Newton’s Method and Secant Method for Set-Valued Mappings

  • Robert Baier
  • Mirko Hessel-von Molo
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7116)


For finding zeros or fixed points of set-valued maps, the fact that the space of convex, compact, nonempty sets of ℝ n 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 with convex, compact images (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 usual nonconvex sets in ℝ n .


set-valued Newton’s method set-valued secant method Gauß-Newton method directed sets embedding of convex compact sets 


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  1. 1.
    Aubin, J.-P.: Mutational and Morphological Analysis. Tools for Shape Evolution and Morphogenesis. Systems & Control: Foundations & Applications. Birkhäuser Boston Inc., Boston (1999)Google Scholar
  2. 2.
    Baier, R., Dellnitz, M., Hessel-von Molo, M., Kevrekidis, I.G., Sertl, S.: The computation of invariant sets via Newton’s method, 21 pages (May 2010) (submitted)Google Scholar
  3. 3.
    Baier, R., Farkhi, E.: Differences of Convex Compact Sets in the Space of Directed Sets. Part I: The Space of Directed Sets. Set-Valued Anal. 9(3), 217–245 (2001); Part II: Visualization of Directed Sets 9(3), 247–252 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Baier, R., Perria, G.: Set-valued Hermite interpolation. J. Approx. Theory 163(10), 1349–1372 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Dontchev, A.L., Hager, W.W., Veliov, V.M.: Uniform convergence and mesh independence of Newton’s method for discretized variational problems. SIAM J. Control Optim. 39(3), 961–980 (2000) (electronic)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Dontchev, A.L., Veliov, V.M.: Metric regularity under approximations. Control Cybernet. 38(4B), 1283–1303 (2009)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Li, C., Zhang, W.H., Jin, X.Q.: Convergence and uniqueness properties of Gauss-Newton’s method. Comput. Math. Appl. 47(6-7), 1057–1067 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Perria, G.: Set-valued interpolation. Bayreuth. Math. Schr. 79, 154 pages (2007)MathSciNetGoogle Scholar
  9. 9.
    Potra, F.-A.: An error analysis for the secant method. Numer. Math. 38(3), 427–445 (1981/1982)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Saint-Pierre, P.: Newton and other continuation methods for multivalued inclusions. Set-Valued Anal. 3(2), 143–156 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Schmidt, J.W.: Eine Übertragung der Regula Falsi auf Gleichungen in Banachräumen. II. Nichtlineare Gleichungssysteme. Z. Angew. Math. Mech. 43(3), 97–110 (1963)CrossRefGoogle Scholar
  12. 12.
    Yamamoto, T.: A method for finding sharp error bounds for Newton’s method under the Kantorovich assumptions. Numer. Math. 49(2-3), 203–220 (1986)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Robert Baier
    • 1
  • Mirko Hessel-von Molo
    • 2
  1. 1.Chair of Applied Math.University of BayreuthBayreuthGermany
  2. 2.Chair of Applied Math.University of PaderbornPaderbornGermany

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