A Higher Order Chebyshev-Halley-Type Family of Iterative Methods for Multiple Roots
Abstract
:1. Introduction
2. Construction of the Higher Order Scheme
3. Numerical Experiments
4. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
- Traub, J.F. Iterative Methods for the Solution of Equations; Prentice-Hall: Englewood Cliffs, NJ, USA, 1964. [Google Scholar]
- Gutiérrez, J.M.; Hernández, M.A. A family of Chebyshev–Halley type methods in Banach spaces. Bull. Aust. Math. Soc. 1997, 55, 113–130. [Google Scholar] [CrossRef]
- Kanwar, V.; Singh, S.; Bakshi, S. Simple geometric constructions of quadratically and cubically convergent iterative functions to solve nonlinear equations. Numer. Algorithms 2008, 47, 95–107. [Google Scholar] [CrossRef]
- Potra, F.A.; Pták, V. Nondiscrete Induction and Iterative Processes; Research Notes in Mathematics; Pitman: Boston, MA, USA, 1984; Volume 103. [Google Scholar]
- Sharma, J.R. A composite third order Newton-Steffensen method for solving nonlinear equations. Appl. Math. Comput. 2005, 169, 242–246. [Google Scholar]
- Argyros, I.K.; Ezquerro, J.A.; Gutiérrez, J.M.; Hernández, M.A.; Hilout, S. On the semilocal convergence of efficient Chebyshev-Secant-type methods. J. Comput. Appl. Math. 2011, 235, 3195–3206. [Google Scholar] [CrossRef]
- Xiaojian, Z. Modified Chebyshev–Halley methods free from second derivative. Appl. Math. Comput. 2008, 203, 824–827. [Google Scholar] [CrossRef]
- Ostrowski, A.M. Solutions of Equations and System of Equations; Academic Press: New York, NY, USA, 1960. [Google Scholar]
- Amat, S.; Hernández, M.A.; Romero, N. A modified Chebyshev’s iterative method with at least sixth order of convergence. Appl. Math. Comput. 2008, 206, 164–174. [Google Scholar] [CrossRef]
- Kou, J.; Li, Y. Modified Chebyshev–Halley method with sixth-order convergence. Appl. Math. Comput. 2007, 188, 681–685. [Google Scholar] [CrossRef]
- Li, D.; Liu, P.; Kou, J. An improvement of Chebyshev–Halley methods free from second derivative. Appl. Math. Comput. 2014, 235, 221–225. [Google Scholar] [CrossRef]
- Sharma, J.R. Improved Chebyshev–Halley methods with sixth and eighth order convergence. Appl. Math. Comput. 2015, 256, 119–124. [Google Scholar] [CrossRef]
- Neta, B. Extension of Murakami’s high-order non-linear solver to multiple roots. Int. J. Comput. Math. 2010, 87, 1023–1031. [Google Scholar] [CrossRef]
- Zhou, X.; Chen, X.; Song, Y. Constructing higher-order methods for obtaining the multiple roots of nonlinear equations. J. Comput. Math. Appl. 2011, 235, 4199–4206. [Google Scholar] [CrossRef] [Green Version]
- Hueso, J.L.; Martínez, E.; Teruel, C. Determination of multiple roots of nonlinear equations and applications. J. Math. Chem. 2015, 53, 880–892. [Google Scholar] [CrossRef]
- Behl, R.; Cordero, A.; Motsa, S.S.; Torregrosa, J.R. On developing fourth-order optimal families of methods for multiple roots and their dynamics. Appl. Math. Comput. 2015, 265, 520–532. [Google Scholar] [CrossRef]
