Abstract
In this paper we discuss the proofs of some results comparing existence statements for the solutions of nonlinear equations. In one important case we correct a proof already published.
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Alefeld, G., Frommer, A., Heindl, G., and Mayer, J.: On the Existence Theorems of Kantorovich, Miranda, and Borsuk, Electronic Transactions of Numerical Analysis, accepted.
Alefeld, G. and Herzberger, J.: Introduction to Interval Computations, Academic Press, New York, 1983.
Alefeld, G., Potra, F. A., and Shen, Z.: On the Existence Theorems of Kantorovich, Moore, and Miranda, in: Alefeld G. and Chen X. (eds), Topics in Numerical Analysis, Computing Supplementum 15, Springer Verlag, 2001, pp. 21–28.
Alefeld, G. and Shen, Z.: Miranda’s Theorem and the Verification of Linear Complementarity Problems, Preprint Nr. 01/05, Institut für Wissenschaftliches Rechnen und Mathematische Modellbildung, University of Karlsruhe, 2001.
Deuflhard, P. and Heindl, G.: Affine Invariant Convergence Theorems for Newton’s Method and Extensions to Related Methods, SIAM J. Numer. Analysis 16 (1979), pp. 1–10.
Frommer, A.: A Hierarchy of Existence Theorems for Nonlinear Equations, Talk presented at Dagstuhl Seminar on Numerical Software with Result Verification, 2003, http://www.cs.utep.edu/interval-comp/dagstuhl.03/from.pdf.
Hansen, E. R.: Interval Forms of Newton’s Method, Computing 20 (1978), pp. 153–163.
Hansen, E. R.: Global Optimization Using Interval Analysis, Marcel Dekker, New York, 1992.
Heuser, H.: Lehrbuch der Analysis Teil 2, B.G. Teubner Verlag, Stuttgart, 9. Auflage, 1995.
Miranda, C.: Un’osservatione su un theorema di Brouwer, Bolletino Unione Mathematica Italiana (1940), pp. 5–7.
Moore, R. E.: A Test for Existence of Solutions to Nonlinear Systems, SIAM J. Numer. Analysis (1977), pp. 611–615.
Neumaier, A. and Shen, Z.: The Krawczyk Operator and Kantorovich’s Theorem, Journal of Math. Anal. and Appl. 149 (1990), pp. 437–443.
Ortega, J. M. and Rheinboldt, W. C.: Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York and London, 1970.
Rall, L. B.: A Comparison of the Existence Theorems of Kantorovich and Moore, SIAM J. Numer. Anal. 17 (1980), pp. 148–161.
Schnurr, M.: Die Sätze von Kantorovich, Moore und Miranda, Diploma thesis, University of Karlsruhe, 2002.
Shen, Z. and Wolfe, M. A.: A Note on the Comparison of the Kantorovich and Moore Theorems, Nonlinear Analysis, Theory, Methods and Appl. 15 (1990), pp. 229–232.
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Schnurr, M. On the Proofs of Some Statements Concerning the Theorems of Kantorovich, Moore, and Miranda. Reliable Comput 11, 77–85 (2005). https://doi.org/10.1007/s11155-005-5944-6
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DOI: https://doi.org/10.1007/s11155-005-5944-6