A study on the local convergence and the dynamics of Chebyshev-Halley-type methods free from second derivative. (English) Zbl 1335.65047
A local convergence analysis for at least five-order variants of Chebyshev-Halley-type methods in order to approximate a solution of a nonlinear equation is conducted. Conditions which contain up to the first derivative are invented. The dynamical analyses of these methods are also given. Results of numerical experiments are exposed.
Reviewer: Anton Iliev (Plovdiv)
MSC:
| 65H05 | Numerical computation of solutions to single equations |
Keywords:
Chebyshev-Halley-type methods; local convergence; order of convergence; dynamics; numerical experimentReferences:
| [1] | Amat, S., Busquier, S.: Plaza, Dynamics of the King and Jarratt iterations. Aequationes Math. 69(3), 212-223 (2005) · Zbl 1068.30019 · doi:10.1007/s00010-004-2733-y |
| [2] | Amat, S., Busquier, S.: Plaza, Chaotic dynamics of a third-order Newton-type method. J. Math. Anal. Appl. 366(1), 24-32 (2010) · Zbl 1187.65050 · doi:10.1016/j.jmaa.2010.01.047 |
| [3] | Amat, M.A., Hernández, N., Romero, A.: A modified Chebyshev’s iterative method with at least sixth order of convergence. Appl. Math. Comput. 206(1), 164-174 (2008) · Zbl 1157.65369 · doi:10.1016/j.amc.2008.08.050 |
| [4] | Argyros, I.K.: Convergence and Application of Newton-type Iterations. Springer (2008) · Zbl 1153.65057 |
| [5] | Argyros, I.K, Hilout, S.: Numerical methods in Nonlinear Analysis. World Scientific Publ. Comp, New Jersey (2013) · Zbl 1262.65061 |
| [6] | Behl, R.: Development and analysis of some new iterative methods for numerical solutions of nonlinear equations (PhD Thesis). Punjab University (2013) · Zbl 1068.30019 |
| [7] | Bruns, D.D., Bailey, J.E.: Nonlinear feedback control for operating a nonisothermal CSTR near an unstable steady state. Chem. Eng. Sci 32, 257-264 (1977) · doi:10.1016/0009-2509(77)80203-0 |
| [8] | Candela, V., Marquina, A.: Recurrence relations for rational cubic methods I: The Halley method. Computing 169-184, 44 (1990) · Zbl 0701.65043 |
| [9] | Candela, V., Marquina, A.: Recurrence relations for rational cubic methods II: The Chebyshev method. Computing 45(4), 355-367 (1990) · Zbl 0714.65061 · doi:10.1007/BF02238803 |
| [10] | Chicharro, F., Cordero, A., Torregrosa, J.R.: Drawing dynamical and parameters planes of iterative families and methods. The Scientific World Journal Volume Article ID 780153 (2013) · Zbl 1305.70018 |
| [11] | Chun, C.: Some improvements of Jarratt’s method with sixth-order convergence. Appl Math. Comput 190(2), 1432-1437 (1990) · Zbl 1122.65329 · doi:10.1016/j.amc.2007.02.023 |
| [12] | Cordero, A., García-Maimó, J., Torregrosa, J.R., Vassileva, M.P., Vindel, P.: Chaos in King’s iterative family. Appl. Math. Lett. 26, 842-848 (2013) · Zbl 1370.37155 |
| [13] | Cordero, A., Torregrosa, J.R., Vindel, P.: Dynamics of a family of Chebyshev-Halley type methods. Appl. Math. Comput. 219, 8568-8583 (2013) · Zbl 1288.65065 · doi:10.1016/j.amc.2013.02.042 |
| [14] | Cordero, A., Torregrosa, J.R.: Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686-698 (2007) · Zbl 1122.65350 · doi:10.1016/j.amc.2007.01.062 |
| [15] | Ezquerro, J. A., Hernández, M.A.: Recurrence relations for Chebyshev-type methods. Appl. Math. Optim. 41(2), 227-236 (2000) · Zbl 0952.47050 · doi:10.1007/s002459911012 |
