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Continuation and path following

Published online by Cambridge University Press:  07 November 2008

Eugene L. Allgower  and
Kurt Georg
Eugene L. Allgower
Affiliation:
Department of MathematicsColorado State UniversityFt. Collins, CO 80523, USA, E-mail: Georg@Math.ColoState.Edu
Kurt Georg
Affiliation:
Department of MathematicsColorado State UniversityFt. Collins, CO 80523, USA, E-mail: Georg@Math.ColoState.Edu
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Abstract

The main ideas of path following by predictor–corrector and piecewise-linear methods, and their application in the direction of homotopy methods and nonlinear eigenvalue problems are reviewed. Further new applications to areas such as polynomial systems of equations, linear eigenvalue problems, interior methods for linear programming, parametric programming and complex bifurcation are surveyed. Complexity issues and available software are also discussed.

Type
Research Article
Information
Acta Numerica , Volume 2 , January 1993 , pp. 1 - 64
Copyright
Copyright © Cambridge University Press 1993

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References

REFERENCES

Abraham, R. and Robbin, J. (1967), Transversal Mappings and Flows, W. A. Benjamin (New York).Google Scholar
Adler, I., Resende, G. C., Veiga, G. and Karmarkar, N. (1989), ‘An implementation of Karmarkar's algorithm for linear programming’, Math. Programming 44, 297–335.CrossRefGoogle Scholar
Alexander, J. C. (1987), ‘Average intersection and pivoting densities’, SIAM J. Numer. Anal. 24, 129–146.CrossRefGoogle Scholar
Alexander, J. C. and Yorke, J. A. (1978), ‘Homotopy continuation method: Numerically implementable topological procedures’, Trans. Amer. Math. Soc. 242, 271–284.CrossRefGoogle Scholar
Allgower, E. L. (1984), ‘Bifurcations arising in the calculation of critical points via homotopy methods’, in Numerical Methods for Bifurcation Problems Vol. 70 of ISNM (Küpper, T., Mittelmann, H. D. and Weber, H., eds), Birkhäuser (Basel) 15–28.CrossRefGoogle Scholar
Allgower, E. L. and Chien, C.-S. (1986), ‘Continuation and local perturbation for multiple bifurcations’, SIAM J. Sci. Statist. Comput. 7, 1265–1281.CrossRefGoogle Scholar
Allgower, E. L. and Georg, K. (1979), ‘Generation of triangulations by reflections’, Utilitas Math. 16, 123–129.Google Scholar
Allgower, E. L. and Georg, K. (1983a), Predictor-corrector and simplicial methods for approximating fixed points and zero points of nonlinear mappings, in Mathematical Programming: The State of the Art (Bachem, A., Grötschel, M. and Korte, B., eds), Springer (Berlin, Heidelberg, New York), 15–56.CrossRefGoogle Scholar
Allgower, E. L. and Georg, K. (1983b), Relationships between deflation and global methods in the problem of approximating additional zeros of a system of nonlinear equations, in Homotopy Methods and Global Convergence (Eaves, B. C., Gould, F. J., Peitgen, H.-O. and Todd, M. J., eds), Plenum (New York), 31–42.CrossRefGoogle Scholar
Allgower, E. L. and Georg, K. (1990), Numerical Continuation Methods: An Introduction, Vol. 13 of Series in Computational Mathematics, Springer (Berlin, Heidelberg, New York), 388.CrossRefGoogle Scholar
Allgower, E. L. and Gnutzmann, S. (1987), ‘An algorithm for piecewise linear approximation of implicitly denned two-dimensional surfaces’, SIAM J. Numer. Anal. 24, 452–469.CrossRefGoogle Scholar
Allgower, E. L. and Gnutzmann, S. (1991), ‘Simplicial pivoting for mesh generation of implicitly defined surfaces’, Computer Aided Geometric Design 8, 305–325.CrossRefGoogle Scholar
Allgower, E. L. and Schmidt, P. H. (1985), ‘An algorithm for piecewise-linear approximation of an implicitly denned manifold’, SIAM J. Numer. Anal. 22, 322–346.CrossRefGoogle Scholar
Allgower, E. L., Böhmer, K. and Golubitsky, M., eds (1992a), Bifurcation and Symmetry, Vol. 104 of ISNM, Birkhäuser (Basel).CrossRefGoogle Scholar
Allgower, E. L., Böhmer, K. and Mei, Z. (1991a), ‘A complete bifurcation scenario for the 2-D nonlinear Laplacian with Neumann boundary conditions’, in Bifurcation and Chaos: Analysis, Algorithms, Applications (Seydel, R., Schneider, F. W., Küpper, T. and Troger, H., eds), Vol. 97 of ISNM, Birkhäuser (Basel), 1–18.Google Scholar
Allgower, E. L., Böhmer, K. and Mei, Z. (1991b), ‘On new bifurcation results for semi-linear elliptic equations with symmetries’, in The Mathematics of Finite Elements and Applications VII (Whiteman, J., ed.), MAFELAP 1990, Academic (Brunei), 487–494.Google Scholar
Allgower, E. L., Böhmer, K., Georg, K. and Miranda, R. (1992b), ‘Exploiting symmetry in boundary element methods’, SIAM J. Numer. Anal. 29, 534–552.CrossRefGoogle Scholar
Allgower, E. L., Chien, C.-S., Georg, K. and Wang, C.-F. (1991c), ‘Conjugate gradient methods for continuation problems’, J. Comput. Appl. Math. 38, 1–16.CrossRefGoogle Scholar
Allgower, E. L., Georg, K. and Widmann, R. (1991d), ‘Volume integrals for boundary element methods’, J. Comput. Appl. Math. 38, 17–29.CrossRefGoogle Scholar
Allison, D. C. S., Harimoto, S. and Watson, L. T. (1989), ‘The granularity of parallel homotopy algorithms for polynomial systems of equations’, Int. J. Comput. Math. 29, 21–37.CrossRefGoogle Scholar
Arun, V., Reinholtz, C. F. and Watson, L. T. (1990), ‘Enumeration and analysis of variable geometry truss manipulators’, Technical Report TR 90–10, Virginia Polytechnic Institute, Blacksburg.Google Scholar
Bank, B., Guddat, J., Klatte, D., Kummer, B. and Tammer, K. (1983), Non-Linear Parametric Optimization, Birkhäuser (Basel).Google Scholar
Bank, R. E. and Chan, T. F. (1986), ‘PLTMGC: A multi-grid continuation program for parameterized nonlinear elliptic systems’, SIAM J. Sci. Statist. Comput. 7, 540–559.CrossRefGoogle Scholar
Bayer, D. and Lagarias, J. C. (1989), ‘The nonlinear geometry of linear programming, I: Affine and projective scaling trajectories, II: Legendre transform coordinates and central trajectories’, Trans. Amer. Math. Soc. 314, 499–581.Google Scholar
Ben-Israel, A. and Greville, T. N. E. (1974), Generalized Inverses: Theory and Applications, John Wiley (New York).Google Scholar
Bernstein, S. (1910), ‘Sur la généralisation du problème de Dirichlet’, Math. Ann. 69, 82–136.CrossRefGoogle Scholar
Bixby, R. E., Gregory, J. W., Lustig, I. J., Marsten, R. E. and Shanno, D. F. (1991), ‘Very large-scale linear programming: A case study in combining interior point and simplex methods’, Technical Report RRR 34–91, Rutgers University, New Brunswick, NJ.CrossRefGoogle Scholar
Bolstad, J. H. and Keller, H. B. (1986), ‘A multigrid continuation method for elliptic problems with folds’, SIAM J. Sci. Statist. Comput. 7, 1081–1104.CrossRefGoogle Scholar
Branin, F. H. (1972), ‘Widely convergent method for finding multiple solutions of simultaneous nonlinear equations’, IBM J. Res. Develop. 16, 504–522.CrossRefGoogle Scholar
Branin, F. H. and Hoo, S. K. (1972), ‘A method for finding multiple extrema of a function of n variables’, in Numerical Methods for Non-linear Optimization (Lootsma, F. A., ed.), Academic (New York, London), 231–237.Google Scholar
