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)
Three-step iterative methods with eighth-order convergence for solving nonlinear equations. (English) Zbl 1161.65039
For solving a nonlinear scalar equation, the authors propose a one parameter family of three-step iterative methods of eight-order. The new methods are based on the classical Newton method, on King’s methods [see {\it R. F. King}, SIAM J. Numer. Anal. 10, 876--879 (1973; Zbl 0266.65040)], and on the methods of {\it C. Chun} and {\it Y. Ham} [Appl. Math. Comp. 193, 389--394 (2007)]. The iterations are expressed by means of divided difference of orders two and three and by means of a given real-valued function. To apply these methods for solving equations, three evaluations of the function from the left hand side of the equation and one evaluation of its first derivative are required. Using the definition of efficiency index, the family of derived methods has the index 1.682, better than the index of Newton’s method, of King’s methods and of Chun’s methods. Numerical examples are performed and comparison of various iterative methods under the same total number of function evaluations are made.

65H05Single nonlinear equations (numerical methods)
65Y20Complexity and performance of numerical algorithms
Full Text: DOI
[1] Ortega, J. M.; Rheinbolt, W. C.: Iterative solution of nonlinear equations in several variables, (1970) · Zbl 0241.65046
[2] King, R. F.: A family of fourth order methods for nonlinear equations, SIAM J. Numer. anal. 10, 876-879 (1973) · Zbl 0266.65040 · doi:10.1137/0710072
[3] Ostrowski, A. M.: Solution of equations in Euclidean and Banach spaces, (1960) · Zbl 0115.11201
[4] Traub, J. F.: Iterative methods for the solution of equations, (1964) · Zbl 0121.11204
[5] Grau, M.; Díaz-Barrero, J. L.: An improvement to Ostrowski root-finding method, Appl. math. Comput. 173, 450-456 (2006) · Zbl 1090.65053 · doi:10.1016/j.amc.2005.04.043
[6] Sharma, J. R.; Guha, R. K.: A family of modified Ostrowski methods with accelerated sixth-order convergence, Appl. math. Comput. 190, 111-115 (2007) · Zbl 1126.65046 · doi:10.1016/j.amc.2007.01.009
[7] Chun, C.; Ham, Y.: Some sixth-order variants of Ostrowski root-finding methods, Appl. math. Comput. 193, 389-394 (2007) · Zbl 1193.65055 · doi:10.1016/j.amc.2007.03.074
[8] Kou, J.; Li, Y.; Wang, X.: Some variants of Ostrowski’s method with seventh-order convergence, J. comput. Appl. math. 209, 153-159 (2007) · Zbl 1130.41006 · doi:10.1016/j.cam.2006.10.073
[9] Kung, H. T.; Traub, J. F.: Optimal order of one-point and multipoint iteration, J. assoc. Comput. math. 21, 634-651 (1974) · Zbl 0289.65023 · doi:10.1145/321850.321860
[10] Quarteroni, A.; Sacco, R.; Saleri, F.: Numerical mathematics, (2000) · Zbl 0957.65001
[11] Gautschi, W.: Numerical analysis: an introduction, (1997) · Zbl 0877.65001
[12] Weerakoon, S.; Fernando, T. G. I.: A variant of Newton’s method with accelerated third-order convergence, Appl. math. Lett. 13, 87-93 (2000) · Zbl 0973.65037 · doi:10.1016/S0893-9659(00)00100-2