H?lder条件下求解重根的Traub算法的收敛半径
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摘要
2010年任宏民和Argyros~([1])对求解重根的牛顿法的收敛半径进行了分析,首次给出了重根迭代局部收敛性的分析方法.2011年毕惟红等人基于该思路计算了Halley算法的收敛半径,2018年JoséL等人给出了Osada算法的收敛半径.而对于非线性方程重根迭代算法中,Traub算法至今无人研究,本文将基于任红民和Argyros给出的基本思路计算Traub算法的收敛半径,并通过具体实例进行分析.
In 2010, Ren Hongmin and Argeryros [1] analyzed the convergence radius of Newton's method for solving multiple roots, and gave the local convergence analysis method for iteration of multiple roots for the first time. In 2011, Bi Weihong and others calculated the convergence radius of Halley algorithm based on this idea. In 2018, Jos L and others gave the convergence radius of Osada algorithm. Traub algorithm has not been studied in the iteration algorithm of multiple roots for non-linear equations. This paper will calculate the convergence radius of Traub algorithm based on Ren Hongmin and Arguyros' basic ideas, and analyze it through specific examples.
In 2010, Ren Hongmin and Argeryros [1] analyzed the convergence radius of Newton's method for solving multiple roots, and gave the local convergence analysis method for iteration of multiple roots for the first time. In 2011, Bi Weihong and others calculated the convergence radius of Halley algorithm based on this idea. In 2018, Jos L and others gave the convergence radius of Osada algorithm. Traub algorithm has not been studied in the iteration algorithm of multiple roots for non-linear equations. This paper will calculate the convergence radius of Traub algorithm based on Ren Hongmin and Arguyros' basic ideas, and analyze it through specific examples.
引文
[1]Ren H, Argyro. I. Convergence radius of the modified Newton method for multiple zeros under Holder continuous derivative[J]. Appl. Math. Comput, 2010, 217:612-621.
[2]Bi W, Ren H, Wu Q. Convergence of the modified Halley's method for multiple zeros under Holder continuous derivative[J]. Numer. Algor, 2011, 58:497-512.
[3]Jose L H, Eulalia M, D.K.G, Fabricio C. Convergence radius of Osada's method under H(O|¨)lder continuous condition[J]. Appl.Math. Coput, 2018, 321:689-699.
[4]Kung H, Traub J. Optimal order of one-point and multipoint iteration[J]. J. ACM, 1974, 21:643-651.
[5]Victory H, Neta B. A higher order method for multiple zeros of nonlinear functions[J]. Int.J. Comput.Math, 1983, 12:329-335.
[6]Zhou X, Chen X, Song Y. Families of third and fourth order methods for multiple roots of nonlinear equations[J]. Appl. Math. Comput, 2013, 219:6030-6038.
[7]Liu B, Zhou X. A New Family of Optimal Fourth-order Methods for Multiple Roots of Nonlinear Equations[J]. Nonlinear Anal-Model, 2013, 18:143-152.
[8]Traub J.Iterative Methods for the Solution of Equations[M]. New York:Chelsea Publishing Company, 1977.
[9]Osada N. An optimal multiple root-finding method of order three[J]. Comput. Appl. Math, 1994, 51:131-133.
[10]Hansen E, Patrick M. A family of root finding methods[J]. Numer. Math, 1977, 27:257-269.
[11]Ren H, Argyros I. Convergence radius of the modified Newton method for multiple zeros under Holder continuous derivative[J]. Appl. Math. Comput, 2010, 217:612-621.
[12]Traub J,Wozniakowski H. Convergence and complexity of Newton iteration for operator equation[J]. J.ACM, 1979, 26:250-258.
[13]Argyros I. On the convergence and application of Newton's method under weak H(o|¨)lder Continuity assumptions[J]. Int. J. Comput. Math, 2003, 80:767-780.
[14]Zhou X, Song Y. Convergence Radius of Osada's Method for Multiple Roots under Holder and Center-Holder Continuous Conditions[J]. ICNAAM 2011, AIP Conference Proceedings, 2011, 1389:1836-1839.
[15]Bi W, Ren H, Wu Q. Convergence of the modified Halley's method for multiple zeros under Holder continuous derivative[J]. Numer. Algor, 2011, 58:497-512.