Brezzi, F., Rappaz, J. and Raviart, P. A. (1980a), ‘Finite dimensional approximation of nonlinear problems. Part 1: Branches of nonsingular solutions’, Numer. Math. 36, 1–25.CrossRefGoogle Scholar
Brezzi, F., Rappaz, J. and Raviart, P. A. (1980b), ‘Finite dimensional approximation of nonlinear problems. Part 2: Limit points’, Numer. Math. 37, 1–28.CrossRefGoogle Scholar
Brezzi, F., Rappaz, J. and Raviart, P. A. (1981), ‘Finite dimensional approximation of nonlinear problems. Part 3: Simple bifurcation points’, Numer. Math. 38, 1–30.CrossRefGoogle Scholar
Brouwer, L. E. J. (1912), ‘Über Abbildung von Mannigfaltigkeiten’, Math. Ann. 71, 97–115.CrossRefGoogle Scholar
Brown, K. M. and Gearhart, W. B. (1971), ‘Deflation techniques for the calculation of further solutions of a nonlinear system’, Numer. Math. 16, 334–342.CrossRefGoogle Scholar
Carpenter, T. J. and Shanno, D. F. (1991), ‘An interior point method for quadratic programs based on conjugate projected gradients’, Technical Report RRR 55–91, Rutgers University, New Brunswick, NJ.Google Scholar
Chan, T. F. (1984a), ‘Deflation techniques and block-elimination algorithms for solving bordered singular systems’, SIAM J. Sci. Statist. Comput. 5, 121–134.CrossRefGoogle Scholar
Chan, T. F. (1984b), ‘Newton-like pseudo-arclength methods for computing simple turning points’, SIAM J. Sci. Statist. Comput. 5, 135–148.CrossRefGoogle Scholar
Chien, C.-S. (1989), ‘Secondary bifurcations in the buckling problem’, J. Comput. Appl. Math. 25, 277–287.CrossRefGoogle Scholar
Chow, S. N. and Hale, J. K. (1982), Methods of Bifurcation Theory, Springer (New York).CrossRefGoogle Scholar
Chow, S. N., Mallet-Paret, J. and Yorke, J. A. (1978), ‘Finding zeros of maps: Homotopy methods that are constructive with probability one’, Math. Comput. 32, 887–899.CrossRefGoogle Scholar
Chow, S. N., Mallet-Paret, J. and Yorke, J. A. (1979), A homotopy method for locating all zeros of a system of polynomials, in Functional Differential Equations and Approximation of Fixed Points (Peitgen, H.-O. and Walther, H.-O., eds), Vol. 730 of Lecture Notes in Mathematics, Springer (Berlin, Heidelberg, New York), 77–88.CrossRefGoogle Scholar
Chu, M. T. (1984a), ‘The generalized Toda flow, the QR algorithm and the center manifold theory’, SIAM J. Alg. Disc. Meth. 5, 187–201.CrossRefGoogle Scholar
Chu, M. T. (1984b), ‘A simple application of the homotopy method to symmetric eigenvalue problems’, Lin. Alg. Appl. 59, 85–90.CrossRefGoogle Scholar
Chu, M. T. (1986), ‘A continuous approximation to the generalized Schur's decomposition’, Lin. Alg. Appl. 78, 119–132.CrossRefGoogle Scholar
Chu, M. T. (1988), ‘On the continuous realization of iterative processes’, SIAM Rev. 30, 375–387.CrossRefGoogle Scholar
Chu, M. T. (1990), ‘Solving additive inverse eigenvalue problems for symmetric matrices by the homotopy method’, IMA J. Numer. Anal. 9, 331–342.CrossRefGoogle Scholar
Chu, M. T. (1991), ‘A continuous Jacobi-like approach to the simultaneous reduction of real matrices’, Lin. Alg. Appl. 147, 75–96.CrossRefGoogle Scholar
Chu, M. T., Li, T.-Y. and Sauer, T. (1988), ‘Homotopy methods for general λ-matrix problems’, SIAM J. Matrix Anal. Appl. 9, 528–536.CrossRefGoogle Scholar
Cliffe, K. A. and Winters, K. H. (1986), ‘The use of symmetry in bifurcation calculations and its application to the Bénard problem’, J. Comput. Phys. 67, 310–326.CrossRefGoogle Scholar
Cottle, R. W. (1974), ‘Solution rays for a class of complementarity problems’, Math. Programming Study 1, 58–70.Google Scholar
Cottle, R. W. and Dantzig, G. B. (1968), ‘Complementary pivot theory of mathematical programming’, Lin. Alg. Appl. 1, 103–125.CrossRefGoogle Scholar
Cottle, R. W., Gianessi, F. and Lions, J. L., eds (1980), Variational Inequalities and Complentarity Problems, John Wiley (London).Google Scholar
Cottle, R. W., Golub, G. H. and Sacher, R. S. (1978), ‘On the solution of large structured linear complementarity problems: The block partitioned case’, Appl. Math. Optim. 4, 347–363.CrossRefGoogle Scholar
Coxeter, H. S. M. (1934), ‘Discrete groups generated by reflections’, Ann. Math. 6, 13–29.Google Scholar
Coxeter, H. S. M. (1973), Regular Polytopes, third edition, Dover (New York).Google Scholar
Crandall, M. G. and Rabinowitz, P. H. (1971), ‘Bifurcation from simple eigenvalues’, J. Fund. Anal. 8, 321–340.CrossRefGoogle Scholar
Crouzeix, M. and Rappaz, J. (1990), 'On Numerical Approximation in Bifurcation Theory, RMA, Masson (Paris).Google Scholar
Dai, R.-X. and Rheinboldt, W. C. (1990), ‘On the computation of manifolds of fold points for parameter-dependent problems’, SIAM J. Numer. Anal. 27, 437–446.CrossRefGoogle Scholar
Dai, Y. and Yamamoto, Y. (1989), ‘The path following algorithm for stationary point problems on polyhedral cones’, J. Op. Res. Soc. Japan 32, 286–309.Google Scholar
Dai, Y., Sekitani, K. and Yamamoto, Y. (1992), ‘A variable dimension algorithm with the Dantzig-Wolfe decomposition for structured stationary point problems’, ZOR — Methods and Models of Operations Research 36, 23–53.CrossRefGoogle Scholar
Dantzig, G. B. (1963), Linear Programming and Extensions Princeton University (Princeton, NJ).Google Scholar
Davidenko, D. (1953), ‘On a new method of numerical solution of systems of non-linear equations’, Dokl. Akad. Nauk USSR 88, 601–602. In Russian.Google Scholar
Dellnitz, M. and Werner, B. (1989), ‘Computational methods for bifurcation problems with symmetries — with special attention to steady state and Hopf bifurcation points’, J. Comput. Appl. Math. 26, 97–123.CrossRefGoogle Scholar
Heijer, C. Den and Rheinboldt, W. C. (1981), ‘On steplength algorithms for a class of continuation methods’, SIAM J. Numer. Anal. 18, 925–948.CrossRefGoogle Scholar
deSa, C., Irani, K. M., Ribbens, C. G., Watson, L. T. and Walker, H. F. (1992), ‘Preconditioned iterative methods for homotopy curve tracking’, SIAM J. Sci. Statist. Comput. 13, 30–46.CrossRefGoogle Scholar
Deuflhard, P. and Heindl, G. (1979), ‘Affine invariant convergence theorems for Newton's method and extensions to related methods’, SIAM Numer. Anal. 16, 1–10.CrossRefGoogle Scholar
Deuflhard, P., Fiedler, B. and Kunkel, P. (1987), ‘Efficient numerical pathfollowing beyond critical points’, SIAM J. Numer. Anal. 24, 912–927.CrossRefGoogle Scholar
Doedel, E. and Kernévez, J. P. (1986), AUTO: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations, California Institute of Technology.Google Scholar
Doedel, E., Keller, H. B. and Kernévez, J. P. (1991a), ‘Numerical analysis and control of bifurcation problems. Part I: Bifurcation in finite dimensions’, Int. J. Bifurcation and Chaos.Google Scholar
Doedel, E., Keller, H. B. and Kernévez, J. P. (1991b), ‘Numerical analysis and control of bifurcation problems. Part II: Bifurcation in infinite dimensions’, Int. J. Bifurcation and Chaos.Google Scholar
Eaves, B. C. (1972), ‘Homotopies for the computation of fixed points’, Math. Programming 2, 1–22.CrossRefGoogle Scholar
