On the Order of Convergence of a Determinantal Family of Root-Finding Methods

BIT Numerical Mathematics

Volumn 39, Issue 1, 1999, Pages 96-109

  Kalantari, Bahman ^a  

^a RUTGERS UNIVERSITY   (United States)

Author keywords

Generalized Fibonacci numbers; Halley's method; Interpolation; Iteration functions; Newton's method; Roots; Taylor's theorem

Indexed keywords

EID: 0004967376     PISSN: 00063835     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1022321325108     Document Type: Article

References (34)

1. G. Alefeld, On the convergence of Halley's Method, Amer. Math. Monthly, 88 (1981), pp. 530-536.
2. K. E. Atkinson, An Introduction to Numerical Analysis, 2nd ed., Wiley, New York, NY, 1989.
3. H. Bateman, Halley's method for solving equations, Amer. Math. Monthly, 45 (1938), pp. 11-17.
4. G. E. Bergum, A. N. Philippou, and A. F. Hordam, Applications of Fibonacci Numbers, vol. 5, Kluwer Academic Publishers, Dordrecht, 1992.
5. E. Bodewig, On types of convergence and on the behavior of approximations in the neighborhood of a multiple root of an equation, Quart. Appl. Math., 7 (1949), pp. 325-333.
6. G. H. Brown, Jr., On Halley's variation of Newton's method, Amer. Math. Monthly, 84 (1977), pp. 726-728.
7. M. Davies, and B. Dawson, On the global convergence of Halley's iteration formula, Numer. Math., 24 (1975), pp. 133-135.
8. M. Feinberg, Fibonacci-Tribonacci, Fibonacci Quarterly, 1 (1963), pp. 71-74.
9. W. F. Ford, and J. A. Pennline, Accelerated convergence in Newton's method, SIAM Review, 38 (1996), pp. 658-659.
10. J. S. Frame, The solution of equations by continued fraction, Amer. Math. Monthly, 60 (1953), pp. 293-305.
11. W. Gander, On Halley's iteration method, Amer. Math. Monthly, 92 (1985), pp. 131-134.
12. J. Gerlach, Accelerated convergence in Newton's method, SIAM Review, 36 (1994), pp. 272-276.
13. G. Golub and C. Van Loan, Matrix Computations, 3rd ed., Johns Hopkins University Press, Baltimore, MD, 1996.
14. E. Halley, A new, exact, and easy method of finding roots of any equations generally, and that without any previous reduction, Philos. Trans. Roy. Soc. London, 18 (1694), pp. 136-145.
15. H. J. Hamilton, A type of variation of Newton's method, Amer. Math. Monthly, 57 (1950), pp. 517-522.
16. E. Hansen, and M. Patrick, A family of root finding methods, Numer. Math., 27 (1977), pp. 257-269.
17. P. Henrici, Applied and Computational Complex Analysis, vol. I, Wiley, New York, 1974.
18. A. S. Householder, The Numerical Treatment of a Single Nonlinear Equation, McGraw-Hill, New York, 1970.
19. B. Kalantari, and I. Kalantari, High order iterative methods for approximating square roots, BIT, 36 (1996), pp. 395-399.
20. B. Kalantari, I. Kalantari, and R. Zaare-Nahandi, A basic family of iteration functions for polynomial root finding and its characterizations, J. Comput. Appl. Math., 80 (1997), pp. 209-226.
21. B. Kalantari, Generalization of Taylor's theorem and Newton's method via a new family of determinantal interpolation formulas, Technical Report DCS-TR 328, Department of Computer Science, Rutgers University, New Brunswick, NJ, 1997.
22. E. P. Miles, Generalized Fibonacci numbers and associated matrices, Amer. Math. Monthly, 6 (1960), pp. 745-757.
23. A. M. Ostrowski, Solution of Equations and System of Equations, 2nd ed., Academic Press, New York, 1966.
24. V. Y. Pan, Solving a polynomial equation: some history and recent progress, SIAM Review, 39 (1997), pp. 187-220.
25. D. B. Popovski, A family of one-point iteration formulae for finding roots, Internat. J. Computer Math., 8 (1980), pp. 85-88.
26. T. R. Scavo, and J. B. Thoo, On the geometry of Halley's method, Amer. Math. Monthly, 102 (1995), pp. 417-426.
27. M. Shub and S. Smale, On the geometry of polynomials and a theory of cost: Part I, Ann. Scient. Ec. Norm. Sup., 18 (1985), pp. 107-142.
28. S. Smale, Newton's method estimates from data at one point, in The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics, R. E. Ewing, K. I. Gross, C. F. Martin, eds., Springer-Verlag, New York, 1986, pp. 185-196.
29. R. W. Snyder, One more correction formula, Amer. Math. Monthly, 62 (1955), pp. 722-725.
30. J. K. Stewart, Another variation of Newton's method, Amer. Math. Monthly, 58 (1951), pp. 331-334.
31. J. F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall, Englewood Cliffs, NJ, 1964.
32. J. F. Traub, A class of globally convergent iteration functions for the, solution of polynomial equations, Math. Comp., 20 (1966), pp. 113-138.
33. H. S. Wall, A modification of Newton's method, Amer. Math. Monthly, 55 (1948), pp. 90-94.
34. T. J. Ypma, Historical development of Newton Raphson method, SIAM Review, 37 (1995), pp. 531-551.