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The homotopy continuation method: Numerically implementable topological procedures. (English) Zbl 0424.58003


MSC:

58C15 Implicit function theorems; global Newton methods on manifolds
55M20 Fixed points and coincidences in algebraic topology
55M25 Degree, winding number
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
65J15 Numerical solutions to equations with nonlinear operators

Citations:

Zbl 0286.47037
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Full Text: DOI

References:

[1] J. F. Adams, Vector fields on spheres, Ann. of Math. (2) 75 (1962), 603 – 632. · Zbl 0112.38102
[2] J. C. Alexander, Bifurcation of zeroes of parametrized functions, J. Funct. Anal. 29 (1978), no. 1, 37 – 53. · Zbl 0385.47038
[3] J. C. Alexander, The additive inverse eigenvalue problem and topological degree, Proc. Amer. Math. Soc. 70 (1978), no. 1, 5 – 7. · Zbl 0386.15007
[4] J. C. Alexander and James A. Yorke, Global bifurcations of periodic orbits, Amer. J. Math. 100 (1978), no. 2, 263 – 292. · Zbl 0386.34040
[5] -, Parameterized functions, bifurcation, and vector fields on spheres (to appear).
[6] -, Calculating bifurcation invariants as homotopy elements of the general linear group, J. Pure Appl. Algebra (to appear).
[7] Shui Nee Chow, John Mallet-Paret, and James A. Yorke, Finding zeroes of maps: homotopy methods that are constructive with probability one, Math. Comp. 32 (1978), no. 143, 887 – 899. · Zbl 0398.65029
[8] B. Curtis Eaves, Homotopies for computation of fixed points, Math. Programming 3 (1972), 1 – 22. · Zbl 0276.55004
[9] B. Curtis Eaves, A short course in solving equations with PL homotopies, Nonlinear programming (Proc. SIAM-AMS Sympos., NewYork, 1975) Amer. Math. Soc., Providence, R. I., 1976, pp. 73 – 143. SIAM-AMS Proc., Vol. IX.
[10] B. Curtis Eaves and Herbert Scarf, The solution of systems of piecewise linear equations, Math. Oper. Res. 1 (1976), no. 1, 1 – 27. · Zbl 0458.65056
[11] Shmuel Friedland, On inverse multiplicative eigenvalue problems for matrices, Linear Algebra and Appl. 12 (1975), no. 2, 127 – 137. · Zbl 0329.15003
[12] Victor Guillemin and Alan Pollack, Differential topology, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974. · Zbl 0361.57001
[13] Morris W. Hirsch, A proof of the nonretractibility of a cell onto its boundary, Proc. Amer. Math. Soc. 14 (1963), 364 – 365. · Zbl 0113.16704
[14] Morris W. Hirsch, Differential topology, Springer-Verlag, New York-Heidelberg, 1976. Graduate Texts in Mathematics, No. 33. · Zbl 0356.57001
[15] D. Husemoller, Fiber bundles, Springer-Verlag, New York, 1974.
[16] R. B. Kellogg, T. Y. Li and J. A. Yorke, A method of continuation for calculating a Brouwer fixed point, Fixed Points, Algorithms, and Applications , Academic Press, New York, 1977. · Zbl 0426.90094
[17] 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. 13 (1976), no. 4, 473 – 483. · Zbl 0355.65037
[18] Herbert B. Keller, Numerical solution of bifurcation and nonlinear eigenvalue problems, Applications of bifurcation theory (Proc. Advanced Sem., Univ. Wisconsin, Madison, Wis., 1976) Academic Press, New York, 1977, pp. 359 – 384. Publ. Math. Res. Center, No. 38.
[19] Harold W. Kuhn, Simplicial approximation of fixed points, Proc. Nat. Acad. Sci. U.S.A. 61 (1968), 1238 – 1242. · Zbl 0191.54904
[20] T. Y. Li, A rigorous algorithm for fixed point computation (to appear).
[21] T. Y. Li and J. A. Yorke, A rigorous algorithm for solving equations when existence is proved using topological degree (in preparation).
[22] Yung Chen Lu, Singularity theory and an introduction to catastrophe theory, Springer-Verlag, New York, 1976. With an introduction by Peter Hilton; Universitext. · Zbl 0354.58008
[23] Gunter H. Meyer, On solving nonlinear equations with a one-parameter operator imbedding., SIAM J. Numer. Anal. 5 (1968), 739 – 752. · Zbl 0182.48701
[24] John W. Milnor, Topology from the differentiable viewpoint, Based on notes by David W. Weaver, The University Press of Virginia, Charlottesville, Va., 1965. · Zbl 0136.20402
[25] -, Morse theory, Ann. of Math. Studies, no. 51, Princeton Univ. Press, Princeton, N. J., 1963.
[26] L. Nirenberg, Topics in nonlinear functional analysis, Courant Institute of Mathematical Sciences, New York University, New York, 1974. With a chapter by E. Zehnder; Notes by R. A. Artino; Lecture Notes, 1973 – 1974.
[27] Paul H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis 7 (1971), 487 – 513. · Zbl 0212.16504
[28] Werner C. Rheinboldt, Numerical continuation methods for finite element applications, Formulations and computational algorithms in finite element analysis (U.S.-Germany Sympos., Mass. Inst. Tech., Cambridge, Mass., 1976) M.I.T. Press, Cambridge, Mass., 1977, pp. 599 – 631.
[29] R. Saigal, Fixed point computing methods, Encyclopedia of Computer Science and Technology, Dekker, New York (to appear). · Zbl 0489.90082
[30] -, On the convergence rate of algorithms for solving equations that are based on methods of complementary pivoting, Math. Operations Research (to appear). · Zbl 0395.90082
[31] Herbert Scarf, The approximation of fixed points of a continuous mapping, SIAM J. Appl. Math. 15 (1967), 1328 – 1343. · Zbl 0153.49401
[32] Stephen Smale, Exchange processes with price adjustment, J. Math. Econom. 3 (1976), no. 3, 211 – 226. · Zbl 0366.90013
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