Eaves, B. C. (1976), ‘A short course in solving equations with pi homotopies’, in Nonlinear Programming (Cottle, R. W. and Lemke, C. E., eds), Vol. 9 of SIAM-AMS Proc., American Mathematical Society (Providence, RI), 73–143.Google Scholar
Eaves, B. C. (1984), A Course in Triangulations for Solving Equations with Deformations, Vol. 234 of Lecture Notes in Economics and Mathematical Systems, Springer (Berlin, Heidelberg, New York).CrossRefGoogle Scholar
Eaves, B. C. and Saigal, R. (1972), ‘Homotopies for computation of fixed points on unbounded regions’, Math. Programming 2, 225–237.CrossRefGoogle Scholar
Eaves, B. C. and Scarf, H. (1976), ‘The solution of systems of piecewise linear equations’, Math. Oper. Res. 1, 1–27.CrossRefGoogle Scholar
Eaves, B. C. and Yorke, J. A. (1984), ‘Equivalence of surface density and average directional density’, Math. Oper. Res. 9, 363–375.CrossRefGoogle Scholar
Fiacco, A. V. (1983), Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Academic (New York).Google Scholar
Fiacco, A. V., ed. (1984), Sensitivity, Stability and Parametric Analysis, Vol. 21 of Mathematical Programming Study, North-Holland (Amsterdam).CrossRefGoogle Scholar
Fiacco, A. V. and McCormick, G. (1968), Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wiley (New York, NY).Google Scholar
Fink, J. P. and Rheinboldt, W. C. (1983), ‘On the discretization error of parametrized nonlinear equations’, SIAM J. Numer. Anal. 20, 732–746.CrossRefGoogle Scholar
Fink, J. P. and Rheinboldt, W. C. (1984), ‘Solution manifolds and submanifolds of parametrized equations and their discretization errors’, Numer. Math. 45, 323–343.CrossRefGoogle Scholar
Fink, J. P. and Rheinboldt, W. C. (1985), ‘Local error estimates for parametrized nonlinear equations’, SIAM J. Numer. Anal. 22, 729–735.CrossRefGoogle Scholar
Fink, J. P. and Rheinboldt, W. C. (1986), ‘Folds on the solution manifold of a parametrized equation’, SIAM J. Numer. Anal. 23, 693–706.CrossRefGoogle Scholar
Fink, J. P. and Rheinboldt, W. C. (1987), ‘A geometric framework for the numerical study of singular points’, SIAM J. Numer. Anal. 24, 618–633.CrossRefGoogle Scholar
Freudenthal, H. (1942), ‘Simplizialzerlegungen von beschränkter Flachheit’, Ann. Math. 43, 580–582.CrossRefGoogle Scholar
Freund, R. M. (1984a), ‘Variable dimension complexes. Part I: Basic theory’, Math. Oper. Res. 9, 479–497.CrossRefGoogle Scholar
Freund, R. M. (1984b), ‘Variable dimension complexes. Part II: A unified approach to some combinatorial lemmas in topology’, Math. Oper. Res. 9, 498–509.CrossRefGoogle Scholar
Freund, R. M. (1991), ‘A potential-function reduction algorithm for solving a linear program directly from an infeasible ‘warm start’’, in Interior Point Methods for Linear Programming: Theory and Practice (Roos, C. and Vial, J.-P., eds), Vol. 52 of Math. Programming, Ser. B, Mathematical Programming Society (North-Holland, Amsterdam), 441–446.Google Scholar
Freund, R. W., Golub, G. H. and Nachtigal, N. M. (1992), ‘Iterative solution of linear systems’, Acta Numerica 1, 57–100.CrossRefGoogle Scholar
Garcia, C. B. and Gould, F. J. (1978), ‘A theorem on homotopy paths’, Math. Oper. Res. 2, 282–289.CrossRefGoogle Scholar
Garcia, C. B. and Gould, F. J. (1980), ‘Relations between several path following algorithms and local and global Newton methods’, SIAM Rev. 22, 263–274.CrossRefGoogle Scholar
Garcia, C. B. and Zangwill, W. I. (1979a), ‘An approach to homotopy and degree theory’, Math. Oper. Res. 4, 390–405.CrossRefGoogle Scholar
Garcia, C. B. and Zangwill, W. I. (1979b), ‘Finding all solutions to polynomial systems and other systems of equations’, Math. Programming 16, 159–176.CrossRefGoogle Scholar
Garcia, C. B. and Zangwill, W. I. (1981), Pathways to Solutions, Fixed Points, and Equilibria, Prentice-Hall (Englewood Cliffs, NJ).Google Scholar
Garratt, T. J., Moore, G. and Spence, A. (1991), ‘Two methods for the numerical detection of Hopf bifurcations’, in Bifurcation and Chaos: Analysis, Algorithms, Applications (Seydel, R., Schneider, F. W., Küpper, T. and Troger, H., eds), Vol. 97 of ISNM, Birkhäuser (Basel), 129–134.CrossRefGoogle Scholar
Georg, K. (1981), ‘On tracing an implicitly denned curve by quasi-Newton steps and calculating bifurcation by local perturbation’, SIAM J. Sci. Statist. Comput. 2, 35–50.CrossRefGoogle Scholar
Georg, K. (1982), ‘Zur numerischen Realisierung von Kontinuitätsmethoden mit Prädiktor-Korrektor- oder simplizialen Verfahren’, Habilitationsschrift, University of Bonn (Germany).Google Scholar
Georg, K. (1983), ‘A note on stepsize control for numerical curve following’, in Homotopy Methods and Global Convergence (Eaves, B. C., Gould, F. J., Peitgen, H.-O. and Todd, M. J., eds), Plenum (New York), 145–154.CrossRefGoogle Scholar
Georg, K. (1990), ‘An introduction to PL algorithms’, in Computational Solution of Nonlinear Systems of Equations (Allgower, E. L. and Georg, K., eds), Vol. 26 of Lectures in Applied Mathematics, American Mathematical Society (Providence, RI), 207–236.Google Scholar
Georg, K. (1991), ‘Approximation of integrals for boundary element methods’, SIAM J. Sci. Statist. Comput. 12, 443–453.CrossRefGoogle Scholar
Georg, K. and Miranda, R. (1990), ‘Symmetry aspects in numerical linear algebra with applications to boundary element methods’, Preprint, Colorado State University.Google Scholar
Georg, K. and Miranda, R. (1992), ‘Exploiting symmetry in solving linear equations’, in Bifurcation and Symmetry (Allgower, E. L., Böhmer, K. and Golubitsky, M., eds), Vol. 104 of ISNM, Birkhäuser (Basel), 157–168.CrossRefGoogle Scholar
Gfrerer, H., Guddat, J. and Wacker, H.-J. (1983), ‘A globally convergent algorithm based on imbedding and parametric optimization’, Computing 30, 225–252.CrossRefGoogle Scholar
Gfrerer, H., Guddat, J., Wacker, H.-J. and Zulehner, W. (1985), ‘Path-following for Kuhn-Tucker curves by an active set strategy’, in Systems and Optimization (Baghi, A. and Jongen, H. T., eds), Vol. 66 of Lecture Notes Contr. Inf. Sc., Springer (Berlin, Heidelberg, New York), 111–132.CrossRefGoogle Scholar
Gill, P. E., Murray, W., Saunders, M. A., Tomlin, J. A. and Wright, M. H. (1986), ‘On projected Newton barrier methods for linear programming and an equivalence to Karmarkar's projective method’, Math. Programming 36, 183–209.CrossRefGoogle Scholar
Glowinski, R., Keller, H. B. and Reinhart, L. (1985), ‘Continuation-conjugate gradient methods for the least squares solution of nonlinear boundary value problems’, SIAM J. Sci. Statist. Comput. 6, 793–832.CrossRefGoogle Scholar
Gnutzmann, S. (1989), ‘Stückweise lineare Approximation implizit definierter Mannigfaltigkeiten’, PhD thesis, University of Hamburg, Germany.Google Scholar
Goldfarb, D. and Liu, S. (1991), ‘An algorithm for solving linear programming problems in O(n 3L) operations’, Math. Programming 49, 325–340.CrossRefGoogle Scholar
Golub, G. H. and van Loan, C. F. (1989), Matrix Computations, second edn, J. Hopkins University (Baltimore, London).Google Scholar
