Newton's method and generation of a determinantal family of iteration functions

[摘要] It is well known that Halley's method can be obtained by applying Newton's method to the function f/root f'. Gerlach (SIAM Rev. 36 (1994) 272-276) gives a generalization of this approach, and for each m greater than or equal to 2, recursively defines an iteration function G(m)(x) having order m. Kalantari et al. (J. Comput. Appl. Math. 80 (1997) 209-226) and Kalantari (Technical Report DCS-TR 328, Department of Computer Science, Rutgers University, New Brunswick, NJ, 1997) derive and characterize a determinantal family of iteration functions, called the Basic Family, B-m(x), m greater than or equal to 2. In this paper we prove, G(m)(x) = B-m(x). On the one hand, this implies that G(m)(x) enjoys the previously derived properties of B-m(x), i.e., the closed formula, its efficient computation, an expansion formula which gives its precise asymptotic constant, as well as its multipoint versions. On the other, this gives a new insight on the Basic Family and Newton's method. (C) 2000 Elsevier Science B.V. All rights reserved.

[发布日期] 2000-04-01

[关键词] Newton's method;rootfinding;polynomial roots