On the Proofs of Some Statements Concerning the Theorems of Kantorovich, Moore, and Miranda

Reliable Computing

Volumn 11, Issue 1, 2005, Pages 77-85

  Schnurr, Marco ^a  

^a UNIVERSITY OF KARLSRUHE   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

MATRIX ALGEBRA; PROBLEM SOLVING; RELIABILITY; SET THEORY; THEOREM PROVING; VECTORS;

INTERVAL ARITHMETIC EVALUATION; KANTOROVICH'S THEOREM; MIRANDA'S THEOREM; MOORE'S THEOREM;

NONLINEAR EQUATIONS;

EID: 17444391690     PISSN: 13853139     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11155-005-5944-6     Document Type: Article

References (16)

1. Alefeld, G., Frommer, A., Heindl, G., and Mayer, J.: On the Existence Theorems of Kantorovich, Miranda, and Borsuk, Electronic Transactions of Numerical Analysis, accepted.
2. Alefeld, G. and Herzberger, J.: Introduction to Interval Computations, Academic Press, New York, 1983.
3. 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.
4. 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.
5. 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.
6. 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.
7. Hansen, E. R.: Interval Forms of Newton's Method, Computing 20 (1978), pp. 153-163.
8. Hansen, E. R.: Global Optimization Using Interval Analysis, Marcel Dekker, New York, 1992.
9. Heuser, H.: Lehrbuch der Analysis Teil 2, B.G. Teubner Verlag, Stuttgart, 9. Auflage, 1995.
10. Miranda, C.: Un' osservatione su un theorema di Brouwer, Bolletino Unione Mathematica Italiana (1940), pp. 5-7.
11. Moore, R. E.: A Test for Existence of Solutions to Nonlinear Systems, SIAM J. Numer. Analysis (1977), pp. 611-615.
12. Neumaier, A. and Shen, Z.: The Krawczyk Operator and Kantorovich's Theorem, Journal of Math. Anal. and Appl. 149 (1990), pp. 437-443.
13. Ortega, J. M. and Rheinboldt, W. C.: Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York and London, 1970.
14. Rail, L. B.: A Comparison of the Existence Theorems of Kantorovich and Moore, SIAM J. Numer. Anal. 17 (1980), pp. 148-161.
15. Schnurr, M.: Die Sätze von Kantorovich, Moore und Miranda, Diploma thesis, University of Karlsruhe, 2002.
16. 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.