Extension of Murakami’s high-order non-linear solver to multiple roots. (English) Zbl 1192.65052

Summary: Several one-parameter families of fourth-order methods for finding multiple zeros of non-linear functions are developed. The methods are based on T. Murakami’s fifth-order method (for simple roots) [J. Inf. Process. 1, 138–139 (1978; Zbl 0394.65015)] and they require one evaluation of the function and three evaluations of the derivative. The informational efficiency of the methods is the same as for the previously developed methods of lower order. For a double root, the method is more efficient than all previously known schemes. All these methods require the knowledge of multiplicity.


65H05 Numerical computation of solutions to single equations
65Y20 Complexity and performance of numerical algorithms


Zbl 0394.65015


Full Text: DOI


[1] DOI: 10.1080/00207168708803576 · Zbl 0656.65050
[2] DOI: 10.1016/j.amc.2005.04.043 · Zbl 1090.65053
[3] DOI: 10.1098/rstl.1694.0029
[4] DOI: 10.1007/BF01396176 · Zbl 0361.65041
[5] DOI: 10.1016/j.cam.2004.07.027 · Zbl 1063.65037
[6] DOI: 10.1090/S0025-5718-66-99924-8
[7] Jarratt P., Comput. J. 8 pp 398– (1966)
[8] DOI: 10.1137/0710072 · Zbl 0266.65040
[9] Murakami T., J. Inform. Process 1 pp 138– (1978)
[10] Neta B., Numerical Methods for the Solution of Equations (1983) · Zbl 0514.65029
[11] B. Neta and A.N. Johnson,High order nonlinear solver for multiple roots, Computers and Mathematics with Applications, accepted for publication, doi:10.1016/j.camwa.2007.09.001
[12] Ostrowski A. M., Solutions of Equations and System of Equations (1960) · Zbl 0115.11201
[13] DOI: 10.1007/BF02165226 · Zbl 0163.38702
[14] Redfern D., The Maple Handbook (1994) · Zbl 0820.68002
[15] DOI: 10.1016/j.amc.2004.10.040 · Zbl 1084.65054
[16] DOI: 10.1080/00207160500113306 · Zbl 1094.65048
[17] DOI: 10.1007/BF01444024 · JFM 02.0042.02
[18] Traub J. F., Iterative Methods for the Solution of Equations (1964) · Zbl 0121.11204
[19] DOI: 10.1080/00207168208803346 · Zbl 0499.65026
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.