Semilocal convergence of a modified Newton method for nonlinear systems with singular Jacobians

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Abstract

In this work, we were interested in approximating the solutions of nonlinear systems with singular Jacobian matrices. It is well known that the Newton iterative method loses its quadratic convergence when the Jacobian matrix of the nonlinear operator is singular in a neighborhood of the solution. We introduced a semilocal convergence analysis for a modification of this iterative method that overcomes this drawback. We performed several dynamical experiments to illustrate the effects of singular Jacobian matrices and solved different nonlinear systems, thereby corroborating the theoretical results obtained.

Keywords

nonlinear systems / singularities / modified Newton method / semilocal convergence / dynamical systems

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References

[1] D. Chen, A basic theorem of constructing an iterative formula of the higher order for computing multiple roots of an equation, Math. Numer. Sin., 4 (1982), 445-450. https://doi.org/10.12286/jssx.1982.4.445

[2] N. Osada, An optimal multiple root finding method of order three, J. Comput. Appl. Math., 51 (1994), 131-133. https://doi.org/10.1016/0377-0427(94)00044-1

[3] B. Neta, New third order nonlinear solvers for multiple roots, Appl. Math. Comput., 202 (2008), 162-170. https://doi.org/10.1016/j.amc.2008.01.031

[4] J. L. Hueso, E. Martínez, C. Teruel, Determination of multiple roots of nonlinear equations and applications, J. Math. Chem., 53 (2015), 880-892. https://doi.org/10.1007/s10910-014-0460-8

[5] J. L. Hueso, E. Martínez, J. R. Torregrosa, Modified Newton’s method for systems of nonlinear equations with singular Jacobian, J. Comput. Appl. Math., 224 (2009), 77-83. https://doi.org/10.1016/j.cam.2008.04.013

[6] X. Wu, Note on the improvement of Newton’s method for system of nonlinear equations, Appl. Math. Comput., 189 (2007), 1476-1479. https://doi.org/10.1016/j.amc.2006.12.035

[7] A. Neamvonk, A hybrid safeguarded quasi-Newton method for nonlinear systems with engineering applications, Burapha Sci. J., 31 (2026), 536-551.

[8] Y. Zheng, Y. Yang, M. Cao, The modified Levenberg-Marquardt method with nonmonotone technique, AIMS Math., 11 (2026), 2527-2546. https://doi.org/10.3934/math.2026102

[9] M. Shams, B. Carpentieri, A high-order parallel framework for simultaneous root- finding in nonlinear systems with multiple solutions, Appl. Math., 6 (2026), 43. https://doi.org/10.3390/appliedmath6030043

[10] L. Chen, F. Wang, X. Chen, R. Ye, J. Li, Quantitative behavior and probabilistic convergence of iterative methods for solving multiple roots: A numerical exploration, Mathematics, 14 (2026), 1929. https://doi.org/10.3390/math14111929

[11] J. Kou, Y. Li, X. Wang, Efficient continuation Newton-like method for solving systems of nonlinear equations, Appl. Math. Comput., 174 (2006), 846-853. https://doi.org/10.1016/j.amc.2005.05.031

[12] X. Wu, A new continuation Newton-like method and its deformation, Appl. Math. Comput., 112 (2000), 75-78. https://doi.org/10.1016/S0096-3003(99)00049-1

[13] M. K. A. Ariyaratne, T. G. I. Fernando, S. Weerakoon, Solving systems of nonlinear equations using a modified firefly algorithm (MODFA), Swarm Evol. Comput., 48 (2019), 72-92. https://doi.org/10.1016/j.swevo.2019.03.010

[14] A. Cordero, J. R. Torregrosa, Variants of Newton’s method using fifth-order quadrature formulas, Appl. Math. Comput., 190 (2007), 686-698. https://doi.org/10.1016/j.amc.2007.01.062