Numerical Algorithms

, Volume 61, Issue 4, pp 567–578

# Modified Chebyshev-Halley type method and its variants for computing multiple roots

• Janak Raj Sharma
• Rajni Sharma
Original Paper

## Abstract

We present two families of third order methods for finding multiple roots of nonlinear equations. One family is based on the Chebyshev-Halley scheme (for simple roots) and includes Halley, Chebyshev and Chun-Neta methods as particular cases for multiple roots. The second family is based on the variant of Chebyshev-Halley scheme and includes the methods of Dong, Homeier, Neta and Li et al. as particular cases. The efficacy is tested on a number of relevant numerical problems. It is observed that the new methods of the families are equally competitive with the well known special cases of the families.

## Keywords

Nonlinear equations Newton method Chebyshev-Halley method Rootfinding Multiple roots Order of convergence

65H05 65B99

## References

1. 1.
Chapra, S.C., Canale, R.P.: Numerical Methods for Engineers. McGraw-Hill Book Company, New York (1988)Google Scholar
2. 2.
Chun, C., Neta, B.: A third order modification of Newton’s method for multiple roots. Appl. Math. Comput. 211, 474–479 (2009)
3. 3.
Dong, C.: A basic theorem of constructing an iterative formula of the higher order for computing multiple roots of an equation. Math. Numer. Sinica 11, 445–450 (1982)Google Scholar
4. 4.
Dong, C.: A family of multipoint iterative functions for finding multiple roots of equations. Int. J. Comput. Math. 21, 363–367 (1987)
5. 5.
Gautschi, W.: Numerical Analysis: An Introduction. Birkhäuser, Boston, MA (1997)
6. 6.
Gutiérrez, J.M., Hernández, M.A.: A family of Chebyshev-Halley type methods in Banach spaces. Bull. Aust. Math. Soc. 55, 113–130 (1997)
7. 7.
Hansen, E., Patrick, M.: A family of root finding methods. Numer. Math. 27, 257–269 (1977)
8. 8.
Homeier, H.H.H.: On Newton-type methods for multiple roots with cubic convergence. J. Comput. Appl. Math. 231, 249–254 (2009)
9. 9.
Li, S., Li, H., Cheng, L.: Second-derivative free variants of Halley’s method for multiple roots. Appl. Math. Comput. 215, 2192–2198 (2009)
10. 10.
Li, S., Cheng, L., Neta, B.: Some fourth-order nonlinear solvers with closed formulae for multiple roots. Comput. Math. Appl. 59, 126–135 (2010)
11. 11.
Neta, B.: New third order nonlinear solvers for multiple roots. Appl. Math. Comput. 202, 162–170 (2008)
12. 12.
Neta, B.: Extension of Murakami’s high-order nonlinear solver to multiple roots. Int. J. Comput. Math. 87, 1023–1031 (2010)
13. 13.
Neta, B., Johnson, A.N.: High order nonlinear solver for multiple roots. Comput. Math. Appl. 55, 2012–2017 (2008)
14. 14.
Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)
15. 15.
Osada, N.: An optimal multiple root-finding method of order three. J. Comput. Appl. Math. 51, 131–133 (1994)
16. 16.
Schröder, E.: Über unendlich viele Algorithmen zur Auflösung der Gleichungen. Math. Ann. 2, 317–365 (1870)
17. 17.
Sharma, J.R., Sharma, R.: New third and fourth order nonlinear solvers for computing multiple roots. Appl. Math. Comput. 217, 9756–9764 (2011)
18. 18.
Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice-Hall, Englewood Cliffs, NJ (1964)
19. 19.
Victory, H.D., Neta, B.: A higher order method for multiple zeros of nonlinear functions. Int. J. Comput. Math. 12, 329–335 (1983)
20. 20.
Wolfram, S.: The Mathematica Book, 5th edn. Wolfram Media (2003)Google Scholar