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On Convergence of an Accelerated Modified Newton Method for Nonlinear Equations

This paper introduces a modified and stable Newton's iteration algorithm that addresses convergence issues caused by near-zero derivatives while reducing computational costs and improving efficiency, supported by a theoretical analysis of its convergence properties.

Original authors: Sanwar Ahmad, Joy Watson, Mohammad Tabanjeh

Published 2026-08-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Sanwar Ahmad, Joy Watson, Mohammad Tabanjeh

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of mathematics, there is a constant need to find the precise point where a curve touches the ground, a moment known as finding a root. This task is fundamental to solving equations that describe everything from the orbit of a planet to the flow of electricity in a circuit. For decades, the standard tool for this job has been a technique called Newton's method. Imagine a hiker trying to find the bottom of a valley in thick fog. The hiker checks the slope of the ground right where they are standing and takes a step downhill in that direction. They repeat this process, checking the slope again and again, until they reach the bottom. This method is famous for being incredibly fast; if the hiker starts close enough to the bottom, they reach it with astonishing speed, doubling their accuracy with every single step. However, this speed comes with a catch: the hiker must be able to measure the slope at every single step. If the ground is flat, the slope is zero, and the hiker gets stuck. Furthermore, if measuring the slope is a slow, difficult, or expensive process, the hiker might spend more time measuring than walking, making the journey inefficient.

A team of researchers at Virginia State University has proposed a different way to navigate this terrain, one that trades the constant checking of the slope for a single, strategic measurement at the start of the journey. In their recent work, they introduced a modified version of the classic algorithm that calculates the slope of the curve only once, at the very beginning, and then uses that same value for every subsequent step. Instead of stopping to measure the changing steepness of the ground at every footfall, the traveler assumes the slope remains constant, based on that initial reading. This approach fundamentally changes the nature of the calculation. While the classic method requires a fresh measurement of the derivative—a mathematical term for the rate of change or slope—at every single iteration, this new method performs that calculation just once. The researchers set out to prove that this shortcut does not lead the traveler astray and to understand exactly how fast this new path leads to the solution.

The researchers began by establishing the mathematical conditions under which this simplified approach is guaranteed to work. They proved that if the starting point is chosen carefully and the function behaves smoothly, the sequence of guesses will inevitably converge to the correct answer. Their analysis showed that while the method is generally linear, meaning it improves the answer by a steady, consistent factor with each step, it can achieve the same rapid, quadratic speed as the classic method under specific circumstances. This happens when the initial guess is close enough to the true root and the shape of the curve does not change drastically from the starting point. The team demonstrated that the method is stable and avoids the common pitfall of dividing by zero, which occurs in the classic method when the slope happens to be flat. By fixing the slope at the beginning, the algorithm sidesteps the danger of getting stuck on a flat patch of ground later in the process.

To test their theory, the researchers ran a series of computer experiments using five different mathematical functions, ranging from simple polynomials to more complex combinations of trigonometric and exponential terms. They compared the performance of their modified method against the traditional Newton method on a standard computer. The results were revealing. In cases where the modified method achieved its fastest possible speed, it consistently finished the job faster than the classic method, even though both took the same number of steps to get there. This is because the modified method spent far less time calculating the slope at each step. In scenarios where the modified method was slightly slower in terms of the number of steps required, it still often completed the task in less total time. This efficiency was particularly pronounced in problems where calculating the slope is a heavy computational burden. For instance, in one test case, the modified method found the solution in 0.018 seconds, while the classic method took 0.021 seconds, despite both finding the same root. In another instance, where the classic method required only seven steps, the modified method needed 117 steps but still finished in less time, taking 0.015 seconds compared to 0.026 seconds.

The study concludes that this modified approach offers a practical and robust alternative for solving nonlinear equations, particularly in situations where calculating the derivative is expensive or difficult. The researchers found that the method is especially effective when the cost of evaluating the function is low, but the cost of finding its slope is high. While the method may sometimes require more steps to reach the final answer, the reduction in computational effort per step often results in a faster overall solution. The authors suggest that this technique could be extended to more complex systems of equations and applied to real-world problems in physics and engineering where computational efficiency is critical. By simplifying the process of finding roots, this work provides a new tool for scientists and engineers who need to solve complex equations quickly and reliably, proving that sometimes, taking a single, careful measurement at the start is more efficient than constantly re-measuring the path ahead.

