Convergence and dynamics of improved Chebyshev-Secant-type methods for non differentiable operators

[EN] In this paper, the convergence and dynamics of improved Chebyshev-Secant-type iterative methods are studied for solving nonlinear equations in Banach space settings. Their semilocal convergence is established using recurrence relations under weaker continuity conditions on first-order divided differences. Convergence theorems are established for the existence-uniqueness of the solutions. Next, center-Lipschitz condition is defined on the first-order divided differences and its influence on the domain of starting iterates is compared with those corresponding to the domain of Lipschitz conditions. Several numerical examples including Automotive Steering problems and nonlinear mixed Hammerstein-type integral equations are analyzed, and the output results are compared with those obtained by some of similar existing iterative methods. It is found that improved results are obtained for all the numerical examples. Further, the dynamical analysis of the iterative method is carried out. It confirms that the proposed iterative method has better stability properties than its competitors.This research was partially supported by Ministerio de Economia y Competitividad under grant PGC2018-095896-B-C22.Kumar, A.; Gupta, DK.; Martínez Molada, E.; Hueso, JL. (2021). Convergence and dynamics of improved Chebyshev-Secant-type methods for non differentiable operators. Numerical Algorithms. 86(3):1051-1070. https://doi.org/10.1007/s11075-020-00922-9S10511070863Hernández, M.A.: Chebyshev’s approximation algorithms and applications. Comput. Math. Appl. 41(3-4), 433–445 (2001)Ezquerro, J.A., Grau-Sánchez, Miquel, Hernández, M.A.: Solving non-differentiable equations by a new one-point iterative method with memory. J. Complex. 28(1), 48–58 (2012)Ioannis , K.A., Ezquerro, J.A., Gutiérrez, J.M., hernández, M.A., saïd Hilout: On the semilocal convergence of efficient Chebyshev-Secant-type methods. J. Comput. Appl. Math. 235(10), 3195–3206 (2011)Hongmin, R., Ioannis, K.A.: Local convergence of efficient Secant-type methods for solving nonlinear equations. Appl. Math. comput. 218(14), 7655–7664 (2012)Ioannis, Ioannis K.A., Hongmin, R.: On the semilocal convergence of derivative free methods for solving nonlinear equations. J. Numer. Anal. Approx. Theory 41 (1), 3–17 (2012)Hongmin, R., Ioannis, K.A.: On the convergence of King-Werner-type methods of order $1+\sqrt {2}$ free of derivatives. Appl. Math. Comput. 256, 148–159 (2015)Kumar, A., Gupta, D.K., Martínez, E., Sukhjit, S.: Semilocal convergence of a Secant-type method under weak Lipschitz conditions in Banach spaces. J. Comput. Appl. Math. 330, 732–741 (2018)Grau-Sánchez, M., Noguera, M., Gutiérrez, J.M.: Frozen iterative methods using divided differences “à la Schmidt–Schwetlick”. J. Optim. Theory Appl. 160 (3), 931–948 (2014)Louis, B.R.: Computational Solution of Nonlinear Operator Equations. Wiley, New York (1969)Blanchard, P.: The dynamics of Newton’s method. Proc. Symp. Appl. Math. 49, 139–154 (1994)Parisa, B., Cordero, A., Taher, L., Kathayoun, M., Torregrosa, J.R.: Widening basins of attraction of optimal iterative methods. Nonlinear Dynamics 87 (2), 913–938 (2017)Chun, C., Neta, B.: The basins of attraction of Murakami’s fifth order family of methods. Appl. Numer. Math. 110, 14–25 (2016)Magreñán, Á. A.: A new tool to study real dynamics: the convergence plane. Appl. Math. Comput. 248, 215–224 (2014)Ramandeep, B., Cordero, A., Motsa, S.S., Torregrosa, J.R.: Stable high-order iterative methods for solving nonlinear models. Appl. Math. Comput. 303, 70–88 (2017)Pramanik, S.: Kinematic synthesis of a six-member mechanism for automotive steering. Trans Ame Soc. Mech. Eng. J. Mech. Des. 124(4), 642–645 (2002

A dynamic convergence control scheme for the solution of the radial equilibrium equation in through-flow analyses

One of the most frequently encountered numerical problems in scientific analyses is the solution of non-linear equations. Often the analysis of complex phenomena falls beyond the range of applicability of the numerical methods available in the public domain, and demands the design of dedicated algorithms that will approximate, to a specified precision, the mathematical solution of specific problems. These algorithms can be developed from scratch or through the amalgamation of existing techniques. The accurate solution of the full radial equilibrium equation (REE) in streamline curvature (SLC) through-flow analyses presents such a case. This article discusses the development, validation, and application of an 'intelligent' dynamic convergence control (DCC) algorithm for the fast, accurate, and robust numerical solution of the non-linear equations of motion for two-dimensional flow fields. The algorithm was developed to eliminate the large extent of user intervention, usually required by standard numerical methods. The DCC algorithm was integrated into a turbomachinery design and performance simulation software tool and was tested rigorously, particularly at compressor operating regimes traditionally exhibiting convergence difficulties (i.e. far off-design conditions). Typical error histories and comparisons of simulated results against experimental are presented in this article for a particular case study. For all case studies examined, it was found that the algorithm could successfully 'guide' the solution down to the specified error tolerance, at the expense of a slightly slower iteration process (compared to a conventional Newton-Raphson scheme). This hybrid DCC algorithm can also find use in many other engineering and scientific applications that require the robust solution of mathematical problems by numerical instead of analytical means