Golubitsky, M. and Schaeffer, D. G. (1985), Singularities and Groups in Bifurcation Theory, Vol. 1, Springer (Berlin, Heidelberg, New York).CrossRefGoogle Scholar
Golubitsky, M., Stewart, I. and Schaeffer, D. G. (1988), Singularities and Groups in Bifurcation Theory, Vol. 2, Springer (Berlin, Heidelberg, New York).CrossRefGoogle Scholar
Gonzaga, C. C. (1988), ‘An algorithm for solving linear programming problems in O(n 3L) operations,, in Progress in Mathematical Programming, Interior Point and Related Methods (Megiddo, N., ed.), Springer (New York), 1–28.Google Scholar
Gonzaga, C. C. (1992), ‘Path-following methods for linear programming’, SIAM Rev. 34, 167–224.CrossRefGoogle Scholar
Gould, F. J. and Tolle, J. W. (1983), Complementary Pivoting on a Pseudomanifold Structure with Applications on the Decision Sciences, Vol. 2 of Sigma Series in Applied Mathematics, Heldermann (Berlin).Google Scholar
Griewank, A. (1985), ‘On solving nonlinear equations with simple singularities or nearly singular solutions’, SIAM Rev. 27, 537–563.CrossRefGoogle Scholar
Griewank, A. and Reddien, G. W. (1984), ‘Characterization and computation of generalized turning points’, SIAM J. Numer. Anal. 21, 176–185.CrossRefGoogle Scholar
Guddat, J., Vasquez, F. Guerra and Jongen, H. T. (1990), Parametric Optimization: Singularities, Path Following, and Jumps, John Wiley (Chichester).CrossRefGoogle Scholar
Guddat, J., Jongen, H. T., Kummer, B. and Nožička, F., eds (1987), Parametric Optimization and Related Topics, Akademie (Berlin).Google Scholar
Haselgrove, C. B. (1961), ‘The solution of nonlinear equations and of differential equations with two-point boundary conditions’, Comput. J. 4, 255–259.CrossRefGoogle Scholar
Healey, T. J. (1988a), ‘Global bifurcation and continuation in the presence of symmetry with an application to solid mechanics’, SIAM J. Math. Anal. 19, 824–840.CrossRefGoogle Scholar
Healey, T. J. (1988b), ‘A group theoretic approach to computational bifurcation problems with symmetry’, Comput. Meth. Appl. Mech. Eng. 67, 257–295.CrossRefGoogle Scholar
Healey, T. J. (1989), ‘Symmetry and equivariance in nonlinear elastostatics. Part I’, Arch. Rational Mech. Anal. 105, 205–228.CrossRefGoogle Scholar
Healey, T. J. and Treacy, J. A. (1991), ‘Exact block diagonalization of large eigenvalue problems for structures with symmetry’, Int. J. Numer. Methods Engrg. 31, 265–285.CrossRefGoogle Scholar
Henderson, M. E. (1985), ‘Complex Bifurcation’, PhD thesis, CALTECH, Pasadena.Google Scholar
Henderson, M. E. and Keller, H. B. (1990), ‘Complex bifurcation from real paths’, SIAM J. Appl. Math. 50, 460–482.CrossRefGoogle Scholar
Hirsch, M. W. (1963), ‘A proof of the nonretractibility of a cell onto its boundary’, Proc. Amer. Math. Soc. 14, 364–365.Google Scholar
Hirsch, M. W. (1976), Differential Topology, Springer (Berlin, Heidelberg, New York).CrossRefGoogle Scholar
Holodniok, M. and Kubíček, M. (1984), ‘DERPER — An algorithm for the continuation of periodic solutions in ordinary differential equations’, J. Comput. Phys. 55, 254–267.CrossRefGoogle Scholar
Hong, B. (1991), ‘Computational methods for bifurcation problems with symmertries on the manifold’, Technical Report ICMA-91–163, University of Pittsburgh, PA.CrossRefGoogle Scholar
Hornung, U. and Mittelmann, H. D. (1991), ‘Bifurcation of axially symmetric capillary surfaces’, J. Coll. Interf. Sci. 146, 219–225.CrossRefGoogle Scholar
Huitfieldt, J. (1991), ‘Nonlinear eigenvalue problems — prediction of bifurcation points and branch switching’, Technical Report 17, University of Göteborg, Sweden.Google Scholar
Huitfieldt, J. and Ruhe, A. (1990), ‘A new algorithm for numerical path following applied to an example from hydrodynamical flow’, SIAM J. Sci. Statist. Comput. 11, 1181–1192.CrossRefGoogle Scholar
Jepson, A. D. and Decker, D. W. (1986), ‘Convergence cones near bifurcation’, SIAM J. Numer. Anal. 23, 959–975.CrossRefGoogle Scholar
Jepson, A. D., Spence, A. and Cliffe, K. A. (1991), ‘The numerical solution of nonlinear equations having several parameters. Part III: Equations with Z2-symmetry’, SIAM J. Numer. Anal. 28, 809–832.CrossRefGoogle Scholar
Jongen, H. T. and Weber, G.-W. (1990), ‘On parametric nonlinear programming’, Ann. Oper. Res. 27, 253–284.CrossRefGoogle Scholar
Jongen, H. T., Jonker, P. and Twilt, F. (1983), Nonlinear Optimization in RN. I. Morse Theory, Chebyshev Approximation, Peter Lang (New York).Google Scholar
Jongen, H. T., Jonker, P. and Twilt, F. (1986), Nonlinear Optimization in RN. II. Transversality, Flows, Parametric Aspects, Peter Lang (New York).Google Scholar
Kaas-Petersen, C. (1989), PATH — User's Guide, University of Leeds, England.Google Scholar
Kamiya, K. and Talman, A. J. J. (1990), ‘Variable dimension simplicial algorithm for balanced games’, Technical Report 9025, Tilburg University, The Netherlands.Google Scholar
Karmarkar, N. K. (1984), ‘A new polynomial-time algorithm for linear programming’, Combinatorica 4, 373–395.CrossRefGoogle Scholar
Karmarkar, N. K. and Ramakrishnan, K. G. (1991), ‘Computational results of an interior point algorithm for large scale linear programming’, in Interior Point Methods for Linear Programming: Theory and Practice (Roos, C. and Vial, J.-P., eds), Vol. 52 of Math. Programming, Ser. B, Mathematical Programming Society, North-Holland (Amsterdam), 555–586.Google Scholar
Kearfott, R. B. (1989), ‘An interval step control for continuation methods’, Math. Comput. to appear.Google Scholar
Kearfott, R. B. (1990), ‘Interval arithmetic techniques in the computational solution of nonlinear systems of equations: Introduction, examples and comparisons’, in Computational Solution of Nonlinear Systems of Equations (Allgower, E. L. and Georg, K., eds), Vol. 26 of Lectures in Applied Mathematics, American Mathematical Society (Providence, RI), 337–357.Google Scholar
Keener, J. P. and Keller, H. B. (1974), ‘Perturbed bifurcation theory’, Arch. Rational Mech. Anal. 50, 159–175.CrossRefGoogle Scholar
Keller, H. B. (1970), ‘Nonlinear bifurcation’, J. Diff. Eq. 7, 417–434.CrossRefGoogle Scholar
Keller, H. B. (1977), ‘Numerical solution of bifurcation and nonlinear eigenvalue problems’, in Applications of Bifurcation Theory (Rabinowitz, P. H., ed.), Academic (New York, London), 359–384.Google Scholar
Keller, H. B. (1978), ‘Global homotopies and Newton methods’, in Recent Advances in Numerical Analysis (de Boor, C. and Golub, G. H., eds), Academic (New York), 73–94.CrossRefGoogle Scholar
Keller, H. B. (1983), ‘The bordering algorithm and path following near singular points of higher nullity’, SIAM J. Sci. Statist. Comput. 4, 573–582.CrossRefGoogle Scholar
Keller, H. B. (1987), Lectures on Numerical Methods in Bifurcation Problems, Springer (Berlin, Heidelberg, New York).Google Scholar
Kellogg, R. B., Li, T.-Y. and Yorke, J. A. (1976), ‘A constructive proof of the Brouwer fixed point theorem and computational results’, SIAM J. Numer. Anal. 13, 473–483.CrossRefGoogle Scholar
Khachiyan, L. G. (1979), ‘A polynomial algorithm in linear programming’, Sov. Math. Dokl. 20, 191–194.Google Scholar
