Annealing approach to root-finding

TL;DR

The paper addresses the challenge of robust and fast root finding by introducing a parameterized Newton-Raphson method with an auxiliary temperature-like variable $\beta$. Grounded in a physical reformulation via a partition function and Adomian decomposition, the authors derive a two-step iterative scheme that reduces to standard NR when $\beta=0$ and connects to Adomian updates at $\beta=1$, while also enabling annealing through a schedule $\beta_n$. They show improved convergence properties (quadratic in general, cubic at $\beta=1$ under suitable conditions), explore function-specific behaviors (e.g., $f(x)=x^{1/3}$), and demonstrate that nonzero $\beta$ enlarges basins of attraction and increases basin entropy, indicating greater exploratory capacity. An annealing strategy $\beta_n$ is proposed to accelerate convergence while managing computational cost, and connections to broader methods such as HAM and topological formulations are discussed. Overall, the approach offers a physically grounded, flexible framework for iterative root finding with practical benefits for challenging nonlinear problems.

Abstract

The Newton-Raphson method is a fundamental root-finding technique with numerous applications in physics. In this study, we propose a parameterized variant of the Newton-Raphson method, inspired by principles from physics. Through analytical and empirical validation, we demonstrate that this novel approach offers increased robustness and faster convergence during root-finding iterations. Furthermore, we establish connections to the Adomian series method and provide a natural interpretation within a series framework. Remarkably, the introduced parameter, akin to a temperature variable, enables an annealing approach. This advancement sets the stage for a fresh exploration of numerical iterative root-finding methodologies.

Figures (3)

• Figure 1: Improved convergence to roots: Comparison between Newton-Raphson method and the revised method. The Newton-Raphson method progresses from an initial point $x_0$ to $x_1$, determined as the solution of $f_L(x_1) = f_0 + f'_0 (x_1 - x_0)=0$, whereas the revised method advances $x_0$ to $x(\beta)$, found as the solution of $\beta f(x) + f_L(x).$ The separation $|x(\beta) - x_0|$ adjusts, either increasing or decreasing, to enhance convergence towards $x_\mathrm{root}$, contingent upon the relationship between $f(x(\beta))$ and $f_0$: (a) when they share the same sign, or (b) when they have opposite signs.
• Figure 2: Improved iteration to roots: Comparison between Newton-Raphson method and the revised method. In the Newton-Raphson method, the progression from an initial point $x_0$ to $x_1 = x_0 - f_0/f'_0$ with a slope $f'_0.$ The revised method can be interpreted to provide an interpolated point, $(\tilde{x}_0, \tilde{f}_0)$, between $(x_0, f_0)$ and $(x_1, f_1)$, where $f_i \equiv f(x_i).$ Then, the next iterated point $\tilde{x}_1$ is determined by a linear function crossing $(\tilde{x}_0, \tilde{f}_0)$ with the same slope $f'_0.$ The convergence behavior depends on the relationship between $f_0$ and $f_1$: (a) when they share the same sign, or (b) when they have opposite signs.
• Figure 3: Newton fractals and basin entropy. Colors represent different basins of attractors (roots). In particular, the colors in the basins of attraction have been shaded such that darker colors correspond to longer iterations to converge to roots. The basins of three functions are computed for $\beta \in \{ -1 , -0.5, 0 , 0.5 , 1\}$: (a) top row $f_2(z) = z^3 -1$, (b) central row $f_7(z) = (z - 1)^3 + 4(z-1)^2 - 10$, and (c) bottom row $f_{14}(z) = z + z^2 \sin(2/z)$. The corresponding basin entropy $S_b$ in the right plot quantifies the unpredictability of roots as a function of $\beta$ over the range $[-1,1]$ in steps of $0.01$. $S_b$ has been computed on a grid of $1000$ times $1000$ initial points with a covering of boxes of size $20$ times $20$. The method to determine the final root is described in Sec. \ref{['sec:results']}.