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Volume 15
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Article

Local Convergence of the Gauss–Newton–Broyden Method for Solving Nonlinear Least Squares Problems

by
Stepan Shakhno
1,* and
Halyna Yarmola
2
1
Department of Theory of Optimal Processes, Ivan Franko National University of Lviv, Universytetska Str. 1, 79007 Lviv, Ukraine
2
Department of Computational Mathematics, Ivan Franko National University of Lviv, Universytetska Str. 1, 79007 Lviv, Ukraine
*
Author to whom correspondence should be addressed.
Axioms 2026, 15(9), 682; https://doi.org/10.3390/axioms15090682
Submission received: 31 July 2026 / Revised: 9 September 2026 / Accepted: 11 September 2026 / Published: 13 September 2026
(This article belongs to the Special Issue Advances in Nonlinear Dynamics: Theory and Application)
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Abstract

The Gauss–Newton–Broyden method is proposed and investigated for solving a nonlinear least squares problem with operator decomposition. This method is obtained from the Gauss–Newton method by replacing the Jacobian matrix of a nonlinear operator with the sum of the derivative of the differentiable part of the operator and the matrix computed by the Broyden update formula for the other part of the nonlinear vector function. A local convergence theorem for the proposed method, under the classical Lipschitz conditions and for problems with zero residual, is proved. The results of numerical experiments are also presented.
Keywords: nonlinear least squares problem; Gauss–Newton–Broyden method; decomposition of nonlinear operator; local convergence; Lipschitz condition; residual; complementarity problem nonlinear least squares problem; Gauss–Newton–Broyden method; decomposition of nonlinear operator; local convergence; Lipschitz condition; residual; complementarity problem

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MDPI and ACS Style

Shakhno, S.; Yarmola, H. Local Convergence of the Gauss–Newton–Broyden Method for Solving Nonlinear Least Squares Problems. Axioms 2026, 15, 682. https://doi.org/10.3390/axioms15090682

AMA Style

Shakhno S, Yarmola H. Local Convergence of the Gauss–Newton–Broyden Method for Solving Nonlinear Least Squares Problems. Axioms. 2026; 15(9):682. https://doi.org/10.3390/axioms15090682

Chicago/Turabian Style

Shakhno, Stepan, and Halyna Yarmola. 2026. "Local Convergence of the Gauss–Newton–Broyden Method for Solving Nonlinear Least Squares Problems" Axioms 15, no. 9: 682. https://doi.org/10.3390/axioms15090682

APA Style

Shakhno, S., & Yarmola, H. (2026). Local Convergence of the Gauss–Newton–Broyden Method for Solving Nonlinear Least Squares Problems. Axioms, 15(9), 682. https://doi.org/10.3390/axioms15090682

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