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)
Chaotic dynamics of a third-order Newton-type method. (English) Zbl 1187.65050

For solving a nonlinear scalar equation $f\left(x\right)=0$ by a third-order Newton-type method, the authors discuss the behaviour of the dynamics of iterations for this method, when it is applied to polynomial equations of degree two or three. It is well known that the classical Newton method converges quadratically in a neighbourhood of a simple root of the equation, and if the left hand side of the equation is a polynomial, then the corresponding Newton iterative function is a rational map (a quotient of two polynomials without common factors). The third-order Newton-type method, obtained by a composition of two Newton iterative function, requires more computational cost and this type of methods is applied only in some cases. Moreover, each iteration consists in two steps of the Newton method having the same derivative.

The main interest of the authors is the study of the dynamics of the discrete dynamical system defined by this third-order Newton-type method. The analysis is based on the so called “scaling theorem”, the main theorem proved in this paper. For several simple quadratic polynomials, the analysis shows that the dynamics of the method is chaotic. For cubic polynomials, the results of analysis show that bifurcations and chaos appear. From the numerical point of view, this represents a great difficulty to determine the region of convergence of the method to the solution of the given equation.

MSC:
 65H05 Single nonlinear equations (numerical methods) 65H04 Roots of polynomial equations (numerical methods) 65P20 Numerical chaos
References:
 [1] Amat, S.; Busquier, S.: A two-step Steffensen’s method under modified convergence conditions, J. math. Anal. appl. 324, No. 1 – 15, 1084-1092 (2006) · Zbl 1103.65060 · doi:10.1016/j.jmaa.2005.12.078 [2] Amat, S.; Busquier, S.: Third-order iterative methods under kantarovich conditions, J. math. Anal. appl. 336, No. 1 – 1, 243-261 (2007) · Zbl 1128.65036 · doi:10.1016/j.jmaa.2007.02.052 [3] Amat, S.; Busquier, S.: Convergence and numerical analysis of a family of two-step Steffensen’s methods, Comput. math. Appl. 49, No. 1, 13-22 (2005) · Zbl 1075.65080 · doi:10.1016/j.camwa.2005.01.002 [4] Amat, S.; Busquier, S.; Candela, V. F.: Third order iterative methods without using second Fréchet derivative, J. comput. Math. 22, No. 3, 341-346 (2005) · Zbl 1054.65056 [5] Amat, S.; Busquier, S.; El Kebir, D.; Molina, J.: A fast Chebyshev’s method for quadratic equations, Appl. math. Comput. 148, No. 2, 461-474 (2004) · Zbl 1038.65045 · doi:10.1016/S0096-3003(02)00914-1 [6] Amat, S.; Busquier, S.; Candela, V. F.: Third-order iterative methods without using any Fréchet derivative, J. comput. Appl. math. 158, No. 1, 11-18 (2003) · Zbl 1037.65058 · doi:10.1016/S0377-0427(03)00460-6 [7] Amat, S.; Busquier, S.; Gutiérrez, J. M.: Geometric constructions of iterative functions to solve nonlinear equations, J. comput. Appl. math. 157, No. 1, 197-205 (2003) · Zbl 1024.65040 · doi:10.1016/S0377-0427(03)00420-5 [8] Argyros, I. K.: Quadratic equations and applications to Chandrasekhar’s and related equations, Bull. austral. Math. soc. 32, 275-292 (1985) · Zbl 0607.47063 · doi:10.1017/S0004972700009953 [9] Gutiérrez, J. M.; Hernández, M. A.; Salanova, M. A.: Quadratic equations in Banach spaces, Numer. funct. Anal. optim. 7, No. 1 – 2, 113-121 (1996) · Zbl 0849.47038 · doi:10.1080/01630569608816686 [10] Hernández, M. A.; Romero, N.: On a characterization of some Newton-like methods of R-order at least three, J. comput. Appl. math. 183, No. 1, 53-66 (2005) · Zbl 1087.65057 · doi:10.1016/j.cam.2005.01.001 [11] Hurley, M.: Attracting orbits in Newton’s method, Trans. amer. Math. soc. 297, No. 1, 143-158 (1986) · Zbl 0632.65059 · doi:10.2307/2000461 [12] Hurley, M.; Martin, C.: Newton’s algorithm and chaotic dynamical systems, SIAM J. Math. anal. 15, No. 2, 238-252 (1984) · Zbl 0588.65033 · doi:10.1137/0515020 [13] Potra, F. A.; Pták, V.: Nondiscrete induction and iterative processes, Res. notes math. 103 (1984) · Zbl 0549.41001 [14] Saari, D.; Urenko, J.: Newton’s method, circle maps, and chaotic motion, Amer. math. Monthly 91, 3-17 (1984) · Zbl 0532.58016 · doi:10.2307/2322163 [15] Traub, J. F.: Iterative methods for the solution of equations, (1964) · Zbl 0121.11204 [16] Wong, S.: Newton’s method and symbolic dynamics, Proc. amer. Math. soc. 91, No. 2, 245-253 (1994) · Zbl 0554.65038 · doi:10.2307/2044635