- Behl, R.; Cordero, A.; Motsa, S.S.; Torregrosa, J.R.; Kanwar, V. An optimal fourth-order family of methods for multiple roots and its dynamics. Numer. Algorithms 2016, 71, 775–796. [Google Scholar] [CrossRef]
- Geum, Y.H.; Kim, Y.I.; Neta, B. A class of two-point sixth-order multiple-zero finders of modified double-Newton type and their dynamics. Appl. Math. Comput. 2015, 270, 387–400. [Google Scholar] [CrossRef] [Green Version]
- Geum, Y.H.; Kim, Y.I.; Neta, B. A sixth-order family of three-point modified Newton-like multiple-root finders and the dynamics behind their extraneous fixed points. Appl. Math. Comput. 2016, 283, 120–140. [Google Scholar] [CrossRef] [Green Version]
- Behl, R.; Cordero, A.; Motsa, S.S.; Torregrosa, J.R. An eighth-order family of optimal multiple root finders and its dynamics. Numer. Algorithms 2018, 77, 1249–1272. [Google Scholar] [CrossRef]
- Zafar, F.; Cordero, A.; Motsa, S.S.; Torregrosa, J.R. Optimal iterative methods for finding multiple roots of nonlinear equations using free parameters. J. Math. Chem. 2018, 56, 1884–1901. [Google Scholar] [CrossRef]
- Behl, R.; Alshomrani, A.S.; Motsa, S.S. An optimal scheme for multiple roots of nonlinear equations with eighth-order convergence. J. Math. Chem. 2018, 56, 2069–2084. [Google Scholar] [CrossRef]
- Behl, R.; Zafar, F.; Alshomrani, A.S.; Junjuaz, M.; Yasmin, N. An optimal eighth-order scheme for multiple zeros of univariate functions. Int. J. Comput. Methods 2018, 1843002. [Google Scholar] [CrossRef]
- McNamee, J.M. A comparison of methods for accelerating convergence of Newton’s method for multiple polynomial roots. ACM Signum Newsl. 1998, 33, 17–22. [Google Scholar] [CrossRef]
- Cordero, A.; Torregrosa, J.R. Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 2007, 190, 686–698. [Google Scholar] [CrossRef]
f | n | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
1 | 2.3 (−3) | 8.4 (−4) | 9.3 (−5) | 3.5 (−5) | * | 3.6 (−5) | 1.6 (−4) | 2.3 (−4) | 7.6 (−5) | 3.7 (−5) | |
2 | 2.0 (−16) | 9.0 (−20) | 8.8 (−28) | 2.0 (−37) | * | 1.4 (−29) | 4.2 (−31) | 8.9 (−30) | 2.6 (−34) | 5.0 (−37) | |
3 | 9.7 (−95) | 1.3 (−115) | 6.4 (−166) | 2.5 (−295) | * | 5.4 (−173) | 1.0 (−243) | 5.5 (−233) | 5.4 (−270) | 5.7 (−292) | |
5.9997 | 6.0000 | 6.0001 | 8.0000 | * | 6.0000 | 8.0000 | 8.0000 | 8.0000 | 8.0000 | ||
1 | 1.3 (−3) | 8.2 (−4) | 4.0 (−3) | 3.5 (−4) | 9.5 (−4) | 3.9 (−4) | 3.9 (−4) | 4.1 (−3) | 2.7 (−4) | 2.6 (−4) | |
2 | 2.5 (−10) | 4.2 (−12) | 6.4 (−16) | 8.7 (−18) | 2.7 (−11) | 1.0 (−14) | 5.2 (−17) | 9.8 (−17) | 1.1 (−18) | 1.4 (−19) | |
3 | 2.0 (−50) | 8.7 (−62) | 6.5 (−87) | 1.5 (−126) | 2.0 (−56) | 3.9 (−78) | 5.9 (−120) | 1.2 (−117) | 6.3 (−134) | 1.0 (−141) | |
5.9757 | 5.9928 | 6.0214 | 7.9963 | 5.9836 | 5.9975 | 7.9945 | 7.9941 | 7.9971 | 8.0026 | ||
1 | 9.1 (−5) | 3.6 (−5) | 8.0 (−6) | 6.0 (−6) | 8.5 (−5) | 4.8 (−5) | 4.9 (−6) | 5.2 (−6) | 2.0 (−6) | 1.8 (−6) | |
2 | 1.8 (−28) | 1.4 (−31) | 9.8 (−38) | 2.0 (−47) | 1.0 (−28) | 5.0 (−31) | 6.0 (−48) | 1.0 (−47) | 1.5 (−51) | 2.8 (−52) | |
