Department of Applied Mathematics Academic Program Review, Self Study / June 2010

The Department of Applied Mathematics has a multi-faceted mission to provide an exceptional mathematical education focused on the unique needs of NPS students, to conduct relevant research, and to provide service to the broader community. A strong and vibrant Department of Applied Mathematics is essential to the university's goal of becoming a premiere research university. Because research in mathematics often impacts science and engineering in surprising ways, the department encourages mathematical explorations in a broad range of areas in applied mathematics with specific thrust areas that support the mission of the school

A family of optimal fourth-order methods for multiple roots of nonlinear equations

[EN] Newton-Raphson method has always remained as the widely used method for finding simple and multiple roots of nonlinear equations. In the past years, many new methods have been introduced for finding multiple zeros that involve the use of weight function in the second step, thereby, increasing the order of convergence and giving a flexibility to generate a family of methods satisfying some underlying conditions. However, in almost all the schemes developed over the past, the usual way is to use Newton-type method at the first step. In this paper, we present a new two-step optimal fourth-order family of methods for multiple roots (m > 1). The proposed iterative family has the flexibility of choice at both steps. The development of the scheme is based on using weight functions. The first step can not only recapture Newton's method for multiple roots as special case but is also capable of defining new choices of first step. A stability analysis of some particular cases is also given to explain the dynamical behavior of the new methods around the multiple roots and decide the best values of the free parameters involved. Finally, we compare our methods with the existing schemes of the same order with a real life application as well as standard test problems. From the numerical results, we find that our methods can be considered as a better alternative for the existing procedures of same order.This research was partially supported by Ministerio de Economía y Competitividad MTM2014¿52016¿C2¿2¿P, Generalitat ValencianaPROMETEO/2016/089, and Schlumberger Foundation¿Faculty for Future ProgramZafar, F.; Cordero Barbero, A.; Torregrosa Sánchez, JR. (2020). A family of optimal fourth-order methods for multiple roots of nonlinear equations. Mathematical Methods in the Applied Sciences. 43(14):7869-7884. https://doi.org/10.1002/mma.5384S786978844314Neta, B., & Johnson, A. N. (2008). High-order nonlinear solver for multiple roots. Computers & Mathematics with Applications, 55(9), 2012-2017. doi:10.1016/j.camwa.2007.09.001Neta, B. (2010). Extension of Murakami’s high-order non-linear solver to multiple roots. International Journal of Computer Mathematics, 87(5), 1023-1031. doi:10.1080/00207160802272263Li, S. G., Cheng, L. Z., & Neta, B. (2010). Some fourth-order nonlinear solvers with closed formulae for multiple roots. Computers & Mathematics with Applications, 59(1), 126-135. doi:10.1016/j.camwa.2009.08.066Zhou, X., Chen, X., & Song, Y. (2011). Constructing higher-order methods for obtaining the multiple roots of nonlinear equations. Journal of Computational and Applied Mathematics, 235(14), 4199-4206. doi:10.1016/j.cam.2011.03.014Sharifi, M., Babajee, D. K. R., & Soleymani, F. (2012). Finding the solution of nonlinear equations by a class of optimal methods. Computers & Mathematics with Applications, 63(4), 764-774. doi:10.1016/j.camwa.2011.11.040Soleymani, F., Babajee, D. K. R., & Lotfi, T. (2013). On a numerical technique for finding multiple zeros and its dynamic. Journal of the Egyptian Mathematical Society, 21(3), 346-353. doi:10.1016/j.joems.2013.03.011Soleymani, F., & Babajee, D. K. R. (2013). Computing multiple zeros using a class of quartically convergent methods. Alexandria Engineering Journal, 52(3), 531-541. doi:10.1016/j.aej.2013.05.001Hueso, J. L., Martínez, E., & Teruel, C. (2014). Determination of multiple roots of nonlinear equations and applications. Journal of Mathematical Chemistry, 53(3), 880-892. doi:10.1007/s10910-014-0460-8Behl, R., Cordero, A., Motsa, S. S., Torregrosa, J. R., & Kanwar, V. (2015). An optimal fourth-order family of methods for multiple roots and its dynamics. Numerical Algorithms, 71(4), 775-796. doi:10.1007/s11075-015-0023-5Behl, R., Cordero, A., Motsa, S. S., & Torregrosa, J. R. (2017). Multiplicity anomalies of an optimal fourth-order class of iterative methods for solving nonlinear equations. Nonlinear Dynamics, 91(1), 81-112. doi:10.1007/s11071-017-3858-6Blanchard, P. (1984). Complex analytic dynamics on the Riemann sphere. Bulletin of the American Mathematical Society, 11(1), 85-141. doi:10.1090/s0273-0979-1984-15240-6Beardon, A. F. (Ed.). (1991). Iteration of Rational Functions. Graduate Texts in Mathematics. doi:10.1007/978-1-4612-4422-6Chicharro, F. I., Cordero, A., & Torregrosa, J. R. (2013). Drawing Dynamical and Parameters Planes of Iterative Families and Methods. The Scientific World Journal, 2013, 1-11. doi:10.1155/2013/780153Jay, L. O. (2001). Bit Numerical Mathematics, 41(2), 422-429. doi:10.1023/a:102190282570

