Finding zeroes of maps: homotopy methods that are constructive with probability one
Authors:
Shui Nee Chow, John MalletParet and James A. Yorke
Journal:
Math. Comp. 32 (1978), 887899
MSC:
Primary 55M25; Secondary 47H10, 65H10, 90C99
MathSciNet review:
492046
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Abstract: We illustrate that most existence theorems using degree theory are in principle relatively constructive. The first one presented here is the Brouwer Fixed Point Theorem. Our method is "constructive with probability one" and can be implemented by computer. Other existence theorems are also proved by the same method. The approach is based on a transversality theorem.
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 [1]
 H. SCARF, "The approximation of fixed points of a continuous mapping," SIAM J. Appl. Math., v. 15, 1967, pp. 13281343. MR 0242483 (39:3814)
 [2]
 B. C. EAVES, "An odd theorem," Proc. Amer. Math. Soc., v. 26, 1970, pp. 509513. MR 0270757 (42:5645)
 [3]
 H. W. KUHN, "Simplicial approximation of fixed points," Proc. Nat. Acad. Sci. U.S.A., v. 61, 1968, pp. 12381242. MR 0488010 (58:7589)
 [4]
 B. C. EAVES, "Homotopies for the computation of fixed points," Math. Programming, v. 3, 1972, pp. 122. MR 0303953 (46:3089)
 [5]
 B. C. EAVES & R. SAIGAL, "Homotopies for the computation of fixed points on unbounded regions," Math. Programming, v. 3, 1972, pp. 225237. MR 0314028 (47:2580)
 [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. 133147.
 [8]
 R. B. KELLOGG, T. Y. LI & J. A. YORKE, "A constructive proof of the Brouwer Fixed Point Theorem and computational results," SIAM J. Numer. Anal., v. 13, 1976, pp. 473483. MR 0416010 (54:4087)
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 S. SMALE, "A convergent process of price adjustment and global Newton methods," J. Math. Econom., v. 3, 1976, pp. 114. MR 0411577 (53:15310)
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 M. HIRSCH & S. SMALE, Personal communication.
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 [13]
 T. Y. LI, "A rigorous algorithm for fixed point computation." (To appear.)
 [14]
 L. WATSON, "Finding fixed points of maps by using homotopy methods," Computation and Appl. Math. (To appear.)
 [15]
 B. C. EAVES & H. SCARF, "The solution of systems of piecewise linear equations," Math. of Oper. Res., v. 1, 1976, pp. 127. MR 0445792 (56:4126)
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 W. C. RHEINBOLDT, "Numerical continuation methods for finite element applications," Formulation and Computational Algorithms in Finite Element Analysis, (Proc. U. S.German Sympos.), M.I.T. Press, Cambridge, Mass. (To appear.) MR 0474782 (57:14415)
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 R. ABRAHAM & J. ROBBIN, Transversal Mappings and Flows, Benjamin, New York, 1967. MR 0240836 (39:2181)
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 J. MILNOR, Topology from the Differentiate Viewpoint, Univ. of Virginia Press, Charlottesville, Va., 1965. MR 0226651 (37:2239)
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 A. MASCOLELL, "A note on a theorem of F. Browder," Math. Programming, v. 6, 1974, pp. 229233. MR 0341225 (49:5975)
 [20]
 F. E. BROWDER, "On the continuity of fixed points under deformations of continuous mappings," Summa Brasil. Math., v. 4, 1960, pp. 183191. MR 0130683 (24:A543)
 [21]
 J. L. LIONS, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969.
 [22]
 J. C. ALEXANDER, "The additive inverse eigenvalue problem and topological degree," Proc. Amer. Math. Soc., v. 70, 1978, pp. 57. MR 487546 (80a:55002)
 [23]
 S. FRIEDLAND, "Inverse eigenvalue problems," Linear Algebra and Appl., v. 17, 1977, pp. 1551. MR 0472861 (57:12550)
 [24]
 G. SCORZADRAGONI, "Sul problema dei valori ai limiti per i system di equazioni differenziali del secondo ordine," Boll. Un. Mat. Ital., v. 14, 1935, pp. 225230.
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 A. LASOTA & J. A. YORKE, "Existence of solutions of twopoint boundary value problems for nonlinear systems," J. Differential Equations, v. 11, 1972, pp. 509518. MR 0299867 (45:8915)
 [26]
 M. A. KRASNOSELSKII, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press, New York, 1964. MR 0159197 (28:2414)
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 J. M. ORTEGA & W. C. RHEINBOLDT, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. MR 0273810 (42:8686)
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 J. DAVIS, The Solution of Nonlinear Operator Equations with Critical Points, Ph. D. thesis, Oregon State Univ., 1966.
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 G. MEYER, "On solving nonlinear equations with a oneparameter operator imbedding," SIAM J. Numer. Anal., v. 5, 1968, pp. 739752. MR 0242366 (39:3697)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197804920469
PII:
S 00255718(1978)04920469
Keywords:
Brouwer Fixed Point Theorem,
constructive proof,
Transversality theorem,
degree theory,
vector fields on spheres
Article copyright:
© Copyright 1978 American Mathematical Society