Technical Summary: Convergence of an Accelerated Modified Newton Method for Nonlinear Equations

Problem Statement
The paper addresses the limitations of the classical Newton's method for solving nonlinear equations of the form $f(x) = 0$ . While the classical method is renowned for its simplicity and quadratic convergence, it faces two primary challenges:

Computational Cost: It requires the evaluation of both the function $f(x)$ and its derivative $f'(x)$ at every iteration. In scenarios where analytical derivatives are unavailable or computationally expensive to evaluate, this becomes a significant bottleneck.
Stability: The method may fail to converge or encounter division by zero if the derivative $f'(x_k)$ approaches zero at any iteration step.

Methodology
The authors propose a Modified Newton Method (referred to as an accelerated modified Newton method in the title) that mitigates these issues by decoupling the derivative evaluation from the iterative loop.

Algorithm: Instead of computing $f'(x_k)$ at every step, the derivative is calculated once at the initial approximation $x_0$ (where $f'(x_0) \neq 0$ ) and held constant throughout the process. The iteration formula is defined as:
$x_{k+1} = x_k - \frac{f(x_k)}{f'(x_0)}$
Geometric Interpretation: Geometrically, this corresponds to finding the zero of a line passing through $(x_k, f(x_k))$ that is parallel to the tangent line of $f(x)$ at the initial point $x_0$ .
Convergence Analysis: The paper provides a rigorous theoretical analysis based on fixed-point iteration theory in Banach spaces.
- Convergence Criterion: Theorem 3.1 establishes conditions for convergence using the Lipschitz constant $C$ of $f'$ and the initial error term. Specifically, convergence is guaranteed if $h = C \|[f'(x_0)]^{-1}\| \|[f'(x_0)]^{-1} f(x_0)\| \leq 1/4$ .
- Rate of Convergence: The analysis demonstrates that the method generally converges linearly. However, under specific conditions—particularly when the initial guess $x_0$ is sufficiently close to the root such that $f'(x) \approx f'(x_0)$ —the method can achieve quadratic convergence. The paper notes that if the tangent at the root is nearly vertical or for functions with repeated roots, convergence may be slow or fail.

Key Contributions

Algorithmic Simplification: The paper introduces a stable variant of Newton's method that eliminates the need for direct computation of derivatives at each iteration, thereby reducing the computational overhead per iteration.
Theoretical Framework: It derives a specific convergence criterion (Theorem 3.1) and proves that the modified iteration defines a contraction mapping within a specific radius $\delta$ around the initial guess, ensuring the existence of a unique solution.
Convergence Characterization: The authors clarify the convergence behavior, distinguishing between the general linear convergence and the potential for quadratic convergence depending on the proximity of $x_0$ to the root and the behavior of the derivative.

Results
The authors evaluated the proposed method against the classical Newton method using five test functions in MATLAB 2023a.

Performance Metrics: The study compared the number of iterations ( $N$ ), execution time, and the estimated rate of convergence.
Findings:
- In cases where quadratic convergence was achieved (e.g., $x_0 = 0$ for $f(x) = x^3 - 3x^2 + 3x - 2$ ), the modified method required the same number of iterations as the classical method but demonstrated faster execution times due to the single derivative evaluation.
- In cases where the modified method exhibited linear or superlinear convergence (e.g., $x_0 = -1$ ), it sometimes required more iterations than the classical method. However, the reduced cost per iteration often resulted in a lower total runtime.
- The method proved particularly advantageous for problems where derivative evaluation is the dominant computational cost.

Significance and Claims
The paper claims that the modified Newton method offers a practical and efficient alternative to the classical approach, particularly in contexts where derivative computation is expensive or analytically difficult.

Efficiency: The primary significance lies in the trade-off: while the method may require more iterations in some scenarios, the reduction in computational cost per step (evaluating the derivative only once) often leads to superior overall performance in terms of wall-clock time.
Robustness: By fixing the derivative, the method avoids the instability associated with $f'(x_k)$ approaching zero during the iteration process.
Scope: The authors modestly frame the work as an analysis of a specific modification, noting that future research could extend this approach to systems of equations or develop adaptive strategies for updating the derivative, but the current paper focuses on the single-variable case and its theoretical and numerical validation.

Technical Summary: Convergence of an Accelerated Modified Newton Method for Nonlinear Equations

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