Klein, F. (1882–1883), ‘Neue Beiträge zur Riemannschen Funktionentheorie’, Math. Ann.Google Scholar
Kojima, M. (1974), ‘Computational methods for solving the nonlinear complementarity problem’, Keio Engrg. Rep. 27, 1–41.Google Scholar
Kojima, M. (1979), ‘A complementarity pivoting approach to parametric programming’, Math. Oper. Res. 4, 464–77.CrossRefGoogle Scholar
Kojima, M. and Hirabayashi, R. (1984), ‘Sensitivity, stability and parametric analysis’, Vol. 21 of Mathematical Programming Study, North-Holland (Amsterdam), 150–198.Google Scholar
Kojima, M. and Yamamoto, Y. (1982), ‘Variable dimension algorithms: Basic theory, interpretation, and extensions of some existing methods’, Math. Programming 24, 177–215.CrossRefGoogle Scholar
Kojima, M. and Yamamoto, Y. (1984), ‘A unified approach to the implementation of several restart fixed point algorithms and a new variable dimension algorithm’, Math. Programming 28, 288–328.CrossRefGoogle Scholar
Kojima, M., Megiddo, N. and Mizuno, S. (1990a), ‘A general framework of continuation methods for complementarity problems’, Technical report, IBM Almaden Research Center, San Jose, CA.Google Scholar
Kojima, M., Megiddo, N. and Mizuno, S. (1991a), ‘Theoretical convergence of large-step primal-dual interior point algorithms for linear programming’, preprint, Tokyo Inst. of Techn.Google Scholar
Kojima, M., Megiddo, N. and Noma, T. (1991b), ‘Homotopy continuation methods for nonlinear complementarity problems’, Math. Oper. Res. 16, 754–774.CrossRefGoogle Scholar
Kojima, M., Megiddo, N. and Ye, Y. (1992), ‘An interior point potential reduction algorithm for the linear complementarity problem’, Math. Programming 54, 267–279.CrossRefGoogle Scholar
Kojima, M., Megiddo, N., Noma, T. and Yoshise, A. (1991c), A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems, Vol. 538 of Lecture Notes in Computer Science, Springer (Berlin).CrossRefGoogle Scholar
Kojima, M., Mizuno, S. and Noma, T. (1990b), ‘Limiting behavior of trajectories generated by a continuation method for monotone complementarity problems’, Math. Oper. Res. 15, 662–675.CrossRefGoogle Scholar
Kojima, M., Mizuno, S. and Yoshise, A. (1988), ‘A primal-dual interior point algorithm for linear programming’, in Progress in Mathematical Programming, Interior Point and Related Methods (Megiddo, N., ed.), Springer (New York), 29–47.Google Scholar
Kojima, M., Mizuno, S. and Yoshise, A. (1989), ‘A polynomial-time algorithm for a class of linear complementarity problems’, Math. Programming 44, 1–26.CrossRefGoogle Scholar
Kojima, M., Mizuno, S. and Yoshise, A. (1991d), ‘An iteration potential reduction algorithm for linear complementarity problems’, Math. Programming 50, 331–342.CrossRefGoogle Scholar
Kojima, M., Nishino, H. and Sekine, T. (1976), ‘An extension of Lemke's method to the piecewise linear complementarity problem’, SIAM J. Appl. Math. 31, 600–613.CrossRefGoogle Scholar
Krasnosel'skĭ, M. A. (1964), Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon (New York).Google Scholar
Kubíček, M. (1976), ‘Algorithm 502. Dependence of solutions of nonlinear systems on a parameter’, ACM Trans. Math. Software 2, 98–107.CrossRefGoogle Scholar
Kuhn, H. W. (1969), ‘Approximate search for fixed points’, in Computing Methods in Optimization Problems 2 (Zadek, L. A., Neustat, L. W. and Balakrishnan, A. V., eds), Academic (New York), 199–211.Google Scholar
Lahaye, E. (1934), ‘Une méthode de resolution d'une categorie d'equations transcendantes’, C. R. Acad. Sci. Paris 198, 1840–1842.Google Scholar
Lemke, C. E. (1965), ‘Bimatrix equilibrium points and mathematical programming’, Management Sci. 11, 681–689.CrossRefGoogle Scholar
Lemke, C. E. (1980), ‘A survey of complementarity theory’, in Variational Inequalities and Complentarity Problems (Cottle, R. W., Gianessi, F. and Lions, J. L., eds), John Wiley (London).Google Scholar
Lemke, C. E. and Grotzinger, S. J. (1976), ‘On generalizing Shapley's index theory to labelled pseudo manifolds’, Math. Programming 10, 245–262.CrossRefGoogle Scholar
Lemke, C. E. and Howson, J. T. (1964), ‘Equilibrium points of bimatrix games’, SIAM J. Appl. Math. 12, 413–423.CrossRefGoogle Scholar
Leray, J. and Schauder, J. (1934), ‘Topologie et Équations fonctionelles’, Ann. Sci. École Norm. Sup. 51, 45–78.CrossRefGoogle Scholar
Li, K. and Li, T.-Y. (1992), ‘An algorithm for symmetric tridiagonal eigen-problems — divide and conquer with homotopy continuation’, Preprint, Michigan State University.Google Scholar
Li, T.-Y. and Rhee, N. H. (1989), ‘Homotopy algorithm for symmetric eigenvalue problems’, Numer. Math. 55, 265–280.CrossRefGoogle Scholar
Li, T.-Y. and Wang, X. (1992a), ‘Nonlinear homotopies for solving deficient polynomial systems with parameters’, SIAM J. Numer. Anal, to appear.CrossRefGoogle Scholar
Li, T.-Y. and Wang, X. (1992b), ‘Solving real polynomial systems with real homotopies’, Math. Comput. to appear.Google Scholar
Li, T.-Y. and Zeng, Z. (1992), ‘Homotopy-determinant algorithm for solving non-symmetric eigenvalue problems’, Math. Comput. to appear.CrossRefGoogle Scholar
Li, T.-Y., Sauer, T. and Yorke, J. A. (1987), ‘Numerical solution of a class of deficient polynomial systems’, SIAM J. Numer. Anal. 24, 435–451.CrossRefGoogle Scholar
Li, T.-Y., Sauer, T. and Yorke, J. A. (1989), ‘The cheater's homotopy: An efficient procedure for solving systems of polynomial equations’, SIAM J. Numer. Anal. 26, 1241–1251.CrossRefGoogle Scholar
Li, T.-Y., Zeng, Z. and Cong, L. (1992), ‘Solving eigenvalue problems of real nonsymmetric matrices with real homotopies’, SIAM J. Numer. Anal. 29, 229–248.CrossRefGoogle Scholar
Li, T.-Y., Zhang, H. and Sun, X.-H. (1991), ‘Parallel homotopy algorithm for the symmetric tridiagonal eigenvalue problem’, SIAM J. Sci. Statist. Comput. 12, 469–487.CrossRefGoogle Scholar
Liu, J. L. and Rheinboldt, W. C. (1991), ‘A posteriori error estimates for parametrized nonlinear equations’, in Nonlinear Computational Mechanics (Wriggers, P. and Wagner, W., eds), Springer (Heidelberg, Germany), 31–46.Google Scholar
Lundberg, B. N. and Poore, A. B. (1991), ‘Variable order Adams-Bashforth predictors with error-stepsize control for continuation methods’, SIAM J. Sci. Statist. Comput. 12, 695–723.CrossRefGoogle Scholar
Lundberg, B. N. and Poore, A. B. (1993), ‘Numerical continuation and singularity detection methods for parametric nonlinear programming’, SIAM J. Optim. to appear.CrossRefGoogle Scholar
Lustig, I. J., Marsten, R. E. and Shanno, D. F. (1991), ‘The interaction of algorithms and architectures for interior point methods’, Technical Report RRR 36–91, Rutgers Univ., New Brunswick, NJ.Google Scholar
Mackens, W. (1989), ‘Numerical differentiation of implicitly denned space curves’, Computing 41, 237–260.CrossRefGoogle Scholar
Marsten, R., Subramanian, R., Saltzman, M., Lustig, I. J. and Shanno, D. F. (1990), ‘Interior point methods for linear programming: Just call Newton, Lagrange and Fiacco and Mccormick!’, Interfaces 20, 105–116.CrossRefGoogle Scholar
