zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Simple geometric constructions of quadratically and cubically convergent iterative functions to solve nonlinear equations. (English) Zbl 1137.65030
The authors derive the solution for a one parameter family of geometric construction methods using assumptions and theorems to solve nonlinear equations. The developed method is provided with solid mathematical fundamentals and numerical experiments are performed for illustration. Some of the remarks made in the paper are quite encouraging for the researchers in the area.
MSC:
65H05Single nonlinear equations (numerical methods)
References:
[1]Ostrowski, A.M.: Solutions of Equations and Systems of Equations. Academic Press, New York (1960)
[2]Traub, J.F.: Iterative Methods for Solution of Equations. Prentice-Hall, Englewood Cliffs, NJ (1964)
[3]Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)
[4]Halley, E.: A new, exact and easy method for finding the roots of any equations generally, without any previous reduction (Latin). Philos. Trans. R. Soc. Lond. 18, 136–148 (1694) · doi:10.1098/rstl.1694.0029
[5]Gutiérrez, J.M., Hernández, M.A.: A family of Chebyshev–Halley type methods in Banach spaces. Bull. Aust. Math. Soc. 55, 113–130 (1997) · Zbl 0893.47043 · doi:10.1017/S0004972700030586
[6]Melman, A.: Geometry and convergence of Euler’s and Halley’s method. SIAM Rev. 39(4), 728–735 (1997) · Zbl 0907.65045 · doi:10.1137/S0036144595301140
[7]Salehov, G.S.: On the convergence of the process of tangent hyperbolas. Dokl. Akad. Nauk. SSSR 82, 525–528 (1952)
[8]Scavo, T.R., Thoo, J.B.: On the geometry of Halley’s method. Am. Math. Mon. 102, 417–426 (1995) · Zbl 0830.01005 · doi:10.2307/2975033
[9]Gander, W.: On Halley’s iteration method. Am. Math. Mon. 92, 131–134 (1985) · Zbl 0574.65041 · doi:10.2307/2322644
[10]Amat, S., Busquier, S., Candela, V.F., Potra, F.A.: Convergence of third order iterative methods in Banach spaces, Preprint, Vol. 16, U.P. Caratgena, 2001
[11]Gutiérrez, J.M., Hernández, M.A.: An acceleration of Newton’s method: Super-Halley method. Appl. Math. Comput. 117, 223–239 (2001) · Zbl 1023.65051 · doi:10.1016/S0096-3003(99)00175-7
[12]Amat, S., Busquier, S., Gutiérrez, J.M.: Geometric constructions of iterative functions to solve nonlinear equations. J. Comput. Appl. Math. 157, 197–205 (2003) · Zbl 1024.65040 · doi:10.1016/S0377-0427(03)00420-5
[13]Kanwar, V., Singh, Sukhjit, Guha, R.K., Mamta, : On method of osculating circle for solving nonlinear equations. Appl. Math. Comput. 176, 379–382 (2006) · Zbl 1108.65047 · doi:10.1016/j.amc.2005.09.026
[14]Sharma, J.R.: A family of third-order methods to solve nonlinear equations by quadratic curves approximation. Appl. Math. Comput. 184, 210–215 (2007) · Zbl 1114.65050 · doi:10.1016/j.amc.2006.05.193
[15]Chun, C.: A one-parameter family of third-order methods to solve nonlinear equations. Appl. Math. Comput. 189, 126–130 (2007) · Zbl 1122.65323 · doi:10.1016/j.amc.2006.11.058
[16]Jiang, D., Han, D.: Some one-parameter families of third-order methods for solving nonlinear equations. Appl. Math. Comput. DOI 10.1016/j.amc.2007.04.100
[17]Ben-Israel, A.: Newton’s method with modified functions. Contemp. Math. 204, 39–50 (1997)
[18]Kanwar, V., Tomar, S.K.: Modified families of Newton, Halley and Chebyshev methods. Appl. Math. Comput. 192, 20–26 (2007) · Zbl 1193.65065 · doi:10.1016/j.amc.2007.02.119
[19]Wu, X., Wu, H.W.: On a class of quadratic convergence iteration formulae without derivatives. Appl. Math. Comput. 10(7), 77–80 (2000) · Zbl 1023.65042 · doi:10.1016/S0096-3003(98)10009-7
[20]Kanwar, V., Tomar, S.K.: Modified families of multi-point iterative methods for solving nonlinear equations. Numer. Algor 44, 381–389 (2007) · Zbl 1216.65057 · doi:10.1007/s11075-007-9120-4