×

Cubic convergence of parameter-controlled Newton-secant method for multiple zeros. (English) Zbl 1194.65073

Authors’ abstract: Let \(\mathbb C\to\mathbb C\) have a multiple zero a with integer multiplicity \(m\geq 1\) and be analytic in a sufficiently small neighborhood of \(\alpha\). For parameter-controlled Newton-secant method defined by
\[ x_{n+1}=x_n-\frac{\lambda f(x_n)^2}{f'(x_n)\cdot \{f(x_n)-f(x_n-\mu f(x_n)/f'(x_n))\}'}\;,\quad n= 0,1,2,\dots \]
we investigate the maximal order of convergence and the theoretical asymptotic error constant by seeking the relationship between parameters \(\lambda\) and \(\mu\). For various test functions, the numerical method has shown a satisfactory result with high-precision mathematica programming.

MSC:

65H05 Numerical computation of solutions to single equations
65H04 Numerical computation of roots of polynomial equations

Software:

Mathematica
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Traub, J. F., Iterative Methods for the Solution of Equations (1982), Chelsea Publishing Company · Zbl 0472.65040
[2] Bathi Kasturiarachi, A., Leap-frogging Newton’s Method, Int. J. Math. Educ. Sci. Technol., 33, 4, 521-527 (2002)
[3] Gamelin, Theodore W., Complex Analysis (2003), Springer · Zbl 0978.30001
[4] Mathews, John H., Basic Complex Variables for Mathematics and Engineering (1982), Allyn and bacon, Inc.
[5] Cheney, Ward; Kincaid, David, Numerical Mathematics and Computing (1980), Brooks/Cole Publishing Company: Brooks/Cole Publishing Company Monterey, California · Zbl 0487.65001
[6] Conte, S. D.; de Boor, Carl, Elementary Numerical Analysis (1980), McGraw-Hill Inc · Zbl 0496.65001
[7] Geum, Y. H., The asymptotic error constant of leap-frogging Newtons method locating a simple real zero, Appl. Math. Computat., 189, 217, 963-969 (2007) · Zbl 1122.65331
[8] Stoer, J.; Bulirsh, R., Introduction to Numerical Analysis (1980), Springer-Verlag: Springer-Verlag New York Inc., pp. 244-313
[9] Chun, Changbum, On the construction of iterative methods with at least cubic convergence, Appl. Math. Computat., 189, 2, 1384-1392 (2007) · Zbl 1122.65326
[10] Golbabai, A.; Javidi, M., A third-order Newton type method for nonlinear equations based on modified homotopy perturbation method, Appl. Math. Computat., 191, 1, 199-205 (2007) · Zbl 1193.65058
[11] Kou, Jisheng, A third-order modification of Newton method for systems of non-linear equations, Appl. Math. Computat., 191, 1, 117-121 (2007) · Zbl 1193.65077
[12] Petkovic, L. D.; Petkovic, M. S.; Zivkovic, D., Hansen-Patrick’s family is of Laguerre’s type, Novi Sad J. Math., 33, 1, 109-115 (2003) · Zbl 1274.65151
[13] Clark, Allan, Elements of Abstract Algebra (1971), Dover Publications, Inc.
[14] Marsden, Jerrold E., Elementary Classical Analysis (1974), W. H. Freeman and Company · Zbl 0285.26005
[15] Wolfram, Stephen, The Mathematica Book (1999), Cambridge University Press · Zbl 0924.65002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.