| [16] | Ezquerro, J. A., Hernández, M.A.: New iterations of R-order four with reduced computational cost. BIT Numer. Math. 49, 325-342 (2009) · Zbl 1170.65038 · doi:10.1007/s10543-009-0226-z |
| [17] | Ezquerro, J. A., Hernández, M.A.: On the R-order of the Halley method. J. Math. Anal. Appl. 303, 591-601 (2005) · Zbl 1079.65064 · doi:10.1016/j.jmaa.2004.08.057 |
| [18] | Gutiérrez, J.M., Hernández, M.A.: Recurrence relations for the super-Halley method. Comput. Math. Applic. 36(7), 1-8 (1998) · Zbl 0933.65063 · doi:10.1016/S0898-1221(98)00168-0 |
| [19] | Ganesh, M., Joshi, M.C.: Numerical solvability of Hammerstein integral equations of mixed type. IMA J. Numer. Anal. 11, 21-31 (1991) · Zbl 0719.65093 · doi:10.1093/imanum/11.1.21 |
| [20] | Hernández, M.A.: Chebyshev’s approximation algorithms and applications. Comput. Math. Applic. 41(3-4), 433-455 (2001) · Zbl 0985.65058 · doi:10.1016/S0898-1221(00)00286-8 |
| [21] | Hernández, M.A., Salanova, M.A.: Sufficient conditions for semilocal convergence of a fourth order multipoint iterative method for solving equations in Banach spaces. Southwest J. Pure Appl. Math. 1, 29-40 (1999) · Zbl 0940.65064 |
| [22] | Jarratt, P.: Some fourth order multipoint methods for solving equations. Math. Comput. 20(95), 434-437 (1966) · Zbl 0229.65049 · doi:10.1090/S0025-5718-66-99924-8 |
| [23] | Kou, J., Li, Y.: An improvement of the Jarratt method. Appl. Math. Comput. 189, 1816-1821 (2007) · Zbl 1122.65338 · doi:10.1016/j.amc.2006.12.062 |
| [24] | Li, D., Liu, P., Kou, J.: An improvement of the Chebyshev-Halley methods free from second derivative. Appl. Math. Comput. 235, 221-225 (2014) · Zbl 1334.65086 · doi:10.1016/j.amc.2014.02.083 |
| [25] | Magreñán, Á. A.: Different anomalies in a Jarratt family of iterative root-finding methods. Appl. Math. Comput. 233, 29-38 (2014) · Zbl 1334.65083 · doi:10.1016/j.amc.2014.01.037 |
| [26] | Magreñán, Á.A.: A new tool to study real dynamics: The convergence plane. Appl. Math. Comput. 248, 215-224 (2014) · Zbl 1338.65277 · doi:10.1016/j.amc.2014.09.061 |
| [27] | Parhi, S.K., Gupta, D.K.: Recurrence relations for a Newton-like method in Banach spaces. J. Comput. Appl. Math. 206(2), 873-887 (2007) · Zbl 1119.47063 · doi:10.1016/j.cam.2006.08.027 |
| [28] | Rall, L.B.: Computational solution of nonlinear operator equations. In: Robert E. Krieger (ed.). New York (1979) · Zbl 0476.65033 |
| [29] | Ren, H., Wu, Q., Bi, W.: New variants of Jarratt method with sixth-order convergence. Numer. Algorithm. 52(4), 585-603 (2009) · Zbl 1187.65052 · doi:10.1007/s11075-009-9302-3 |
| [30] | Rheinboldt, W.C.: An adaptive continuation process for solving systems of nonlinear equations. Pol. Acad. Sci., Banach. Ctr. Publ. 3, 129-142 (1978) · Zbl 0378.65029 |
| [31] | Traub, J.F.: Iterative methods for the solution of equations. Prentice- Hall Series in Automatic Computation Englewood Cliffs, N. J. (1964) · Zbl 1119.47063 |
| [32] | Wang, X., Kou, J., Gu, C.: Semilocal convergence of a sixth-order Jarratt method in Banach spaces. Numer. Algorithm. 57, 441-456 (2011) · Zbl 1234.65030 · doi:10.1007/s11075-010-9438-1 |
| [33] | Kou, J., Wang, X.: Semilocal convergence of a modified multi-point Jarratt method in Banach spaces under general continuity conditions. Numer. Algorithm. 60, 369-390 (2012) · Zbl 1250.65074 · doi:10.1007/s11075-011-9519-9 |
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