Newton's method and secant method for set-valued mappings

R. Baier, M. Hessel-von Molo: Newton's method and secant method for set-valued mappings
in: Proceedings on the 8th International Conference on "Large-Scale Scientific Computations" (LSSC 2011), Sozopol, Bulgaria, June 6-10, 2011. Revised Selected Papers, I. Lirkov, S. Margenov, J. Wanśiewski (eds.)
Lecture Notes in Computer Science 7116, Springer-Verlag, Berlin-Heidelberg, 2012, 91 - 98

ISBN/ISSN/ISMV Nummer: 978-3-642-29842-4
DOI: 10.1007/978-3-642-29843-1_9
MR Nummer: 2955111
Zentralblattnummer: 06056174
Keywords: set-valued Newton's method; set-valued secant method; Gauß-Newton method; directed sets; embedding of convex compact sets
Mathematics Subject Classification Code: 65J15 (52A20 65H10 90C56 26E25 54C60)
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For finding zeros or fixed points of set-valued maps, the fact that the space of convex, compact, nonempty sets of Rn is not a vector space presents a major disadvantage. Therefore, fixed point iterations or variants of Newton's method, in which the derivative is applied only to a smooth single-valued part of the set-valued map, are often applied for calculations. We will embed the set-valued map (i.e. by embedding its images) and shift the problem to the Banach space of directed sets. This Banach space extends the arithmetic operations of convex sets and allows to consider the Fréchet-derivative or divided differences of maps that have embedded convex images. For the transformed problem, Newton's method and the secant method in Banach spaces are applied via directed sets. The results can be visualized as usually nonconvex sets in Rn.


1. Introduction
1.1 Directed Sets
2. Fréchet-Derivative for Set-Valued Maps
3. Newton's Method and Secant Method
3.1 Newton's Method for Directed Sets
3.2 Secant Method Based on Directed Sets
4. Examples

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