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

Bahman Kalantari, Jürgen Gerlach

Research output: Contribution to journalArticle

21 Citations (Scopus)

Abstract

It is well known that Halley's method can be obtained by applying Newton's method to the function f/f′. Gerlach (SIAM Rev. 36 (1994) 272-276) gives a generalization of this approach, and for each m≥2, recursively defines an iteration function Gm(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, Bm(x), m≥2. In this paper we prove, Gm(x)=Bm(x). On the one hand, this implies that Gm(x) enjoys the previously derived properties of Bm(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.

Original languageEnglish (US)
Pages (from-to)195-200
Number of pages6
JournalJournal of Computational and Applied Mathematics
Volume116
Issue number1
DOIs
StatePublished - Apr 1 2000

Fingerprint

Iteration Function
Newton-Raphson method
Newton Methods
Halley's Method
Precise Asymptotics
Computer science
Computer Science
Imply
Closed
Family

All Science Journal Classification (ASJC) codes

  • Computational Mathematics
  • Applied Mathematics

Cite this

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Newton's method and generation of a determinantal family of iteration functions. / Kalantari, Bahman; Gerlach, Jürgen.

In: Journal of Computational and Applied Mathematics, Vol. 116, No. 1, 01.04.2000, p. 195-200.

Research output: Contribution to journalArticle

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