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A new iterative three-point method for solving systems of nonlinear equations

This paper presents a new sixth-order iterative three-point method for solving systems of nonlinear equations, derived by adapting a scalar method and validated through theoretical analysis, numerical experiments, and efficiency comparisons with existing techniques.

Original authors: Carlos E. Cadenas R., Yorman J. Mendoza N

Published 2026-01-23

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Carlos E. Cadenas R., Yorman J. Mendoza N

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

Imagine you are trying to find the exact center of a massive, twisting maze. You can't see the whole maze at once, so you have to take steps, check your surroundings, and adjust your path until you finally hit the bullseye. In the world of mathematics, this "maze" is a system of nonlinear equations, and the "bullseye" is the solution where everything balances out perfectly.

This paper introduces a new, super-fast way to navigate that maze. Here is the breakdown of what the authors did, using simple analogies.

1. The Problem: The Slow Walker

Usually, mathematicians use a standard method called Newton's Method to solve these puzzles. Think of Newton's method as a very smart hiker. Every time they take a step, they look at the ground, calculate the slope, and take one big, confident step toward the center. It works well, but it can be slow if the maze is huge or very twisty.

2. The Solution: The "Three-Step" Sprinter

The authors created a new method that acts like a three-step sprinter. Instead of just looking once and stepping, this new method takes three distinct "looks" at the problem before making its final move.

Step 1 (The Scout): It takes a quick look at where it is and makes a rough guess.
Step 2 (The Refiner): It checks that guess against a second point to get a better idea of the terrain.
Step 3 (The Sprint): Using all that gathered information, it makes a massive, highly accurate leap to the solution.

The authors took a method that was already famous for solving simple, single-variable puzzles (like finding the root of one equation) and "translated" it to handle complex, multi-variable systems (like solving a whole web of equations at once).

3. The Magic: Speed and Precision

The most exciting part of this paper is the speed.

In math, "order of convergence" is like a speed rating. A standard method might be a "quadratic" runner (speed rating of 2).
This new method is a sixth-order runner.

The Analogy: Imagine you are trying to guess a number between 1 and 100.

A quadratic method might cut the remaining possibilities in half, then in half again, then in half again. It gets there, but it takes a few rounds.
This sixth-order method is like having a magic compass that doesn't just cut the distance in half; it shrinks the remaining distance to almost nothing in a single step. If you are off by a tiny bit, the next step makes you off by a microscopic amount. It converges to the answer incredibly fast.

4. Proving It Works (The Test Drive)

The authors didn't just claim it was fast; they put it through a crash test.

Test 1: They solved a simple two-equation puzzle. The new method found the answer in just 4 steps.
Test 2: They solved a much harder problem involving a "boundary value" (think of it as a complex bridge design problem). Again, it found the answer in 4 steps.

In both cases, the math proved that the method was indeed moving at "sixth-order" speed, just as the theory predicted.

5. The Catch: The Cost of Speed

Here is the trade-off. To get this incredible speed, the method has to do more work per step.

Newton's Method is like a car that gets great gas mileage but is slow.
This New Method is like a Formula 1 car: it goes incredibly fast, but it burns a lot of fuel (computational power) to do it.

The authors calculated the "fuel cost" (computational efficiency). They compared their new car to two other high-performance cars (one 5th-order, one 6th-order).

The Result: Their new car is the most fuel-efficient only if the race is very long (specifically, if the system has 21 or more equations).
For smaller puzzles (fewer than 21 equations), the extra fuel it burns per step makes it slightly less efficient than the competition. But for massive, complex systems, it is the clear winner.

Summary

The paper presents a new mathematical tool that solves complex systems of equations by taking three smart "looks" before moving. It is incredibly fast (sixth-order convergence), but it requires more heavy lifting per step. The authors proved mathematically and with computer tests that for large, complex problems, this extra effort pays off, making it the most efficient way to find the solution.

