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Numerische Mathematik 1 pp 291–364Cite as

Verfahren zur Nullstellenbestimmung. Minimierungsmethoden

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Zusammenfassung

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

$$p\left( x \right) = {a_0}{x^n} + {a_1}{x^{n - 1}} + \cdots + {a_n}$$

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Literatur zu Kapitel 5

  • Baptist, P. Stoer, J. (1977): On the relation between quadratic termination and convergence properties of minimization algorithms. Part II. Applications. Numer. Math. 28 367–391.

    Google Scholar 

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

    Google Scholar 

  • Brent, R.P. (1973): Algorithms for Minimization without Derivatives. Englewood Cliffs, N.J.: Prentice-Hall.

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Broyden, C.G. (1970): The convergence of a class of double rank minimization algorithms. 1. General considerations, 2. The new algorithm. J. Inst. Math. Appl. 6, 76–90, 222–231.

    Article  MathSciNet  Google Scholar 

  • Broyden, C.G., Dennis, J.E., Mori, J.J. (1970): On the local and superlinear convergence of quasi-Newton methods. J. Inst. Math. Appl. 12, 223–245.

    Article  Google Scholar 

  • Collatz, L. (1968): Funktionalanalysis und numerische Mathematik Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen. Bd. 120. BerlinHeidelberg-New York: Springer.

    Google Scholar 

  • Collatz, L., Wetterling, W. (1971): Optimierungsaufgaben. Berlin-Heidelberg-New York: Springer.

    Book  MATH  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Dixon, L.C.W. (1971): 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.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Fletcher, R. (1980): Unconstrained Optimization. New York: Wiley.

    MATH  Google Scholar 

  • Fletcher, R. (1981): Constrained Optimization. New York: Wiley.

    MATH  Google Scholar 

  • Gill, P.E., Golub, G.H., Murray, W., Saunders, M.A. (1974): Methods for modifying matrix factorizations. Math. Comput. 28, 505–535.

    Article  MathSciNet  MATH  Google Scholar 

  • Henrici, P. (1974): Applied and Computional Complex Analysis. Vol. 1. New York: Wiley.

    Google Scholar 

  • Himmelblau, D.M. (1972): Applied Nonlinear Programming. New York: McGraw-Hill.

    MATH  Google Scholar 

  • Householder, A.S. (1970): The Numerical Treatment of a Single Non-linear Equation. New York: McGraw-Hill.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Luenberger, D.G. (1973): Introduction to Linear and Nonlinear Programming. Reading, Mass.: Addison-Wesley.

    MATH  Google Scholar 

  • Maehly, H. (1954): Zur iterativen Auflösung algebraischer Gleichungen. Z. Angew. Math. Physik 5, 260–263.

    Article  MathSciNet  MATH  Google Scholar 

  • Marden, M. (1966): Geometry of Polynomials. Providence, R.I.: Amer. Math. Soc. Nickel, K. (1966): Die numerische Berechnung der Wurzeln eines Polynoms. Numer. Math. 9, 80–98.

    Google Scholar 

  • Oren, S.S., Luenberger, D.G. (1974): Self-scaling variable metric (SSVM) algorithms. I. Criteria and sufficient conditions for scaling a class of algorithms. Manage. Sci. 20, 845–862.

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  • Ortega, J.M., Rheinboldt, W.C. (1970): Iterative Solution of Non-linear Equations in Several Variables. New York: Academic Press.

    Google Scholar 

  • Ostrowski, A.M. (1973): Solution of Equations in Euclidean and Banach Spaces. New York: Academic Press.

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Powell, M.J.D. (1975): Some global convergence properties of a variable metric algorithm for minimization without exact line searches. In: Proc. AMS Symposium on Nonlinear Programming 1975. Amer. Math. Soc.

    Google Scholar 

  • Spellucci, P. (1993): Numerische Verfahren der nichtlinearen Optimierung. Basel: Birkhäuser.

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Stoer, J. (1977): On the relation between quadratic termination and convergence properties of minimization algorithms. Part I. Theory. Numer. Math. 28, 343–366.

    Article  MathSciNet  MATH  Google Scholar 

  • Tornheim, L. (1964): Convergence of multipoint methods. J. Assoc. Comput. Mach. 11, 210–220.

    Article  MathSciNet  MATH  Google Scholar 

  • Traub, J.F. (1964): Iterative Methods for the Solution of Equations. Englewood Cliffs, NJ: Prentice-Hall.

    MATH  Google Scholar 

  • Wilkinson, J.H. (1959): The evaluation of the zeros of ill-conditioned polynomials. Part I. Numer. Math. 1, 150–180.

    Article  MathSciNet  Google Scholar 

  • Wilkinson, J.H. (1969): Rundungsfehler. Heidelberger Taschenbücher, Bd. 44, BerlinHeidelberg-New York: Springer.

    Google Scholar 

  • Wilkinson, J.H. (1965): The Algebraic Eigenvalue Problem. Oxford: Clarendon Press.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Institut für Angewandte Mathematik und Statistik, Universität Würzburg, Am Hubland, D-97074, Würzburg, Deutschland

    Prof. Dr. Josef Stoer

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  1. Prof. Dr. Josef Stoer
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© 1999 Springer-Verlag Berlin Heidelberg

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Stoer, J. (1999). Verfahren zur Nullstellenbestimmung. Minimierungsmethoden. In: Numerische Mathematik 1. Springer-Lehrbuch. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-09021-3_5

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  • DOI: https://doi.org/10.1007/978-3-662-09021-3_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-66154-2

  • Online ISBN: 978-3-662-09021-3

  • eBook Packages: Springer Book Archive

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