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References
R. Abraham and J. Robbin. Transversal Mappings and Flows, Benjamin, New York-Amsterdam, 1967.
J.C. Alexander. The topological theory of an embedding method, Continuation methods, H. Wacker, ed., Academic Press, New York, 1978.
J.C. Alexander and J.A. Yorke. The homotopy continuation method: Numerically implementable topological procedures, Trans.Amer.Math.Soc. 242 (1978), 271–284.
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.
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.
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.
E.L.Allgower, K.Böhmer and S.F.McCormick. Discrete correction methods for operator equations, these proceedings.
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.
E. Bohl. Chord techniques and Newton's method for discrete bifurcation problems, Numer.Math., 34 (1980), 111–124.
E. Bohl. On the bifurcation diagram of discrete analogues for ordinary bifurcation problems, Math.Meth. in the Appl.Sci., 1 (1979), 566–571.
F.J. Branin, Jr. Widely convergent method for finding multiple solutions of simultaneous nonlinear equations, IBM J.Res.Develop. 16 (1972), 504–522.
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).
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.
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.
M.G. Crandall and P.H. Rabinowitz. Bifurcation from simple eigenvalues, J.Func.Anal., 8 (1971), 321–340.
D. Davidenko. On a new method of numerically integrating a system of nonlinear equations, Dokl.Akad.Nauk SSSR, 88 (1953), 601–604. (In Russian)
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.
P. Deufelhard. A step size control for continuation methods and its special application to multiple shooting techniques, Numer.Math. 33 (1979), 115–146.
F.-J. Drexler. Eine Methode zur Berechnung sämtlicher Lösungen von Polynomgleichungssystemen, Numer.Math., 29 (1977), 45–58.
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.
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.
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.
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.
K.Georg. Numerical integration of the Davidenko equation, these proceedings.
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.
C. Haselgrove. Solution of nonlinear equations and of differential equations with two-point boundary conditions, Comput.J. 4 (1961), 255–259.
M. Hirsch and S. Smale. On algorithms for solving f(x) = 0, Comm.Pure Appl.Math., 32 (1979), 281–312.
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.
R.B.Kearfott. A derivative-free arc continuation method and a bifurcation technique, preprint (also see these proceedings).
J.P. Keener and H.B. Keller. Perturbed bifurcation theory, Arch.Rat.Mech.Anal. 50 (1973), 159–175.
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.
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.
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.
E. Lahaye. Une méthode de résolution d'une catégorie d'équations transcendantes, C.R.Acad.Sci., Paris, 198 (1934), 1840–1842.
W.F. Langford. Numerical solution of bifurcation problems for ordinary differential equations, preprint, McGill University, Montreal (1976).
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.
R. Menzel and H. Schwetlick. Über einen Ordnungsbegriff bei Einbettungsalgorithm zur Lösung nichtlinearer Gleichungen, Computing 16 (1976), 187–199.
R. Menzel and H. Schwetlick. Zur Lösung parameterabhängiger nichtlinearer Gleichungen mit singulären Jacobi-Matrizen, Numer.Math., 30 (1978), 65–79.
R. Menzel. Ein implementierbarer Algorithmus zur Lösung nichtlinearer Gleichungssysteme bei schwach singulärer Einbettung, Beiträge zur Numerischen Mathematik, 8 (1980), 99–111.
G. Meyer. On solving nonlinear equations with a one-parameter operator imbedding, SIAM J.Numer.Anal., 5 (1968), 739–752.
J.W. Milnor. Topology from the Differentiable Viewpoint, University Press of Virginia, Charlottesville, VA, 1969.
H.D.Mittelman and H.Weber. Numerical treatment of bifurcation problems, University of Dortmund, preprint, 1979.
Paul Nelson, Jr. Subcriticality for submultiplying steady-state neutron diffusion, in Nonlinear diffusion, ed. John Nohel, Research Notes in Math. 14, Pitman, London.
J.M. Ortega and W.C. Rheinboldt. Iterative Solutions of Nonlinear Equations in Several Variables, Academic Press, New York-London, 1970.
H.O. Peitgen and H.O. Walther, eds., Functional Differential Equations and Approximation of Fixed Points, Springer L.N.730
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.
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.
P.H. Rabinowitz. Some global results for nonlinear eigenvalue problems, J.Func.Anal., 7 (1971), 487–513.
W.C. Rheinboldt. Numerical methods for a class of finite-dimensional bifurcation problems, SIAM J.Numer.Anal., 15 (1978), 1–11.
W.C. Rheinboldt. Solution field of nonlinear equations and continuation methods, SIAM J.Numer.Anal., 17 (1980), 221–237.
E. Riks. The application of Newton's Method to the problem of elastic stability, J.Appl.Mech.Techn.Phys., 39 (1972), 1060–1065.
C.Schmidt. Approximating differential equations that describe homotopy paths, Univ. of Chicago School of Management Science Report 7931.
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.
H. Schwetlick. Ein neues Princip zur Konstruktion implementierbarer, global konvergenter Einbettungsalgorithmen, Beiträge Numer.Math., 4–5 (1975–6), 215–228; 201–206.
L.F. Shampine and M.K. Gordon. Computer Solution of Ordinary Differential Equations: The Initial Value Problem, Freeman Press, San Francisco, 1975.
G.Shearing. Ph.D. Thesis, Manchester (1960).
R.Seydel. Numerische Berechnung von Verzweigungen bei gewöhnlichen Differentialgleichungen, TUM-Math-7736 Technische Universität München, 1977.
S. Smale. A convergent process of price adjustment and global Newton methods, J.Math.Econ., 3 (1976), 1–14.
G.A. Thurston. Continuation of Newton's method through bifurcation points, J.Appl.Mech.Tech.Phys., 36 (1969), 425–430.
H. Wacker. Minimierung des Rechenaufwandes für spezieller Randwertprobleme, Computing, 8 (1972), 275–291.
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.
H. Wacker, ed. Continuation Methods, Academic Press, New York, 1978.
E. Wasserstrom. Numerical solutions by the continuation method, SIAM Review, 15 (1973), 89–119.
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.
L.T. Watson. A globally convergent algorithm for computing fixed points of C maps, Appl.Math. and Computation, 5 (1979), 297–311.
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.
H. Weber. Numerische Behandlung von Verzweigungsproblemen bei gewöhnlichen Differentialgleichungen, Numer.Math., 32 (1979), 17–29.
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Allgower, E.L. (1981). A survey of homotopy methods for smooth mappings. In: Allgower, E.L., Glashoff, K., Peitgen, HO. (eds) Numerical Solution of Nonlinear Equations. Lecture Notes in Mathematics, vol 878. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0090675
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