A family of optimal fourth‐order methods for multiple roots of nonlinear equations

[EN] Newton-Raphson method has always remained as the widely used method for finding simple and multiple roots of nonlinear equations. In the past years, many new methods have been introduced for finding multiple zeros that involve the use of weight function in the second step, thereby, increasing the order of convergence and giving a flexibility to generate a family of methods satisfying some underlying conditions. However, in almost all the schemes developed over the past, the usual way is to use Newton-type method at the first step. In this paper, we present a new two-step optimal fourth-order family of methods for multiple roots (m > 1). The proposed iterative family has the flexibility of choice at both steps. The development of the scheme is based on using weight functions. The first step can not only recapture Newton's method for multiple roots as special case but is also capable of defining new choices of first step. A stability analysis of some particular cases is also given to explain the dynamical behavior of the new methods around the multiple roots and decide the best values of the free parameters involved. Finally, we compare our methods with the existing schemes of the same order with a real life application as well as standard test problems. From the numerical results, we find that our methods can be considered as a better alternative for the existing procedures of same order.This research was partially supported by Ministerio de Economía y Competitividad MTM2014¿52016¿C2¿2¿P, Generalitat ValencianaPROMETEO/2016/089, and Schlumberger Foundation¿Faculty for Future ProgramZafar, F.; Cordero Barbero, A.; Torregrosa Sánchez, JR. (2020). A family of optimal fourth-order methods for multiple roots of nonlinear equations. Mathematical Methods in the Applied Sciences. 43(14):7869-7884. https://doi.org/10.1002/mma.5384786978844314Neta, B., & Johnson, A. N. (2008). High-order nonlinear solver for multiple roots. Computers & Mathematics with Applications, 55(9), 2012-2017. doi:10.1016/j.camwa.2007.09.001Neta, B. (2010). Extension of Murakami’s high-order non-linear solver to multiple roots. International Journal of Computer Mathematics, 87(5), 1023-1031. doi:10.1080/00207160802272263Li, S. G., Cheng, L. Z., & Neta, B. (2010). Some fourth-order nonlinear solvers with closed formulae for multiple roots. Computers & Mathematics with Applications, 59(1), 126-135. doi:10.1016/j.camwa.2009.08.066Zhou, X., Chen, X., & Song, Y. (2011). Constructing higher-order methods for obtaining the multiple roots of nonlinear equations. Journal of Computational and Applied Mathematics, 235(14), 4199-4206. doi:10.1016/j.cam.2011.03.014Sharifi, M., Babajee, D. K. R., & Soleymani, F. (2012). Finding the solution of nonlinear equations by a class of optimal methods. Computers & Mathematics with Applications, 63(4), 764-774. doi:10.1016/j.camwa.2011.11.040Soleymani, F., Babajee, D. K. R., & Lotfi, T. (2013). On a numerical technique for finding multiple zeros and its dynamic. Journal of the Egyptian Mathematical Society, 21(3), 346-353. doi:10.1016/j.joems.2013.03.011Soleymani, F., & Babajee, D. K. R. (2013). Computing multiple zeros using a class of quartically convergent methods. Alexandria Engineering Journal, 52(3), 531-541. doi:10.1016/j.aej.2013.05.001Hueso, J. L., Martínez, E., & Teruel, C. (2014). Determination of multiple roots of nonlinear equations and applications. Journal of Mathematical Chemistry, 53(3), 880-892. doi:10.1007/s10910-014-0460-8Behl, R., Cordero, A., Motsa, S. S., Torregrosa, J. R., & Kanwar, V. (2015). An optimal fourth-order family of methods for multiple roots and its dynamics. Numerical Algorithms, 71(4), 775-796. doi:10.1007/s11075-015-0023-5Behl, R., Cordero, A., Motsa, S. S., & Torregrosa, J. R. (2017). Multiplicity anomalies of an optimal fourth-order class of iterative methods for solving nonlinear equations. Nonlinear Dynamics, 91(1), 81-112. doi:10.1007/s11071-017-3858-6Blanchard, P. (1984). Complex analytic dynamics on the Riemann sphere. Bulletin of the American Mathematical Society, 11(1), 85-141. doi:10.1090/s0273-0979-1984-15240-6Beardon, A. F. (Ed.). (1991). Iteration of Rational Functions. Graduate Texts in Mathematics. doi:10.1007/978-1-4612-4422-6Chicharro, F. I., Cordero, A., & Torregrosa, J. R. (2013). Drawing Dynamical and Parameters Planes of Iterative Families and Methods. The Scientific World Journal, 2013, 1-11. doi:10.1155/2013/780153Jay, L. O. (2001). Bit Numerical Mathematics, 41(2), 422-429. doi:10.1023/a:102190282570