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Numerische Mathematik 1 pp 241–309Cite as

Nullstellenbestimmung durch Iterationsverfahren. Minimierungsverfahren

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Zusammenfassung

Ein wichtiges Problem ist die Bestimmung der Nullstellen ξ einer gegebenen Funktion f: f (ξ) = 0. Man denke dabei nicht nur an das Problem, die Nullstellen eines Polynoms

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaabm % aabaGaamiEaaGaayjkaiaawMcaaiabg2da9iaadggadaWgaaWcbaGa % aGimaaqabaGccqGHRaWkcaWGHbWaaSbaaSqaaiaaigdaaeqaaOGaam % iEaiabgUcaRiaac6cacaGGUaGaaiOlaiabgUcaRiaadggadaWgaaWc % baGaamOBaaqabaGccaWG4bWaaWbaaSqabeaacaWGUbaaaaaa!4807!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$p\left( x \right) = {a_0} + {a_1}x + ... + {a_n}{x^n}$$

zu finden. Je nach Definition der Funktion f: E → F und der Mengen E und F kann man sehr allgemeine Probleme als eine Aufgabe der Nullstellenbestimmung auffassen. Ist z. B. E = F = ℝn so wird eine Abbildung f : ℝn → ℝn durch n reelle Funktionen f i (x l,..., x n) von n reellen Variablen x 1,..., x n beschrieben1:

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabm % aabaGaamiEaaGaayjkaiaawMcaaiabg2da9maadmaabaqbaeqabuqa % aaaabaGaamOzamaaBaaaleaacaaIXaaabeaakmaabmaabaGaamiEam % aaCaaaleqabaGaaGymaaaakiaacYcacaGGUaGaaiOlaiaac6cacaGG % SaGaamiEamaaCaaaleqabaGaamOBaaaaaOGaayjkaiaawMcaaaqaai % aac6caaeaacaGGUaaabaGaaiOlaaqaaiaadAgadaWgaaWcbaGaamOB % aaqabaGcdaqadaqaaiaadIhadaahaaWcbeqaaiaaigdaaaGccaGGSa % GaaiOlaiaac6cacaGGUaGaaiilaiaadIhadaahaaWcbeqaaiaad6ga % aaaakiaawIcacaGLPaaaaaaacaGLBbGaayzxaaGaaiilaiaaysW7ca % WG4bWaaWbaaSqabeaacaWGubaaaOGaeyypa0ZaaeWaaeaacaWG4bWa % aWbaaSqabeaacaaIXaaaaOGaaiilaiaac6cacaGGUaGaaiOlaiaacY % cacaWG4bWaaWbaaSqabeaacaWGUbaaaaGccaGLOaGaayzkaaaaaa!6305!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$f\left( x \right) = \left[ {\begin{array}{*{20}{c}} {{f_1}\left( {{x^1},...,{x^n}} \right)} \\ . \\ . \\ . \\ {{f_n}\left( {{x^1},...,{x^n}} \right)}\end{array}} \right],\;{x^T} = \left( {{x^1},...,{x^n}} \right)$$

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Literatur zu Kapitel 5

  1. Bauer, F. L.: Beiträge zur Entwicklung numerischer Verfahren für programmgesteuerte Rechenanlagen. II. Direkte Faktorisierung eines Polynoms. Bayer Akad. Wiss. Math. Nat. Kl. S.B. 163–203 (1956).

    Google Scholar 

  2. Collatz, L.: Funktionalanalysis und numerische Mathematik. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen. Bd. 120. Berlin-HeidelbergNew York: Springer 1968.

    Google Scholar 

  3. Householder, A. S.: The Numerical treatment of a single non-linear equation. New York: McGraw-Hill 1970.

    Google Scholar 

  4. Jenkins, M. A., Traub, J. F.: A three-stage variable-shift iteration for polynomial zeros and its relation to generalized Rayleigh iteration. Num. Math. 14, 252–263 (1970).

    Article  MathSciNet  MATH  Google Scholar 

  5. Nickel, K.: Die numerische Berechnung der Wurzeln eines Polynoms. Num Math. 9, 80–98 (1966).

    Article  MathSciNet  MATH  Google Scholar 

  6. Marden, M.: Geometry of polynomials. Providence, Rhode Island: Amer. Math. Soc. 1966.

    Google Scholar 

  7. Ortega, J. M., Rheinboldt, W. C.: Iterative solution of non-linear equations in several variables. Computer science and applied mathematics. New York-London: Academic Press 1970.

