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Eine Verallgemeinerung des Newtonschen Iterationsverfahrens

A generalization of the Newton-Iteration method

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Zusammenfassung

Zur Berechnung einer Lösung gewisser nichtlinearer Operatorgleichungen imBanach-Raum wird ein Iterationsverfahren angegeben. Dieses Verfahren konvergiert bei beliebigem Ausgangselement gegen die eindeutige Lösung des Problems. Ferner geht das Verfahren nach endlich vielen Schritten in das quadratisch konvergenteNewton-Verfahren über. Einige Varianten des Iterationsverfahrens werden betrachtet. Das Verfahren wird auf die diskretisierteHammersteinsche Integralgleichung angewandt und ein numerisches Beispiel hierzu angegeben.

Summary

For the construction of a solution of certain nonlinear operator equations in aBanach space an iteration method is given. This method converges to the unique solution of the problem for any initial data. After a finite number of steps the method is identical with theNewton method. Some variants of the iteration method are given. The method is applied to the discretized integral equation ofHammerstein type and a numerical example is given.

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Literatur

  1. Altman, M.: Connection between the method of steepest descent andNewton's method. Bull. Acad. Pol. Sci.5, 1031–1036 (1957).

    MathSciNet  MATH  Google Scholar 

  2. Altman, M.: Generalized gradient methods of minimizing a functional. Bull. Acad. Pol. Sci. Ser. Math. Astr. et Phys.14, 313–318 (1966).

    MathSciNet  MATH  Google Scholar 

  3. Amann, H.: Über die näherungsweise Lösung nichtlinearer Integralgleichungen. Erscheint in Numer. Mathematik.

  4. Browder, F. E.: Approximation-solvability of nonlinear functional equations in normed linear spaces. Arch. Rational Mech. Anal.26, 33–42 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  5. Browder, F. E., andW. V. Petryshyn: Construction of fixed points of non-linear mappings inHilbert space. J. Math. Anal.20, 197–228 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  6. Davidenko, D. F.: Über die angenäherte Lösung von Systemen nichtlinearer Gleichungen. Ukrain. Mat. Zurnal5, 196–206 (1953) (russ.).

    MathSciNet  MATH  Google Scholar 

  7. Davidenko, D. F.: The application of the method of the variation of a parameter to the construction of iteration formulas of heightened precision for the determination of numerical solutions of nonlinear integral equations. Soviet Math. Dokl.6, 702–706 (1965).

    MATH  Google Scholar 

  8. Dennis, J. E.: A stationaryNewton method for nonlinear functional equations. J. SIAM Num. Anal. Ser. B.4, 222–232 (1967).

    Article  MathSciNet  Google Scholar 

  9. Dennis, J. E.: OnNewton-like methods. Num. Mathematik11, 324–330 (1968).

    Article  MathSciNet  Google Scholar 

  10. Gavurin, M. K.: Nichtlineare Funktionalgleichungen und stetige Analoga von Iterationsverfahren. Izv. Vyss. Ucebn. Zaved. Mat.5, 18–31 (1958) (russ.).

    MathSciNet  MATH  Google Scholar 

  11. Goldstein, A. A.: OnNewton's method. Numer. Mathematik7, 391–393 (1965).

    Article  Google Scholar 

  12. Goldstein, A. A.: Minimizing functionals on normed-linear spaces. SIAM. J. Control4, 81–89 (1966).

    Article  MathSciNet  MATH  Google Scholar 

  13. Jakovlev, M. N.: The solution of nonlinear equations by an iteration method. Soviet Math. Dokl.5, 697–699 (1964).

    MathSciNet  Google Scholar 

  14. Kantorowitsch, L. W., andG. P. Akilow: Funktionalanalysis in normierten Räumen. Berlin: Akademie-Verlag. 1964.

    MATH  Google Scholar 

  15. Kleinmichel, H.: Stetige Analoga und Iterationsverfahren für nichtlineare Gleichungen inBanach-Räumen. Math. Nachrichten37, 313–343 (1968).

    Article  MathSciNet  MATH  Google Scholar 

  16. Krasnoselskii, M. A.: Topological methods in the theory of nonlinear integral equations. Oxford-London-New York-Paris: Pergamon Press. 1964.

    Google Scholar 

  17. Ljusternik, L. A. andW. I. Sobolev: Elemente der Funktionalanalysis. Berlin: Akademie-Verlag. 1960.

    MATH  Google Scholar 

  18. Necas, J.: Sur une méthode générale pour la solution des problèmes aux limites non linéaires. Ann. Scuola Norm. Sup. Pisa, Ser. 3,20, 655–674 (1966).

    MathSciNet  MATH  Google Scholar 

  19. Petryshyn, W. V.: Remarks on the approximation-solvability of nonlinear functional equations. Arch. Rational. Mech. Anal.26, 43–49 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  20. Petry, W.: Eine Variante desNewtonschen Iterationsverfahren. ZAMM50 (T-Heft) (1970).

  21. Polyak, B. T.: Gradient methods for solving equations and inequalities. USSR. Comp. math. Phys.4, 17–32 (1964).

    Article  MathSciNet  Google Scholar 

  22. Vainberg, M. M.: On the convergence of the process of steepest descent. Sibir. Mat. Z.2, 201–220 (1961) (russian.).

    MathSciNet  MATH  Google Scholar 

  23. Yosida, K.: Funktional Analysis. Berlin-Heidelberg-New York: Springer-Verlag. 1966.

    Google Scholar 

  24. Collatz, L.: Funktionalanalysis und numerische Mathematik. Berlin-Göttingen-Heidelberg: Springer-Verlag. 1964.

    Book  MATH  Google Scholar 

  25. Moore, R. H.: Newton's method and Variation: Nonlinear integral equations (Ed.:P. M. Anselone). Math. Research Centr. Madison. Univ. of Wisconsin Press. 1964.

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Authors and Affiliations

  1. Kernforschungsanlage Jülich Zentralinstitut für Angewandte Mathematik, Postfach 365, BRD-5170, Jülich, Deutschland

    Walter Petry

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  1. Walter Petry
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Petry, W. Eine Verallgemeinerung des Newtonschen Iterationsverfahrens. Computing 7, 25–45 (1971). https://doi.org/10.1007/BF02279939

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  • DOI: https://doi.org/10.1007/BF02279939