Maurer, H. and Mittelmann, H. D. (1991), ‘The nonlinear beam via optimal control with bounded state variables’, Optimal Control Appl. Methods 12, 19–31.CrossRefGoogle Scholar
McShane, K. A., Monma, C. L. and Shanno, D. F. (1989), ‘An implementation of a primal-dual interior point method for linear programming’, ORSA J. Comput. 1, 70–83.CrossRefGoogle Scholar
Megiddo, N. (1988), ‘Pathways to the optimal set in linear programming’, in Progress in Mathematical Programming, Interior Point and Related Methods (Megiddo, N., ed.), Springer (New York), 131–158.Google Scholar
Megiddo, N. and Shub, M. (1989), ‘Boundary behavior of interior point algorithms in linear programming’, Math. Oper. Res. 14, 97–146.CrossRefGoogle Scholar
Mejia, R. (1986), ‘CONKUB: A conversational path-follower for systems of nonlinear equations’, J. Comput. Phys. 63, 67–84.CrossRefGoogle Scholar
Mejia, R. (1990), ‘Interactive program for continuation of solutions of large systems of nonlinear equations’, in Computational Solution of Nonlinear Systems of Equations (Allgower, E. L. and Georg, K., eds), Vol. 26 of Lectures in Applied Mathematics, American Mathematical Society (Providence, RI), 429–449.Google Scholar
Melhem, R. G. and Rheinboldt, W. C. (1982), ‘A comparison of methods for determining turning points of nonlinear equations’, Computing 29, 201–226.CrossRefGoogle Scholar
Melville, R. C., Trajkovic, L., Fang, S.-C. and Watson, L. T. (1990), ‘Globally convergent homotopy methods for the DC operating point problem’, Technical Report TR 90–61, Virginia Polytechnic Institute, Blacksburg.Google Scholar
Menzel, R. and Schwetlick, H. (1978), ‘Zur Lösung parameterabhängiger nichtlinearer Gleichungen mit singulären Jacobi-Matrizen’, Numer. Math. 30, 65–79.CrossRefGoogle Scholar
Menzel, R. and Schwetlick, H. (1985), ‘Parametrization via secant length and application to path following’, Numer. Math. 47, 401–412.CrossRefGoogle Scholar
Merrill, O. (1972), ‘Applications and extensions of an algorithm that computes fixed points of a certain upper semi-continuous point to set mapping’, Ph.D thesis, University of Michigan, Ann Arbor, MI.Google Scholar
Miersemann, E. and Mittelmann, H. D. (1989), ‘Continuation for parametrized nonlinear variational inequalities’, J. Comput. Appl. Math. 26, 23–34.CrossRefGoogle Scholar
Miersemann, E. and Mittelmann, H. D. (1990a), ‘Extension of Beckert's continuation method to variational inequalities’, Math. Nachr. 148, 183–195.CrossRefGoogle Scholar
Miersemann, E. and Mittelmann, H. D. (1990b), ‘A free boundary problem and stability for the rectangular plate’, Math. Methods Appl. Sci. 12, 129–138.CrossRefGoogle Scholar
Miersemann, E. and Mittelmann, H. D. (1990c), ‘On the stability in obstacle problems with applications to the beam and plate’, Z. Angew. Math. Mech. 71, 311–321.CrossRefGoogle Scholar
Miersemann, E. and Mittelmann, H. D. (1991), ‘Stability and continuation of solutions to obstacle problems’, J. Comput. Appl. Math. 35, 5–31.CrossRefGoogle Scholar
Miersemann, E. and Mittelmann, H. D. (1992), ‘Stability in obstacle problems for the von Karman plate’, SIAM J. Math. Anal, to appear.CrossRefGoogle Scholar
Milnor, J. W. (1969), Topology from the Differentiable Viewpoint, University Press of Virginia (Charlottesville, VA).Google Scholar
Mittelmann, H. D. (1990), ‘Nonlinear parametrized equations: New results for variational problems and inequalities’, in Computational Solution of Nonlinear Systems of Equations (Allgower, E. L. and Georg, K., eds), Vol. 26 of Lectures in Applied Mathematics, American Mathematical Society (Providence, RI), 451–456.Google Scholar
Mittelmann, H. D. and Roose, D., eds (1990), Continuation Techniques and Bifurcation Problems, Vol. 92 of ISNM, Birkhäuser (Berlin).CrossRefGoogle Scholar
Mizuno, S. (1992), ‘A new polynomial time method for a linear complementarity problem’, Math. Programming 56, 31–43.CrossRefGoogle Scholar
Mizuno, S., Todd, M. J. and Ye, Y. (1992), ‘On adaptive-step primal-dual interior-point algorithms for linear programming’, Math. Oper. Res. to appear.Google Scholar
Mizuno, S., Yoshise, A. and Kikuchi, T. (1989), ‘Practical polynomial time algorithms for linear complementarity problems’, J. Oper. Res. Soc. Japan 32, 75–92.Google Scholar
Monteiro, R. C. and Adler, I. (1989), ‘Interior path following primal-dual algorithms, I: Linear programming, II: Convex quadratic programming’, Math. Programming 44, 27–66.CrossRefGoogle Scholar
Moore, G. and Spence, A. (1980), ‘The calculation of turning points of nonlinear equations’, SIAM J. Numer. Anal. 17, 567–576.CrossRefGoogle Scholar
Morgan, A. P. (1986), ‘A transformation to avoid solutions at infinity for polynomial systems’, Appl. Math. Comput. 18, 77–86.Google Scholar
Morgan, A. P. (1987), Solving Polynomial Systems Using Continuation for Engineering and Scientific Problems, Prentice-Hall (Englewood Cliffs, NJ).Google Scholar
Morgan, A. P. and Sommese, A. J. (1987), ‘A homotopy for solving general polynomial systems that respects m-homogeneous structures’, Appl. Math. Comput. 24, 101–113.Google Scholar
Morgan, A. P. and Sommese, A. J. (1989), ‘Coefficient parameter polynomial continuation’, Appl. Math. Comput. 29, 123–160.Google Scholar
Morgan, A. P., Sommese, A. J. and Wampler, C. W. (1991a), ‘Computing singular solutions to nonlinear analytic systems’, Numer. Math. 58, 669–684.CrossRefGoogle Scholar
Morgan, A. P., Sommese, A. J. and Wampler, C. W. (1991b), ‘Computing singular solutions to polynomial systems’, Adv. Appl. Math, to appear.Google Scholar
Morgan, A. P., Sommese, A. J. and Wampler, C. W. (1992), ‘A power series method for computing singular solutions to nonlinear analytic systems’, Numer. Math. to appear.CrossRefGoogle Scholar
Morgan, A. P., Sommese, A. J. and Watson, L. T. (1989), ‘Finding all isolated solutions to polynomial systems using HOMPACK’, A CM Trans. Math. Software 15, 93–122.CrossRefGoogle Scholar
Murtagh, B. A. and Saunders, M. A. (1987), MINOS 5.1 Users Guide, Stanford University, CA.Google Scholar
Ortega, J. M. and Rheinboldt, W. C. (1970), Iterative Solution of Nonlinear Equations in Several Variables, Academic Press (New York, London).Google Scholar
Peitgen, H.-O. (1982), ‘Topologische Perturbationen beim globalen numerischen Studium nichtlinearer Eigenwert- und Verzweigungsprobleme’, Jahresber. Deutsch. Math.-Verein. 84, 107–162.Google Scholar
Peitgen, H.-O. and Siegberg, H. W. (1981), ‘An εε-perturbation of Brouwer's definition of degree’, in Fixed Point Theory (Fadell, E. and Fournier, G., eds), Vol. 886 of Lecture Notes in Mathematics, Springer (Berlin, Heidelberg, New York), 331–366.CrossRefGoogle Scholar
Percell, P. (1980), ‘Note on a global homotopy’, Numer. Fund. Anal. Optim. 2, 99–106.CrossRefGoogle Scholar
Poincaré, H. (1881–1886), Sur les Courbes Definé par une Équation Differentielle. I–IV, Oeuvres I. Gauthier-Villars (Paris).Google Scholar
Polak, E., Higgins, J. E. and Mayne, D. Q. (1992), ‘A barrier function method for minimax problems’, Math. Programming 54, 155–176.CrossRefGoogle Scholar
