# Finding Zeros and Minimum Points by Iterative Methods

• J. Stoer
• R. Bulirsch

## Abstract

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)
$$p\left( x \right) = {a_0} + {a_1}x + \cdots + {a_n}{x^n}$$
(5.0.1)
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.

## Keywords

Iterative Method Newton Method Line Search Real Root Minimum Point

## 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).
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.
4. Broyden, C. G.: A class of methods for solving nonlinear simultaneous equations. Math. Comput. 19, 577–593 (1965).
5. Broyden, C. G.: Quasi-Newton-methods and their application to function minimization. Math. Comput. 21, 368–381 (1967).
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).
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).
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.
10. Davidon, W. C.: Optimally conditioned optimization algorithms without line searches. Math. Programming 9, 1–30 (1975).
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).
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).
14. Gill, P. E., Golub, G. H., Murray, W., Saunders, M. A.: Methods for modifying matrix factorizations. Math. Comput. 28, 505–535 (1974).
15. Henrici, P.: Applied and Computational Complex Analysis. Vol. 1. New York: Wiley 1974.
16. Himmelblau, D. M.: Applied Nonlinear Programming. New York: McGraw-Hill 1972.
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).
19. Luenberger, D. G.: Introduction to Linear and Nonlinear Programming. Reading, Mass.: Addison-Wesley 1973.
20. Maehly, H.: Zur iterativen Auflösung algebraischer Gleichungen. Z. Angew. Math. Physik 5, 260–263 (1954).
21. Marden, M.: The Geometry of the Zeros of a Polynomial in a Complex Variable. Providence, R.I.: Amer. Math. Soc. 1949.
22. Nickel, K.: Die numerische Berechnung der Wurzeln eines Polynoms. Numer. Math. 9, 80–98 (1966).
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).
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.
27. Peters, G., Wilkinson, J. H.: Eigenvalues of A = λ B x with band symmetric A and B. Comput. J. 12, 398–404 (1969).
28. Peters, G., Wilkinson, J. H.: Practical problems arising in the solution of polynomial equations. J. Inst. Math. Appl. 8, 16–35 (1971).
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).
31. Stoer, J.: On the relation between quadratic termination and convergence properties of minimization algorithms. Part I. Theory. Numer. Math. 28, 343–366 (1977).
32. Tornheim, L.: Convergence of multipoint methods. J. Assoc. Comput. Mach. 11, 210–220 (1964).
33. Traub, J. F.: Iterative Methods for the Solution of Equations. Englewood Cliffs, N.J.: Prentice-Hall 1964.
34. Wilkinson, J. H.: The evaluation of the zeros of ill-conditioned polynomials. Part I. Numer. Math. 1, 150–180 (1959).
35. Wilkinson, J. H.: Rounding Errors in Algebraic Processes. Englewood Cliffs, N.J.: Prentice-Hall 1963.
36. Wilkinson, J. H.: The Algebraic Eigenvalue Problem. Oxford: Clarendon Press 1965.

## 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