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Verified numerical computation for nonlinear equations. (English) Zbl 1186.65056
This is a kind of a survey paper on verification methods for solving nonlinear equations. Fixed point iterations, Newton-like methods, and methods for nonsmooth equations are explained and discussed.
MSC:
65G30Interval and finite arithmetic
65H05Single nonlinear equations (numerical methods)
65G20Algorithms with automatic result verification
References:
[1]G. Alefeld, Intervallrechnung über den komplexen Zahlen und einige Anwendungen. Ph.D. thesis, Universität Karlsruhe, Karlsruhe, 1968.
[2]G. Alefeld, Inclusion methods for systems of nonlinear equations–the interval Newton method and modifications. Topics in Validated Computations, J. Herzberger (ed.), Elsevier, Amsterdam, 1994, 7–26.
[3]G. Alefeld, X. Chen and F. Potra, Numerical validation of solutions of linear complementarity problems. Numer. Math.,83 (1999), 1–23. · Zbl 0933.65072 · doi:10.1007/s002110050437
[4]G. Alefeld, X. Chen and F. Potra, Numerical validation of solutions of complementarity problems: the nonlinear case. Numer. Math.,92 (2002), 1–16. · Zbl 1040.65046 · doi:10.1007/s002110100351
[5]G. Alefeld and J. Herzberger, Einführung in die Intervallrechnung. Bibliographisches Institut, Mannheim, 1974.
[6]G. Alefeld and J. Herzberger, Introduction to Interval Computations., Academic Press, New York, 1983.
[7]G. Alefeld and G. Mayer, Interval analysis. Theory and applications. J. Comp. Appl. Math.,121 (2000), 421–464. · Zbl 0995.65056 · doi:10.1016/S0377-0427(00)00342-3
[8]G. Alefeld and U. Schäfer, Iterative methods for linear complementarity problems with interval data. Computing,70 (2003), 235–259.
[9]G. Alefeld, Z. Shen and Z. Wang, Enclosing solutions of linear complementarity problems for H-matrices. Reliable Computing,10 (2004), 423–435. · Zbl 1074.65066 · doi:10.1023/B:REOM.0000047093.79994.8f
[10]G. Alefeld and Z. Wang, Verification of solutions of almost linear complementarity problems. Annals of the European Academy of Sciences 2005, EAS Publishing House, 211–231.
[11]X. Chen, A verification method for solutions of nonsmooth equations. Computing,58 (1997), 281–294. · Zbl 0882.65038 · doi:10.1007/BF02684394
[12]R. Krawczyk, Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing,4 (1969), 187–201. · Zbl 0187.10001 · doi:10.1007/BF02234767
[13]A. Neumaier, Interval Methods for Systems of Equations. University Press, Cambridge, 1990.
[14]H. Schwandt, Schnelle fast global konvergente Verfahren für die Fünf-Punkte-Diskrestisierung der Poissongleichung mit Dirichletschen Randbedingungen auf Rechteckgebieten. Thesis, Fachbereich Mathematik der TU Berlin, Berlin, 1981.