Pönisch, G. and Schwetlick, H. (1981), ‘Computing turning points of curves implicitly defined by nonlinear equations depending on a parameter’, Computing 26, 107–121.CrossRefGoogle Scholar
Poore, A. B. and Tiahrt, C. A. (1987), ‘Bifurcation problems in nonlinear parametric programming’, Math. Programming 39, 189–205.CrossRefGoogle Scholar
Poore, A. B. and Tiahrt, C. A. (1990), ‘A bifurcation analysis of the nonlinear parametric programming problem’, Math. Programming 47, 117–141.Google Scholar
Rabinowitz, P. H. (1971), ‘Some global results for nonlinear eigenvalue problems’, J. Fund. Anal. 7, 487–513.CrossRefGoogle Scholar
Rakowska, J., Haftka, R. T. and Watson, L. T. (1991), ‘An active set algorithm for tracing parametrized optima’, Structural Optimization 2, 29–44.CrossRefGoogle Scholar
Renegar, J. (1985), ‘On the complexity of a piecewise linear algorithm for approximating roots of complex polynomials’, Math. Programming 32, 301–318.CrossRefGoogle Scholar
Renegar, J. (1987), ‘On the efficiency of Newton's method in approximating all zeros of systems of complex polynomials’, Math. Oper. Res. 12, 121–148.CrossRefGoogle Scholar
Renegar, J. (1988a), ‘A polynomial-time algorithm, based on Newton's method, for linear programming’, Math. Programming 40, 59–93.CrossRefGoogle Scholar
Renegar, J. (1988b), ‘Rudiments of an average case complexity theory for piecewise-linear path following algorithms’, Math. Programming 40, 113–163.CrossRefGoogle Scholar
Renegar, J. and Shub, M. (1992), ‘Unified complexity analysis for Newton LP methods’, Math. Programming 53, 1–16.CrossRefGoogle Scholar
Rheinboldt, W. C. (1978), ‘Numerical methods for a class of finite dimensional bifurcation problems’, SIAM J. Numer. Anal. 15, 1–11.CrossRefGoogle Scholar
Rheinboldt, W. C. (1980), ‘Solution fields of nonlinear equations and continuation methods’, SIAM J. Numer. Anal. 17, 221–237.CrossRefGoogle Scholar
Rheinboldt, W. C. (1981), ‘Numerical analysis of continuation methods for nonlinear structural problems’, Comput. and Structures 13, 103–113.CrossRefGoogle Scholar
Rheinboldt, W. C. (1986), Numerical Analysis of Parametrized Nonlinear Equations, John Wiley (New York).Google Scholar
Rheinboldt, W. C. (1987), ‘On a moving-frame algorithm and the triangulation of equilibrium manifolds’, in Bifurcation: Analysis, Algorithms, Applications (Küpper, T., Seydel, R. and Troger, H., eds), Vol. 79 of ISNM, Birkhäuser (Basel), 256–267.CrossRefGoogle Scholar
Rheinboldt, W. C. (1988a), ‘On a theorem of S. Smale about Newton's method for analytic mappings’, Appl. Math. Lett. 1, 69–72.CrossRefGoogle Scholar
Rheinboldt, W. C. (1988b), ‘On the computation of multi-dimensional solution manifolds of parametrized equations’, Numer. Math. 53, 165–182.CrossRefGoogle Scholar
Rheinboldt, W. C. (1991), ‘On the sensitivity of solutions of parametrized equations’, Technical Report ICMA–91–158, University of Pittsburgh.CrossRefGoogle Scholar
Rheinboldt, W. C. (1992a), ‘On the sensitivity of parametrized equations’, SIAM J. Numer. Anal, to appear.Google Scholar
Rheinboldt, W. C. (1992b), ‘On the theory and numerics of differential-algebraic equations’, in Advances in Numerical Analysis (Light, W., ed.), Oxford University Press (Oxford), 237–175.Google Scholar
Rheinboldt, W. C. and Burkardt, J. V. (1983a), ‘Algorithm 596: A program for a locally-parametrized continuation process’, ACM Trans. Math. Software 9, 236–241.CrossRefGoogle Scholar
Rheinboldt, W. C. and Burkardt, J. V. (1983b), ‘A locally-parametrized continuation process’, ACM Trans. Math. Software 9, 215–235.CrossRefGoogle Scholar
Rheinboldt, W. C., Roose, D. and Seydel, R. (1990), ‘Aspects of continuation software’, in Continuation and Bifurcations: Numerical Techniques and Applications (Roose, D., de Dier, B. and Spence, A., eds), Vol. 313 of NATO ASI Series C, Kluwer (Dordrecht), 261–268.CrossRefGoogle Scholar
Robinson, S. M. (1987), ‘Local structure of feasible sets in nonlinear programming, Part III: Stability and sensitivity’, Math. Programming Study 30, 45–66.CrossRefGoogle Scholar
Roos, C. and Vial, J.-P., eds (1991), Interior Point Methods for Linear Programming: Theory and Practice, Vol. 52 of Math. Programming, Ser. B, Mathematical Programming Society, North-Holland (Amsterdam).Google Scholar
Roose, D., de Dier, B. and Spence, A., eds (1990), Continuation and Bifurcations: Numerical Techniques and Applications, Vol. 313 of NATO ASI Series C, Kluwer (Dordrecht).CrossRefGoogle Scholar
Saigal, R. (1971), ‘Lemke's algorithm and a special linear complementarity problem’, Oper. Res. 8, 201–208.Google Scholar
Saigal, R. (1976), ‘Extension of the generalized complementarity problem’, Math. Oper. Res. 1, 260–266.CrossRefGoogle Scholar
Saigal, R. (1977), ‘On the convergence rate of algorithms for solving equations that are based on methods of complementary pivoting’, Math. Oper. Res. 2, 108–124.CrossRefGoogle Scholar
Saigal, R. (1984), ‘Computational complexity of a piecewise linear homotopy algorithm’, Math. Programming 28, 164–173.CrossRefGoogle Scholar
Saigal, R. and Todd, M. J. (1978), ‘Efficient acceleration techniques for fixed point algorithms’, SIAM J. Numer. Anal. 15, 997–1007.CrossRefGoogle Scholar
Saupe, D. (1982), ‘On accelerating PL continuation algorithms by predictor-corrector methods’, Math. Programming 23, 87–110.CrossRefGoogle Scholar
Scarf, H. E. (1967), ‘The approximation of fixed points of a continuous mapping’, SIAM J. Appl. Math. 15, 1328–1343.CrossRefGoogle Scholar
Schwetlick, H. (1984a), ‘Algorithms for finite-dimensional turning point problems from viewpoint to relationships with constrained optimization methods’, in Numerical methods for bifurcation problems (Küpper, T., Mittelmann, H. and Weber, H., eds), Birkhäuser (Basel), 459–479.CrossRefGoogle Scholar
Schwetlick, H. (1984b), ‘Effective methods for computing turning points of curves implicitly defined by nonlinear equations’, in Computational Mathematics (Wakulicz, A., ed.), Vol. 13 of Banach Center Publications, PWN (Polish Scientific Publ.) (Warsaw), 623–645.Google Scholar
Schwetlick, H. and Cleve, J. (1987), ‘Higher order predictors and adaptive stepsize control in path following algorithms’, SIAM J. Numer. Anal. 24, 1382–1393.CrossRefGoogle Scholar
Seydel, R. (1988), From Equilibrium to Chaos. Practical Bifurcation and Stability Analysis, Elsevier (New York).Google Scholar
Seydel, R. (1991a), BIFPACK: A Program Package for Continuation, Bifurcation and Stability Analysis, Version 2.3+, University of Ulm (Germany).Google Scholar
Seydel, R. (1991b), ‘On detecting stationary bifurcations’, Int. J. Bifurcation and Chaos 1, 335–337.CrossRefGoogle Scholar
Seydel, R., Schneider, F. W., Küpper, T. and Troger, H., eds (1991), Bifurcation and Chaos: Analysis, Algorithms, Applications, Vol. 97 of ISNM, Birkhäuser (Basel).CrossRefGoogle Scholar
Shampine, L. F. and Gordon, M. K. (1975), Computer Solutions of Ordinary Differential Equations. The Initial Value Problem, W. H. Freeman (San Francisco).Google Scholar
