zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Julia sets for the super-Newton method, Cauchy’s method, and Halley’s method. (English) Zbl 1080.65532
Summary: We study numerically and dynamically three cubically convergent iterative root-finding algorithms, namely Cauchy’s method, the super-Newton method, and Halley’s method. Using the concept of a universal Julia set (motivated by the results of McMullen), we establish that these algorithms converge when applied to any quadratic with distinct roots. We give examples showing the existence of attracting periodic orbits not associated to a root for the super-Newton method and Halley’s method applied to cubic polynomials. We include computer plots showing the dynamic structure for each algorithm applied to a variety of polynomials.

65H05Single nonlinear equations (numerical methods)
30C15Zeros of polynomials, etc. (one complex variable)
37F50Small divisors, rotation domains and linearization; Fatou and Julia sets
Full Text: DOI