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Finding zeroes of maps: homotopy methods that are constructive with probability one. (English) Zbl 0398.65029

MSC:
65H10 Numerical computation of solutions to systems of equations
54C25 Embedding
34B15 Nonlinear boundary value problems for ordinary differential equations
55M25 Degree, winding number
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[1] Herbert Scarf, The approximation of fixed points of a continuous mapping, SIAM J. Appl. Math. 15 (1967), 1328 – 1343. · Zbl 0153.49401
[2] B. Curtis Eaves, An odd theorem, Proc. Amer. Math. Soc. 26 (1970), 509 – 513. · Zbl 0203.25301
[3] Harold W. Kuhn, Simplicial approximation of fixed points, Proc. Nat. Acad. Sci. U.S.A. 61 (1968), 1238 – 1242. · Zbl 0191.54904
[4] B. Curtis Eaves, Homotopies for computation of fixed points, Math. Programming 3 (1972), 1 – 22. · Zbl 0276.55004
[5] B. Curtis Eaves and Romesh Saigal, Homotopies for computation of fixed points on unbounded regions, Math. Programming 3 (1972), 225 – 237. · Zbl 0258.65060
[6] R. T. WILLMUTH, The Computation of Fixed Points, Ph. D. Thesis, Dept. of Operations Research, Stanford University, 1973.
[7] R. B. KELLOGG, T. Y. LI & J. A. YORKE, ”A method of continuation for calculating a Brouwer fixed point,” Computing Fixed Points with Applications (Proc. Conf., Clemson Univ., 1974), S. Karamadian (editor), Academic Press, New York, 1977, pp. 133-147.
[8] 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
[9] Steve Smale, A convergent process of price adjustment and global Newton methods, J. Math. Econom. 3 (1976), no. 2, 107 – 120. · Zbl 0354.90018
[10] M. HIRSCH & S. SMALE, Personal communication.
[11] L. S. Pontryagin, Smooth manifolds and their applications in homotopy theory, American Mathematical Society Translations, Ser. 2, Vol. 11, American Mathematical Society, Providence, R.I., 1959, pp. 1 – 114. · Zbl 0084.19002
[12] 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
[13] T. Y. LI, ”A rigorous algorithm for fixed point computation.” (To appear.)
[14] L. WATSON, ”Finding fixed points of \( {C^2}\) maps by using homotopy methods,” Computation and Appl. Math. (To appear.)
[15] 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
[16] 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.
[17] Ralph Abraham and Joel Robbin, Transversal mappings and flows, An appendix by Al Kelley, W. A. Benjamin, Inc., New York-Amsterdam, 1967. · Zbl 0171.44404
[18] 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
[19] Andreu Mas-Colell, A note on a theorem of F. Browder, Math. Programming 6 (1974), 229 – 233. · Zbl 0285.90068
[20] Felix E. Browder, On continuity of fixed points under deformations of continuous mappings, Summa Brasil. Mat. 4 (1960), 183 – 191. · Zbl 0102.37807
[21] J. L. LIONS, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969. · Zbl 0189.40603
[22] J. C. Alexander, The additive inverse eigenvalue problem and topological degree, Proc. Amer. Math. Soc. 70 (1978), no. 1, 5 – 7. · Zbl 0386.15007
[23] Shmuel Friedland, Inverse eigenvalue problems, Linear Algebra and Appl. 17 (1977), no. 1, 15 – 51. · Zbl 0358.15007
[24] G. SCORZA-DRAGONI, ”Sul problema dei valori ai limiti per i system di equazioni differenziali del secondo ordine,” Boll. Un. Mat. Ital., v. 14, 1935, pp. 225-230. · JFM 61.1238.08
[25] A. Lasota and James A. Yorke, Existence of solutions of two-point boundary value problems for nonlinear systems, J. Differential Equations 11 (1972), 509 – 518. · Zbl 0263.34016
[26] M. A. Krasnosel’skii, Topological methods in the theory of nonlinear integral equations, Translated by A. H. Armstrong; translation edited by J. Burlak. A Pergamon Press Book, The Macmillan Co., New York, 1964.
[27] J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New York-London, 1970. · Zbl 0241.65046
[28] J. DAVIS, The Solution of Nonlinear Operator Equations with Critical Points, Ph. D. thesis, Oregon State Univ., 1966.
[29] Gunter H. Meyer, On solving nonlinear equations with a one-parameter operator imbedding., SIAM J. Numer. Anal. 5 (1968), 739 – 752. · Zbl 0182.48701
[30] J. C. Alexander and James A. Yorke, The homotopy continuation method: numerically implementable topological procedures, Trans. Amer. Math. Soc. 242 (1978), 271 – 284. · Zbl 0424.58003
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