Shapley, L. S. (1974), ‘A note on the Lemke-Howson algorithm’, in Pivoting and Extensions: In Honor of A. W. Tucker (Balinski, M. L., ed.), Vol. 1 of Math. Programming Study, North-Holland (New York), 175–189.CrossRefGoogle Scholar
Shroff, G. M. and Keller, H. B. (1991), ‘Stabilization of unstable procedures: A hybrid algorithm for continuation’, SIAM J. Numer. Anal. to appear.Google Scholar
Shub, M. and Smale, S. (1991), ‘Complexity of Bezout's theorem, I, geometric aspects’, IBM Research Report.Google Scholar
Smale, S. (1976), ‘A convergent process of price adjustement and global Newton methods’, J. Math. Econom. 2, 1–14.CrossRefGoogle Scholar
Smale, S. (1986), ‘Newton's method estimates from data at one point’, in The Merging of Disciplines in Pure, Applied and Computational Mathematics, Springer (New York), 185–196.Google Scholar
Sonnevend, G. (1985), ‘An analytical center for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming’, Vol. 84 of Lecture Notes in Control and Information Sciences, Springer (New York), 866–876.Google Scholar
Sonnevend, G., Stoer, J. and Zhao, G. (1989), ‘On the complexity of following the central path of linear programs by linear extrapolation’, Meth. Oper. Res. 63, 19–31.Google Scholar
Sonnevend, G., Stoer, J. and Zhao, G. (1991), ‘On the complexity of following the central path of linear programs by linear extrapolation II’, in Interior Point Methods for Linear Programming: Theory and Practice (Roos, C. and Vial, J.-P., eds), Vol. 52 of Math. Programming, Ser. B, Mathematical Programming Society, North-Holland (Amsterdam), 527–553.Google Scholar
Talman, A. J. J. and Yamamoto, Y. (1989), ‘A simplicial algorithm for stationary point problems on polytopes’, Math. Oper. Res. 14, 383–399.CrossRefGoogle Scholar
Todd, M. J. (1976a), The Computation of Fixed Points and Applications, Vol. 124 of Lecture Notes in Economics and Mathematical Systems, Springer (Berlin, Heidelberg, New York).CrossRefGoogle Scholar
Todd, M. J. (1976b), ‘Extensions of Lemke's algorithm for the linear complementarity problem’, J. Optim. Theory Appl. 20, 397–416.CrossRefGoogle Scholar
Todd, M. J. (1976c), ‘Orientation in complementary pivot algorithms’, Math. Oper. Res. 1, 54–66.CrossRefGoogle Scholar
Todd, M. J. (1978), ‘Fixed-point algorithms that allow restarting without extra dimension’, Technical Report, Cornell University, Ithaca, NY.Google Scholar
Todd, M. J. (1981), PLALGO: A FORTRAN Implementation of a Piecewise-linear Homotopy Algorithm for Solving Systems of Nonlinear Equations, School of Operations Research and Industrial Engineering, Cornell University (Ithaca, NY).Google Scholar
Todd, M. J. (1982), ‘On the computational complexity of piecewise-linear homotopy algorithms’, Math. Programming 24, 216–224.CrossRefGoogle Scholar
Todd, M. J. (1986), ‘Polynomial expected behavior of a pivoting algorithm for linear complementarity and linear programming problems’, Math. Programming 35, 173–192.CrossRefGoogle Scholar
Todd, M. J. (1989), ‘Recent developments and new directions in linear programming’, in Mathematical Programming, Recent Developments and Applications (Iri, N. and Tanabe, K., eds), Kluwer (London), 109–157.Google Scholar
Ushida, A. and Chua, L. O. (1984), ‘Tracing solution curves of nonlinear equations with sharp turning points’, Int. J. Circuit Theory Appl. 12, 1–21.CrossRefGoogle Scholar
Vaidya, P. M. (1990), ‘An algorithm for linear programming which requires O(((m+n)n 2 + (m + n)1.5n)L) arithmetic operations’, Math. Programming 47, 175–202.CrossRefGoogle Scholar
van der Laan, G. and Talman, A. J. J. (1979), ‘A restart algorithm for computing fixed points without an extra dimension’, Math. Programming 17, 74–84.CrossRefGoogle Scholar
Vanderbauwhede, A. (1982), Local Bifurcation Theory and Symmetry, Pitman (London).Google Scholar
Vasudevan, G., Lutze, F. H. and Watson, L. T. (1990), ‘A homotopy method for space flight rendezvous problems’, in Astrodynamics 1989 (Thornton, C. L., Proulx, R. J., Prussing, J. E. and Hoots, F. R., eds), Vol. 71 of Advances in the Astronautical Sciences, 533–548.Google Scholar
Wampler, C. W. and Morgan, A. P. (1991), ‘Solving the 6R inverse position problem using a generic-case solution methodology’, Mech. Mach. Theory 26, 91–106.CrossRefGoogle Scholar
Wampler, C. W., Morgan, A. P. and Sommese, A. J. (1990), ‘Numerical continuation methods for solving polynomial systems arising in kinematics’, ASME J. on Design 112, 59–68.CrossRefGoogle Scholar
Wampler, C. W., Morgan, A. P. and Sommese, A. J. (1992), ‘Complete solution of the nine-point path synthesis problem for four-bar linkages’, ASME J. Mech. Des. to appear.CrossRefGoogle Scholar
Watkins, D. S. (1984), ‘Isospectral flows’, SIAM Rev. 26, 379–391.CrossRefGoogle Scholar
Watson, L. T. (1981), ‘Engineering application of the Chow-Yorke algorithm’, Appl. Math. Comput. 9, 111–133.Google Scholar
Watson, L. T. (1986), ‘Numerical linear algebra aspects of globally convergent homotopy methods’, SIAM Rev. 28, 529–545.CrossRefGoogle Scholar
Watson, L. T. and Wang, C. Y. (1981), ‘A homotopy method applied to elastica problems’, Int. J. Solids Structures 17, 29–37.CrossRefGoogle Scholar
Watson, L. T. and Yang, W. H. (1980), ‘Optimal design by a homotopy method’, Applicable Anal. 10, 275–284.CrossRefGoogle Scholar
Watson, L. T., Billups, S. C. and Morgan, A. P. (1987), ‘HOMPACK: A suite of codes for globally convergent homotopy algorithms’, A CM Trans. Math. Software 13, 281–310.CrossRefGoogle Scholar
Watson, L. T., Li, T.-Y. and Wang, C. Y. (1978), ‘Fluid dynamics of the elliptic porous slider’, J. Appl. Mech. 45, 435–436.CrossRefGoogle Scholar
Werner, B. (1992), ‘Test functions for bifurcation points and Hopf points in problems with symmetries’, in Bifurcation and Symmetry (Allgower, E. L., Böhmer, K. and Golubitsky, M., eds), Vol. 104 of ISNM, Birkhäuser (Basel), 317–327.CrossRefGoogle Scholar
Widmann, R. (1990a), ‘An efficient algorithm for the triangulation of surfaces in R3’, Preprint, Colorado State University.Google Scholar
Widmann, R. (1990b), ‘Efficient triangulation of 3-dimensional domains’, Preprint, Colorado State University.Google Scholar
Wright, A. H. (1981), ‘The octahedral algorithm, a new simplicial fixed point algorithm’, Math. Programming 21, 47–69.CrossRefGoogle Scholar
Wright, A. H. (1985), ‘Finding all solutions to a system of polynomial equations’, Math. Comput. 44, 125–133.CrossRefGoogle Scholar
Wright, M. H. (1992), ‘Interior methods for constrained optimization’, Acta Numerica 1, 341–407.CrossRefGoogle Scholar
Yang, Z.-H. and Keller, H. B. (1986), ‘A direct method for computing higher order folds’, SIAM J. Sci. Statist. Comput. 7, 351–361.CrossRefGoogle Scholar
Ye, Y., Güler, O., Tapia, R. A. and Zhang, Y. (1991), ‘A quadratically convergent O()-iteration algorithm for linear programming’, Preprint.CrossRefGoogle Scholar
Yomdin, Y. (1990), ‘Sard's theorem and its improved versions in numerical analysis, in Computational Solution of Nonlinear Systems of Equations (Allgower, E. L. and Georg, K., eds), Vol. 26 of Lectures in Applied Mathematics, American Mathematical Society, Providence, RI, pp. 701–706.Google Scholar