Full Text:   <1928>

CLC number: O241.6

On-line Access: 

Received: 2001-10-15

Revision Accepted: 2002-03-21

Crosschecked: 0000-00-00

Cited: 0

Clicked: 3987

Citations:  Bibtex RefMan EndNote GB/T7714

-   Go to

Article info.
Open peer comments

Journal of Zhejiang University SCIENCE A 2003 Vol.4 No.1 P.80-85

http://doi.org/10.1631/jzus.2003.0080


On the approximate zero of Newton method


Author(s):  HUANG Zheng-da

Affiliation(s):  Department of Mathematics, Zhejiang University, Hangzhou 310028, China

Corresponding email(s):   huangzd@css.zju.edu.cn

Key Words:  Approximate zero, Newton method, Generalized Kantorovich Condition


Share this article to: More

HUANG Zheng-da. On the approximate zero of Newton method[J]. Journal of Zhejiang University Science A, 2003, 4(1): 80-85.

@article{title="On the approximate zero of Newton method",
author="HUANG Zheng-da",
journal="Journal of Zhejiang University Science A",
volume="4",
number="1",
pages="80-85",
year="2003",
publisher="Zhejiang University Press & Springer",
doi="10.1631/jzus.2003.0080"
}

%0 Journal Article
%T On the approximate zero of Newton method
%A HUANG Zheng-da
%J Journal of Zhejiang University SCIENCE A
%V 4
%N 1
%P 80-85
%@ 1869-1951
%D 2003
%I Zhejiang University Press & Springer
%DOI 10.1631/jzus.2003.0080

TY - JOUR
T1 - On the approximate zero of Newton method
A1 - HUANG Zheng-da
J0 - Journal of Zhejiang University Science A
VL - 4
IS - 1
SP - 80
EP - 85
%@ 1869-1951
Y1 - 2003
PB - Zhejiang University Press & Springer
ER -
DOI - 10.1631/jzus.2003.0080


Abstract: 
A judgment criterion to guarantee a point to be a Chen's approximate zero of newton method for solving nonlinear equation is sought by dominating sequence techniques. The criterion is based on the fact that the dominating function may have only one simple positive zero, assuming that the operator is weak Lipschitz continuous, which is much more relaxed and can be checked much more easily than Lipschitz continuous in practice. It is demonstrated that a Chen's approximate zero may not be a Smale's approximate zero. The error estimate obtained indicated the convergent order when we use |f(x)|<ε to stop computation in software. The result can also be applied for solving partial derivative and integration equations.

Darkslateblue:Affiliate; Royal Blue:Author; Turquoise:Article

Reference

[1]Blum,L., Cucker, F., Shub, M., and Smale, S., 1997. Real Complexity and Computation. Preprint, City University of Hong Kong.

[2]Chen, P., 1994. Approximate zeros of quadratically convergent algorithms. Math. Compu., 63: 247-270.

[3]Curry, J.H., 1989. On zero finding methods of higher order from data at one point. J. Complexity, 5: 219-237.

[4]Kim, K., 1988. On approximate zeros and root finding algorithms for a complex polynomial. Math. Comput., 51:707-719.

[5]Ostrowski, A. M., 1973. Solution of Equations in Euclidean and Banach Spaces. Academic Press, New York.

[6]Rheinboldt W., 1998. Methods for Solving Systems of Nonlinear Equations, (2nd Edition), SIAM, Philadelphia.

[7]Shub, M. and Smale, S.,1985. Computational complexity: on the geometry of polynomials and a theory of costs I.Ann. Sci. Ecole Norm. Sup., 18(5):107-142.

[8]Shub, M. and Smale, S., 1986. Computational complexity: on the geometry of polynomials and a theory of costs II.SIAM J.Comput., 15:145-161.

[9]Smale, S., 1981. The fundamental theorem of algebra and complexity theory. Bull. Amer. Math. Soc.(N.S.), 4:1-36.

[10]Smale, S., 1985. On the efficiency of algorithms of analysis. Bull. Amer. Math. Soc. (N.S.), 13: 87-121.

[11]Smale, S., 1986. Newton's Method Estimates from Data at One Point, In: Richard E. Ewing, Kenneth I. Gross, and Clyde F. Martin, (Eds.) The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics, Springer-Verlag, p.185-196.

[12]Smale, S., 1987. Algorithms for Solving Equations. Proc. Internet. Congr. Math.(Berkeley, CA, 1986), Amer. Math. Soc., Providence, RI, p.172-195.

[13]Traub, J.F., 1963. Iterative Methods for The Solution of Equations. Prentice-Hall, Ink., Englwood Cliffs, N.Y.

[14]Tsuchiya, T., 1999. An application of the Kantorovich theorem to nonlinear finite element analysis. Numer. Math., 84:121-141.

[15]Wang, X., 1997. Convergence on the iteration of Halley family in weak conditions. Chinese Science Bulletin, 42:552-554.

[16]Wang, X., 1998a. Convergence of iteration of Halley's family and Smale operator class in Banach space. Science in China, (Ser. A)., 41:700-706.

[17]Wang, X., 1998b. Definite version on precise point estimate. Progress in Natural Science, 8:152-158.

[18]Wang, X., 1999. Convergence of Newton's method and inverse function theorem in Banach space. Math. Comput., 68:169-186.

[19]Wang, X., 2000a. Convergence of Newton's method and uniqueness of the solution of equations in Banach space. IMA J. Numer. Anal., 20:123-134.

[20]Wang, X., 2000b. Convergence of iterations of Euler family under weak condition. Science in China (Ser. A), 43:958-962.

[21]Wang, X. and Han, D., 1990. On dominating sequence method in the point estimate and Smale theorem. Sci. Sinica (Ser. A), 33:135-144.

[22]Wang, X. and Han, D., 1997. Criterion α and Newton's method in the weak conditions. Math. Numer. Sinica, 19:103-112(in Chinese).

[23]Wang, X. and Li,C., 2001. Local and global behavior for algorithms of solving equations. Chinese Science Bulletin, 46:441-448.

[24]Wang, X. and Xuan, X., 1987. Radom polynomial space and computational complexity theory. Sci. Sinica (Ser. A), 30:673-684.

Open peer comments: Debate/Discuss/Question/Opinion

<1>

Please provide your name, email address and a comment





Journal of Zhejiang University-SCIENCE, 38 Zheda Road, Hangzhou 310027, China
Tel: +86-571-87952783; E-mail: cjzhang@zju.edu.cn
Copyright © 2000 - 2022 Journal of Zhejiang University-SCIENCE