# 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)
Third-order methods from quadrature formulae for solving systems of nonlinear equations. (English) Zbl 1050.65055
Authors’ summary: We extend to $p$-dimensional problems a modification of the Newton method, based on quadrature formulas of order at least one, which produces iterative methods with order of convergence three. A general error analysis providing the higher order of convergence is given. These new methods may be more efficient then other third-order methods as they do not require the use of the second-order Fréchet derivative.

##### MSC:
 65H10 Systems of nonlinear equations (numerical methods)
Full Text:
##### References:
 [1] Dennis, J. E.; Schnable, R. B.: Numerical methods for unconstrained optimization and nonlinear equations. (1983) [2] Ford, W. F.; Pennline, I. A.: Accelerated convergence in Newton’s method. SIAM rev. 38, 658-659 (1996) · Zbl 0863.65026 [3] Frontini, M.; Sormani, E.: Some variants of Newton’s method with third-order convergence. Appl. math. Comput. 140, 419-426 (2003) · Zbl 1037.65051 [4] M. Frontini, E. Sormani, Modified Newton’s method with third-order convergence and multiple roots, Comp. Appl. Math., in press · Zbl 1030.65044 [5] Gerlach, J.: Accelerated convergence in Newton’s method. SIAM rev. 2, 272-276 (1994) · Zbl 0814.65046 [6] Halley, E.: Methodus nova, accurata and facilis inveniendi radices aequationum quarumcumque generaliter, sine praevia reductione. Philos. trans. Roy. soc. London 18, 136-148 (1694) [7] Ortega, J. M.; Rheinboldt, W. C.: Iterative solution of nonlinear equations in several variables. (1970) · Zbl 0241.65046 [8] Palacios, M.: Kepler equation and accelerated Newton method. J. comput. Appl. math. 138, 335-346 (2002) · Zbl 0998.65054 [9] Weerakoom, 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 [10] Zheng, S.; Robbie, D.: A note on the convergence of halley’s method for solving operator equations. J. austral. Math. soc. Ser. B 37, 16-25 (1995) · Zbl 0842.65035