×

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
Full Text: DOI

References:

[1] Herbert Scarf, The approximation of fixed points of a continuous mapping, SIAM J. Appl. Math. 15 (1967), 1328 – 1343. · Zbl 0153.49401 · doi:10.1137/0115116
[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 · doi:10.1007/BF01584975
[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 · doi:10.1007/BF01584991
[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 · doi:10.1137/0713041
[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 · doi:10.1016/0304-4068(76)90019-7
[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 · doi:10.1287/moor.1.1.1
[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 · doi:10.1007/BF01580239
[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 · doi:10.1016/0022-0396(72)90063-0
[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 · doi:10.1137/0705057
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.