Technical Summary: A New Iterative Three-Point Method for Systems of Nonlinear Equations

Problem Statement
The paper addresses the construction and analysis of iterative fixed-point methods for solving systems of nonlinear equations of the form $F(x) = 0$ , where $F: D \subseteq \mathbb{R}^n \to \mathbb{R}^n$ . While Newton's method is a standard single-point technique with quadratic convergence, the authors note that applications in science, engineering, and finance often require solving large systems where computational speed is critical. This necessity has driven the development of multipoint methods to improve the order of convergence and computational efficiency beyond the limitations of one-point methods.

Methodology
The authors propose a new three-point iterative method for vector systems by extending a specific scalar nonlinear equation solver constructed by Parhi and Gupta [10] to the vector case ( $\mathbb{R}^n$ ).

Iterative Scheme: The proposed method, denoted as $\phi^{(6)}_1$ , follows a three-step process for each iteration $k$ :
- Step 1 (Newton-like): $y^{(k)} = x^{(k)} - F'(x^{(k)})^{-1}F(x^{(k)})$
- Step 2 (Weighted Average): $z^{(k)} = x^{(k)} - 2[F'(x^{(k)}) + F'(y^{(k)})]^{-1}F(x^{(k)})$
- Step 3 (Final Correction): $x^{(k+1)} = z^{(k)} - [3F'(y^{(k)}) - F'(x^{(k)})]^{-1}[F'(x^{(k)}) + F'(y^{(k)})]F'(x^{(k)})^{-1}F(z^{(k)})$
Convergence Analysis: The theoretical order of convergence is established using Taylor series expansions for functions in $\mathbb{R}^n$ around the solution $\alpha$ . The authors derive the error equation $e_{k+1} = C e_k^6 + O(e_k^7)$ , proving that the method achieves a sixth order of convergence. This derivation was performed manually without symbolic software assistance, as the authors note that tools like Maple and Mathematica struggle with the non-commutative matrix multiplications inherent in the Taylor expansion for vector functions.
Computational Efficiency: The efficiency is evaluated using the Ostrowski efficiency index ( $E = p^{1/C}$ ), where $p$ is the order of convergence and $C$ is the computational cost. The cost model accounts for:
- Evaluations of the function $F$ and its derivative $F'$ .
- Matrix inversions (solved via LU decomposition and triangular system resolutions).
- Matrix-vector and scalar-vector multiplications.
- The ratio of divisions to multiplications ( $l$ ).

Key Results

Numerical Verification: The method was tested on two problems using MATLAB R2019a with 20 significant digits:
1. A system of two nonlinear algebraic equations.
2. A boundary value problem discretized into a system of six nonlinear equations.
  In both cases, the computational order of convergence ( $p$ ) was calculated using the ratio of successive errors. The results confirmed the theoretical sixth-order convergence (observed $p \approx 5.995$ and $5.75$).
Efficiency Comparison: The proposed method ( $\phi^{(6)}_1$ ) was compared against two other three-point methods:
- A fifth-order method ( $\phi^{(5)}_1$ ) from Cordero et al. [16].
- A sixth-order method ( $\phi^{(6)}_2$ ) from Cordero et al. [13].
Findings:
- Vs. Fifth-Order Method: The proposed sixth-order method is more computationally efficient than the fifth-order method when the system size $n \geq 21$ . This is because the higher convergence order outweighs the increased computational cost for larger systems.
- Vs. Sixth-Order Method: When compared to the other sixth-order method ( $\phi^{(6)}_2$ ), the proposed method is less efficient for all $n \geq 2$ . The analysis reveals that the proposed method requires three matrix inversions per iteration, whereas the competing sixth-order method requires fewer operations. The authors explicitly state that the number of matrix inversions negatively impacts efficiency.

Significance and Claims
The paper claims to successfully extend a scalar three-point method to the vector case, achieving a theoretical and computational order of convergence of six. The primary contribution is the rigorous analytical proof of this convergence order and the subsequent efficiency analysis.

The authors maintain a modest tone regarding the method's superiority. They acknowledge that while the method is highly effective for large systems ( $n \geq 21$ ) compared to lower-order methods, it is not the most efficient option among sixth-order methods due to the high cost of multiple matrix inversions. The paper concludes that reducing the number of Jacobian matrix inversions and functional evaluations is a necessary future direction to improve efficiency and simplify the theoretical convergence analysis for vector extensions.