[16]Zhou X, Chen X, Song Y. On the convergence radius of the modified Newton's method for multiple roots under the Center-Holder condition[J]. Numer. Algor, 2014, 65:221-232.
[17]Zhou X, Song Y. Convergence radius of Osada's method under center-Holder continuous condition[J].Appl Math. Comput, 2014, 243:809-816.
[18]Argyros I K, Y. Cho, S. Hilout. On the semilocal convergence of the Halley method using recurrent functions[J]. J. Appl. Math. Comput, 2011, 37:221-246.
[19]Ren H, Argyros I. On the semi-local convergence of Halley's method under a center-Lipschitz condition on the second Frechet derivative[J]. Appl. Math. Comput, 2012, 218(23):11488-11495.
[20]Argyros I. The convergence of a Halley-Chebysheff-type method under Newton Kantorovich hypotheses[J].Appl. Math. Lett, 1993, 6(5):71-74.
[2]Bi W, Ren H, Wu Q. Convergence of the modified Halley's method for multiple zeros under Holder continuous derivative[J]. Numer. Algor, 2011, 58:497-512.
[3]Jose L H, Eulalia M, D.K.G, Fabricio C. Convergence radius of Osada's method under H(O|¨)lder continuous condition[J]. Appl.Math. Coput, 2018, 321:689-699.
[4]Kung H, Traub J. Optimal order of one-point and multipoint iteration[J]. J. ACM, 1974, 21:643-651.
[5]Victory H, Neta B. A higher order method for multiple zeros of nonlinear functions[J]. Int.J. Comput.Math, 1983, 12:329-335.
[6]Zhou X, Chen X, Song Y. Families of third and fourth order methods for multiple roots of nonlinear equations[J]. Appl. Math. Comput, 2013, 219:6030-6038.
[7]Liu B, Zhou X. A New Family of Optimal Fourth-order Methods for Multiple Roots of Nonlinear Equations[J]. Nonlinear Anal-Model, 2013, 18:143-152.
[8]Traub J.Iterative Methods for the Solution of Equations[M]. New York:Chelsea Publishing Company, 1977.
[9]Osada N. An optimal multiple root-finding method of order three[J]. Comput. Appl. Math, 1994, 51:131-133.
[10]Hansen E, Patrick M. A family of root finding methods[J]. Numer. Math, 1977, 27:257-269.
[11]Ren H, Argyros I. Convergence radius of the modified Newton method for multiple zeros under Holder continuous derivative[J]. Appl. Math. Comput, 2010, 217:612-621.
[12]Traub J,Wozniakowski H. Convergence and complexity of Newton iteration for operator equation[J]. J.ACM, 1979, 26:250-258.
[13]Argyros I. On the convergence and application of Newton's method under weak H(o|¨)lder Continuity assumptions[J]. Int. J. Comput. Math, 2003, 80:767-780.
[14]Zhou X, Song Y. Convergence Radius of Osada's Method for Multiple Roots under Holder and Center-Holder Continuous Conditions[J]. ICNAAM 2011, AIP Conference Proceedings, 2011, 1389:1836-1839.
[15]Bi W, Ren H, Wu Q. Convergence of the modified Halley's method for multiple zeros under Holder continuous derivative[J]. Numer. Algor, 2011, 58:497-512.
[16]Zhou X, Chen X, Song Y. On the convergence radius of the modified Newton's method for multiple roots under the Center-Holder condition[J]. Numer. Algor, 2014, 65:221-232.
[17]Zhou X, Song Y. Convergence radius of Osada's method under center-Holder continuous condition[J].Appl Math. Comput, 2014, 243:809-816.
[18]Argyros I K, Y. Cho, S. Hilout. On the semilocal convergence of the Halley method using recurrent functions[J]. J. Appl. Math. Comput, 2011, 37:221-246.
[19]Ren H, Argyros I. On the semi-local convergence of Halley's method under a center-Lipschitz condition on the second Frechet derivative[J]. Appl. Math. Comput, 2012, 218(23):11488-11495.
[20]Argyros I. The convergence of a Halley-Chebysheff-type method under Newton Kantorovich hypotheses[J].Appl. Math. Lett, 1993, 6(5):71-74.