    Google Scholar 

  8. Ostrowski, A. M.: Solution of equations in Euclidean and Banach spaces. New York-London: Academic Press 1973.

    MATH  Google Scholar 

  9. Peters, G., Wilkinson, J. H.: Eigenvalues of Ax =)Bx with band symmetric A and B. Comput. J. 12, 398–404 (1969).

    Article  MathSciNet  MATH  Google Scholar 

  10. Peters, G., Wilkinson, J. H.: Practical problems arising in the solution of polynomial equations. J. Inst. Math. Appl. 8, 16–35 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  11. Traub, J. F.: Iterative methods for the solution of equations. Englewood Cliffs, N.J.: Prentice Hall 1964.

    MATH  Google Scholar 

  12. Wilkinson, J. H.: Rundungsfehler, Deutsche Übersetzung: Heidelberger Taschenbücher, Bd. 44. Berlin-Heidelberg-New York: Springer 1969.

    Google Scholar 

  13. Blum, E., Oettli, W.: Mathematische Optimierung. Berlin-Heidelberg-New York: Springer 1975.

    Book  MATH  Google Scholar 

  14. Broyden, C. G.: A class of methods for solving nonlinear simultaneous equations. Math. Comp. 19, 577–593 (1965).

    Article  MathSciNet  MATH  Google Scholar 

  15. Broyden, C. G.: Quasi-Newton-methods and their application to function minimization. Math. Comp. 21, 368–381 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  16. Broyden, C. G.: The convergence of a class of double rank minimization algorithms 2. The new algorithm. J. Inst. Maths. Applies. 6, 222–231 (1970).

    Article  MathSciNet  MATH  Google Scholar 

  17. Broyden, C. G., Dennis, J. E., Moré, J. J.: On the local and superlinear convergence of quasi-Newton methods. J. Inst. Maths. Applies. 12, 223–245 (1973).

    Article  MATH  Google Scholar 

  18. Collatz, L., Wetterling, W.: Optimierungsaufgaben. Berlin-Heidelberg-New York: Springer 1971.

    Book  MATH  Google Scholar 

  19. Davidon, W. C.: Variable metric methods for minimization. Argonne National Laboratory Report ANL-5990 (1959).

    Google Scholar 

  20. Davidon, W. C.: Optimally conditioned optimization algorithms without line searches. Math. Programming 9, 1–30 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  21. Deufihard, P.: A modified Newton method for the solution of ill-conditioned systems of nonlinear equations with application to multiple shooting. Numer. Math. 22, 289–315 (1974).

    Article  MathSciNet  Google Scholar 

  22. Dixon, L. C. W.: The choice of step length, a crucial factor in the performance of variable metric algorithms. In: Numerical methods for nonlinear optimization (F. A. Lootsma, ed.), 149–170. New York: Academic Press 1971.

    Google Scholar 

  23. Fletcher, R Powell, M. J. D.: A rapidly convergent descent method for minimization. Comp. J. 6 163–168 (1963).

    Google Scholar 

  24. algorithms I: Criteria and sufficient conditions for scaling a class of algorithms. Man. Sci. 20, 845–862 (1974).

    Article  Google Scholar 

  25. Oren, S. S., Spedicato, E.: Optimal conditioning of self-scaling variable metric algorithms. Dept. of Engineering Economic Systems Report ARG-MR 74–5, Stanford University 1974.

    Google Scholar 

  26. Powell, M. J. D.: Some global convergence properties of a variable metric algorithm for minimization without exact line searches. Proc. AMS Symposium on Nonlinear Programming, New York 1975.

    Google Scholar 

  27. Stoer, J.: On the convergence rate of imperfect minimization algorithms in Broyden’s ß-class. Math. Programming 9, 313–335 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  28. Stoer, J.: On the relation between quadratic termination and convergence properties of minimization algorithms. Computer Sci. Dept. Report Stan-CS-75–516, Stanford, 1975.

    Google Scholar 

  29. Brent, R. P.: Algorithms for minimization without derivatives. Englewood Cliffs, N.J.: Prentice Hall 1973.

    MATH  Google Scholar 

  30. Henrici, P.: Applied and computational complex analysis, Vol. 1. New York: Wiley 1974.

    MATH  Google Scholar 

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  1. Institut für Angewandte Mathematik und Statistik, Universität Würzburg, Am Hubland, 8700, Würzburg, Deutschland

    Prof. Dr. Josef Stoer

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© 1989 Springer-Verlag Berlin Heidelberg

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Stoer, J. (1989). Nullstellenbestimmung durch Iterationsverfahren. Minimierungsverfahren. In: Numerische Mathematik 1. Springer-Lehrbuch. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-09024-4_5

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  • DOI: https://doi.org/10.1007/978-3-662-09024-4_5

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