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

Eva G. Villalba; M. A. Hernández-Verón; Eulalia Martínez; Eva G. Villalba; M. A. Hernández-Verón; Eulalia Martínez

doi:10.3934/era.2026296

Electronic Research Archive

2026, Volume 34, Issue 9: 6807-6824. doi: 10.3934/era.2026296

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

1.
Institut Universitari de Matemàtica Multidisciplinar, Universitat Politècnica de València, Spain
2.
Department of Mathematics and Computation, University of La Rioja, Spain

Received: 20 May 2026 Revised: 06 July 2026 Accepted: 23 July 2026 Published: 11 August 2026

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.
- nonlinear systems,
- singularities,
- modified Newton method,
- semilocal convergence,
- dynamical systems
Citation: Eva G. Villalba, M. A. Hernández-Verón, Eulalia Martínez. Semilocal convergence of a modified Newton method for nonlinear systems with singular Jacobians[J]. Electronic Research Archive, 2026, 34(9): 6807-6824. doi: 10.3934/era.2026296

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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.

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