3 | 1.2 (−170) | 4.4 (−190) | 3.3 (−229) | 2.5 (−379) | 3.1 (−172) | 5.8 (−187) | 2.7 (−383) | 2.3 (−381) | 1.4 (−412) | 1.3 (−418) | |
6.0000 | 6.0000 | 6.0000 | 8.0000 | 6.0000 | 6.0000 | 8.0000 | 8.0000 | 8.0000 | 8.0000 | ||
1 | 2.4 (−5) | 7.1 (−6) | 4.2 (−7) | 1.4 (−7) | 1.8 (−5) | 2.0 (−7) | 4.8 (−7) | 6.5 (−7) | 1.9 (−7) | 6.3 (−8) | |
2 | 1.5 (−26) | 1.7 (−30) | 3.9 (−40) | 6.7 (−54) | 1.1 (−26) | 1.8 (−41) | 5.7 (−49) | 8.4 (−48) | 8.0 (−53) | 4.2 (−57) | |
3 | 7.5 (−154) | 3.2 (−178) | 2.6 (−438) | 1.7 (−424) | 6.6 (−154) | 1.0 (−245) | 2.2 (−384) | 6.6 (−375) | 9.6 (−416) | 5.9 (−169) | |
6.0000 | 6.0000 | 6.0000 | 8.0000 | 6.0000 | 6.0000 | 8.0000 | 8.0000 | 8.0000 | 2.2745 |
f | n | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
1 | 2.7 | 1.0 | 1.1 (−1) | 4.2 (−2) | * | 4.4 (−2) | 1.9 (−1) | 2.7 (−1) | 9.2 (−2) | 4.4 (−2) | |
2 | 2.4 (−13) | 1.1 (−16) | 1.1 (−24) | 2.4 (−34) | * | 1.7 (−26) | 5.1 (−28) | 1.1 (−26) | 3.2 (−31) | 6.0 (−34) | |
3 | 1.2 (−91) | 1.6 (−112) | 7.8 (−163) | 3.0 (−292) | * | 5.4 (−173) | 1.2 (−240) | 6.7 (−230) | 6.5 (−267) | 7.0 (−289) | |
1 | 5.0 (−8) | 2.1 (−8) | 4.8 (−9) | 3.6 (−9) | 2.8 (−8) | 4.6 (−9) | 4.6 (−9) | 5.1 (−9) | 2.3 (−9) | 2.0 (−9) | |
2 | 1.8 (−21) | 5.3 (−25) | 1.2 (−32) | 2.3 (−36) | 2.2 (−23) | 3.2 (−30) | 8.0 (−35) | 2.9 (−34) | 3.4 (−38) | 5.9 (−40) | |
3 | 1.2 (−101) | 2.2 (−124) | 1.3 (−174) | 6.9 (−254) | 1.2 (−113) | 4.6 (−157) | 1.1 (−240) | 4.3 (−236) | 1.2 (−268) | 3.1 (−284) | |
1 | 4.9 (−8) | 3.1 (−9) | 3.1 (−11) | 1.4 (−11) | 4.1 (−8) | 7.4 (−9) | 7.8 (−12) | 9.1 (−12) | 5.2 (−13) | 3.6 (−13) | |
2 | 3.9 (−79) | 1.8 (−88) | 6.1 (−107) | 4.9 (−136) | 7.1 (−80) | 8.0 (−87) | 1.4 (−137) | 6.9 (−137) | 2.1 (−148) | 1.5 (−150) | |
3 | 1.0 (−505) | 5.6 (−564) | 2.4 (−681) | 1.1 (−1131) | 1.9 (−510) | 1.2 (−554) | 1.3 (−1143) | 7.5 (−1138) | 1.9 (−1231) | 1.3 (−1249) | |
1 | 1.2 (−207) | 2.7 (−234) | 1.1 (−295) | 3.3 (−319) | 3.5 (−214) | 1.0 (−311) | 6.6 (−293) | 2.3 (−286) | 1.8 (−313) | 6.2 (−337) | |
2 | 1.9 (−1268) | 2.6 (−1465) | 3.8 (−1947) | 1.6 (−2635) | 1.9 (−1274) | 9.8 (−2014) | 3.4 (−2389) | 9.4 (−2331) | 9.8 (−2582) | 1.1 (−2795) | |
3 | 4.2 (−7633) | 2.3 (−8851) | 7.5 (−11856) | 6.1 (−21166) | 6.0 (−7636) | 7.3 (−12226) | 1.6 (−19159) | 7.1 (−18686) | 8.8 (−20728) | 3.4 (−8388) |
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Behl, R.; Martínez, E.; Cevallos, F.; Alarcón, D. A Higher Order Chebyshev-Halley-Type Family of Iterative Methods for Multiple Roots. Mathematics 2019, 7, 339. https://doi.org/10.3390/math7040339
Behl R, Martínez E, Cevallos F, Alarcón D. A Higher Order Chebyshev-Halley-Type Family of Iterative Methods for Multiple Roots. Mathematics. 2019; 7(4):339. https://doi.org/10.3390/math7040339
Chicago/Turabian StyleBehl, Ramandeep, Eulalia Martínez, Fabricio Cevallos, and Diego Alarcón. 2019. "A Higher Order Chebyshev-Halley-Type Family of Iterative Methods for Multiple Roots" Mathematics 7, no. 4: 339. https://doi.org/10.3390/math7040339