Finding Zeros and Minimum Points by Iterative Methods

  • J. Stoer
  • R. Bulirsch


Finding the zeros of a given function f, that is arguments ξ for which f(ξ) = 0, is a classical problem. In particular, determining the zeros of a polynomial (the zeros of a polynomial are also known as its roots)
$$p\left( x \right) = {a_0} + {a_1}x + \cdots + {a_n}{x^n}$$
has captured the attention of pure and applied mathematicians for centuries. However, much more general problems can be formulated in terms of finding zeros, depending upon the definition of the function f: E → F, its domain E, and its range F.


Iterative Method Newton Method Line Search Real Root Minimum Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References for Chapter 5

  1. Baptist, P., Stoer, J.: On the relation between quadratic termination and convergence properties of minimization algorithms. Part II. Applications. Numer. Math. 28, 367–391 (1977).MathSciNetzbMATHCrossRefGoogle Scholar
  2. Bauer, F. L.: Beiträge zur Entwicklung numerischer Verfahren für programmgesteuerte Rechenanlagen. II. Direkte Faktorisierung eines Polynoms. Bayer. Akad. Wiss. Math. Natur. Kl. S.B. 163–203 (1956).Google Scholar
  3. Brent, R. P.: Algorithms for Minimization without Derivatives. Englewood Cliffs, N.J.: Prentice-Hall 1973.zbMATHGoogle Scholar
  4. Broyden, C. G.: A class of methods for solving nonlinear simultaneous equations. Math. Comput. 19, 577–593 (1965).MathSciNetzbMATHCrossRefGoogle Scholar
  5. Broyden, C. G.: Quasi-Newton-methods and their application to function minimization. Math. Comput. 21, 368–381 (1967).MathSciNetzbMATHCrossRefGoogle Scholar
  6. Broyden, C. G.: The convergence of a class of double rank minimization algorithms. 2. The new algorithm. J. Inst. Math. Appl. 6, 222–231 (1970).MathSciNetzbMATHCrossRefGoogle Scholar
  7. Broyden, C. G.: Dennis, J. E., More, J. J.: On the local and superlinear convergence of quasi-Newton methods. J. Inst. Math. Appl. 12, 223–245 (1973).MathSciNetzbMATHCrossRefGoogle Scholar
  8. Collatz, L.: Funktionalanalysis und numerische Mathematik. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen. Bd. 120. Berlin, Heidelberg, New York: Springer 1968.Google Scholar
  9. Davidon, W. C.: Variable metric methods for minimization. Argonne National Laboratory Report ANL-5990, 1959.CrossRefGoogle Scholar
  10. Davidon, W. C.: Optimally conditioned optimization algorithms without line searches. Math. Programming 9, 1–30 (1975).MathSciNetzbMATHCrossRefGoogle Scholar
  11. Deuflhard, P.: A modified Newton method for the solution of ill-conditioned systems of nonlinear equations with applications to multiple shooting. Numer. Math. 22, 289–315 (1974).MathSciNetzbMATHCrossRefGoogle Scholar
  12. 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. Edited by F. A. Lootsma. 149–170. New York: Academic Press 1971.Google Scholar
  13. Fletcher, R., Powell, M. J. D.: A rapidly convergent descent method for minimization. Comput. J. 6, 163–168 (1963).MathSciNetzbMATHCrossRefGoogle Scholar
  14. Gill, P. E., Golub, G. H., Murray, W., Saunders, M. A.: Methods for modifying matrix factorizations. Math. Comput. 28, 505–535 (1974).MathSciNetzbMATHCrossRefGoogle Scholar
  15. Henrici, P.: Applied and Computational Complex Analysis. Vol. 1. New York: Wiley 1974.zbMATHGoogle Scholar
  16. Himmelblau, D. M.: Applied Nonlinear Programming. New York: McGraw-Hill 1972.zbMATHGoogle Scholar
