Visualization of Convergence Behavior of a Generalized Newton Method and Levenberg–Marquardt Algorithm

Regina S. Burachik () and C. Yalçın Kaya ()
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Regina S. Burachik: University of South Australia, Mathematics, UniSA STEM
C. Yalçın Kaya: University of South Australia, Mathematics, UniSA STEM

A chapter in Handbook of Visual, Experimental and Computational Mathematics, 2026, pp 1011-1026 from Springer

Abstract: Abstract A generalized Newton method was proposed by Burachik, Kaya, and Sabach, in 2012, for solving univariate real-valued nonlinear equations. What makes the generalized Newton iteration different from the classical Newton iteration is that the generalized Newton iteration involves an additional function which can be chosen in such a way that (i) the additional function reflects the structure of the problem and, as a result, (ii) the generalized method may have better convergence properties than the classical one. In 2021, Burachik, Caldwell, and Kaya extended the generalized method to solve real-valued systems of equations in multiple variables. In this chapter, we consider the Levenberg–Marquardt versions of the generalized Newton method and illustrate how visualizations of the color-coded number of iterations a method takes to reach a solution can be used to determine which choice of the auxiliary function yields a favorable convergence behavior.

Keywords: Levenberg–Marquardt algorithm; Generalized Newton method; Nonlinear equations; Global behavior; Visualization; Himmelblau function; 49M15; 65H04; 65H10 (search for similar items in EconPapers)
Date: 2026
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DOI: 10.1007/978-3-032-16368-4_16

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