Phys., 39 (1972), 1060–1065.CrossRefzbMATHGoogle Scholar
  52. [52]
    C.Schmidt. Approximating differential equations that describe homotopy paths, Univ. of Chicago School of Management Science Report 7931.Google Scholar
  53. [53]
    W.F. Schmidt. Adaptive step size selection for use with the continuation method, Int'l. J.for Numer.Meths. in Engrg, 12 (1978), 677–694.CrossRefzbMATHGoogle Scholar
  54. [54]
    H. Schwetlick. Ein neues Princip zur Konstruktion implementierbarer, global konvergenter Einbettungsalgorithmen, Beiträge Numer.Math., 4–5 (1975–6), 215–228; 201–206.MathSciNetzbMATHGoogle Scholar
  55. [55]
    L.F. Shampine and M.K. Gordon. Computer Solution of Ordinary Differential Equations: The Initial Value Problem, Freeman Press, San Francisco, 1975.zbMATHGoogle Scholar
  56. [56]
    G.Shearing. Ph.D. Thesis, Manchester (1960).Google Scholar
  57. [57]
    R.Seydel. Numerische Berechnung von Verzweigungen bei gewöhnlichen Differentialgleichungen, TUM-Math-7736 Technische Universität München, 1977.Google Scholar
  58. [58]
    S. Smale. A convergent process of price adjustment and global Newton methods, J.Math.Econ., 3 (1976), 1–14.MathSciNetCrossRefzbMATHGoogle Scholar
  59. [59]
    G.A. Thurston. Continuation of Newton's method through bifurcation points, J.Appl.Mech.Tech.Phys., 36 (1969), 425–430.CrossRefzbMATHGoogle Scholar
  60. [60]
    H. Wacker. Minimierung des Rechenaufwandes für spezieller Randwertprobleme, Computing, 8 (1972), 275–291.CrossRefGoogle Scholar
  61. [61]
    H. Wacker, E. Zarzer and W. Zulehner. Optimal step size control for the globalized Newton methods, in Continuation Methods, ed. H. Wacker, Academic Press, New York, 1978, 249–277.Google Scholar
  62. [62]
    H. Wacker, ed. Continuation Methods, Academic Press, New York, 1978.zbMATHGoogle Scholar
  63. [63]
    E. Wasserstrom. Numerical solutions by the continuation method, SIAM Review, 15 (1973), 89–119.MathSciNetCrossRefGoogle Scholar
  64. [64]
    L.T. Watson. An algorithm that is globally convergent with probability one for a class of nonlinear two-point boundary value problems, SIAM J.Num. Anal., 16 (1979), 394–401.MathSciNetCrossRefzbMATHGoogle Scholar
  65. [65]
    L.T. Watson. A globally convergent algorithm for computing fixed points of C maps, Appl.Math. and Computation, 5 (1979), 297–311.MathSciNetCrossRefzbMATHGoogle Scholar
  66. [66]
    L.T. Watson and D. Fenner. Chow-Yorke algorithm for fixed points or zeros of C2 maps, ACM Trans. on Math. Software, 6 (1980), 252–259.CrossRefzbMATHGoogle Scholar
  67. [67]
    H. Weber. Numerische Behandlung von Verzweigungsproblemen bei gewöhnlichen Differentialgleichungen, Numer.Math., 32 (1979), 17–29.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • E. L. Allgower
    • 1
  1. 1.Mathematics DepartmentColorado State UniversityFort CollinsUSA

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