  17. Householder, A. S.: The Numerical Treatment of a Single Non-linear Equation. New York: McGraw-Hill 1970.Google Scholar
  18. Jenkins, M. A., Traub, J. F.: A three-stage variable-shift iteration for polynomial zeros and its relation to generalized Rayleigh iteration. Numer. Math. 14, 252–263 (1970).MathSciNetzbMATHCrossRefGoogle Scholar
  19. Luenberger, D. G.: Introduction to Linear and Nonlinear Programming. Reading, Mass.: Addison-Wesley 1973.zbMATHGoogle Scholar
  20. Maehly, H.: Zur iterativen Auflösung algebraischer Gleichungen. Z. Angew. Math. Physik 5, 260–263 (1954).MathSciNetzbMATHCrossRefGoogle Scholar
  21. Marden, M.: The Geometry of the Zeros of a Polynomial in a Complex Variable. Providence, R.I.: Amer. Math. Soc. 1949.zbMATHGoogle Scholar
  22. Nickel, K.: Die numerische Berechnung der Wurzeln eines Polynoms. Numer. Math. 9, 80–98 (1966).MathSciNetzbMATHCrossRefGoogle Scholar
  23. Oren, S. S., Luenberger, D. G.: Self-scaling variable metric (SSVM) algorithms. I. Criteria and sufficient conditions for scaling a class of algorithms. Manage. Sci. 20, 845–862 (1974).MathSciNetzbMATHCrossRefGoogle Scholar
  24. Oren, S. S., Luenberger, D. G., Spedicato, E.: Optimal conditioning of self-scaling variable metric algorithms. Stanford University Dept. of Engineering, Economic Systems Report ARG-MR 74–5, 1974.Google Scholar
  25. Ortega, J. M, Rheinboldt, W. C.: Iterative Solution of Non-linear Equations in Several Variables. New York: Academic Press 1970.Google Scholar
  26. Ostrowski, A. M.: Solution of Equations and Systems of Equations. 2d edition. New York: Academic Press 1966.zbMATHGoogle Scholar
  27. Peters, G., Wilkinson, J. H.: Eigenvalues of A = λ B x with band symmetric A and B. Comput. J. 12, 398–404 (1969).MathSciNetzbMATHCrossRefGoogle Scholar
  28. Peters, G., Wilkinson, J. H.: Practical problems arising in the solution of polynomial equations. J. Inst. Math. Appl. 8, 16–35 (1971).MathSciNetzbMATHCrossRefGoogle Scholar
  29. Powell, M. J. D.: 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. 1976.Google Scholar
  30. Stoer, J.: On the convergence rate of imperfect minimization algorithms in Broyden’s β-class. Math. Programming 9, 313–335 (1975).MathSciNetzbMATHCrossRefGoogle Scholar
  31. Stoer, J.: On the relation between quadratic termination and convergence properties of minimization algorithms. Part I. Theory. Numer. Math. 28, 343–366 (1977).MathSciNetzbMATHCrossRefGoogle Scholar
  32. Tornheim, L.: Convergence of multipoint methods. J. Assoc. Comput. Mach. 11, 210–220 (1964).MathSciNetzbMATHCrossRefGoogle Scholar
  33. Traub, J. F.: Iterative Methods for the Solution of Equations. Englewood Cliffs, N.J.: Prentice-Hall 1964.zbMATHGoogle Scholar
  34. Wilkinson, J. H.: The evaluation of the zeros of ill-conditioned polynomials. Part I. Numer. Math. 1, 150–180 (1959).MathSciNetzbMATHCrossRefGoogle Scholar
  35. Wilkinson, J. H.: Rounding Errors in Algebraic Processes. Englewood Cliffs, N.J.: Prentice-Hall 1963.zbMATHGoogle Scholar
  36. Wilkinson, J. H.: The Algebraic Eigenvalue Problem. Oxford: Clarendon Press 1965.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1980

Authors and Affiliations

  • J. Stoer
    • 1
  • R. Bulirsch
    • 2
  1. 1.Institut für Angewandte MathematikUniversität Würzburg am HublandWürzburgFederal Republic of Germany
  2. 2.Institut für MathematikTechnische UniversitätMünchenFederal Republic of Germany

Personalised recommendations