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Multi-point variants of the Newton-Raphson-Simpson method arising from organizing a formal zero according to a function $\phi$

This paper introduces the "max-phi method," a multi-point variant of the Newton-Raphson-Simpson algorithm for $L$ -differentiable functions, by utilizing a specific integer-valued function $\phi$ to define iteration numbers on rooted plane trees and constructing weighted generating functions to derive the iterative scheme.

Original authors: Mario DeFranco

Published 2026-08-14

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Mario DeFranco

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

The Great Mathematical Hunt for Hidden Numbers

Imagine you are a treasure hunter, but instead of gold, you are looking for a specific number hidden inside a complex mathematical equation. This number is called a "root" or a "zero," and it's the secret key that makes the equation balance perfectly to zero. Finding these keys is a fundamental job in science and engineering, from calculating the orbit of a satellite to designing a bridge that won't collapse. For centuries, mathematicians have used a famous tool called the Newton-Raphson method to hunt for these keys. Think of it like a hiker trying to find the bottom of a valley in thick fog. The hiker looks at the slope of the ground right under their feet, takes a step downhill, and repeats the process until they reach the bottom. It's fast and reliable, but sometimes the terrain is tricky, and the hiker needs a better map.

In recent years, mathematicians have realized that the "hiker" doesn't have to look at just one spot to decide where to step. They can look at several spots at once, gathering more information to make a smarter, more powerful leap. This idea of using multiple points to find a solution is the playground of this paper. The author, Mario DeFranco, is working in the world of algebra and calculus, specifically exploring how to organize these multi-step jumps using a strange and beautiful structure called a "rooted plane tree." You can picture these trees not as plants with leaves, but as family trees or organizational charts where one main root branches out into smaller branches, which branch out further. By arranging these trees in a specific way, the paper suggests a new, more flexible way to calculate these hidden numbers, potentially making the hunt for mathematical answers even more efficient.

The Tree-Map Method: A New Way to Find Math's Hidden Keys

In this paper, Mario DeFranco introduces a fresh twist on an old mathematical game. He calls his new approach the max-phi method. To understand it, we first need to meet the characters: the "formal zero" and the "tree."

Imagine you have a function, which is just a fancy machine that takes a number and spits out another number. You want to find the input that makes the machine spit out zero. DeFranco starts with a "formal zero," which is like a giant, infinite recipe for finding that answer. This recipe is built by adding up the weights of thousands of different "rooted plane trees." Think of these trees as a way to organize the steps of a calculation. A single dot is a tree; a dot with two branches coming off it is a tree; a dot with three branches, each of which has its own sub-branches, is also a tree.

In previous work, mathematicians figured out how to count these trees to get a better approximation of the answer. They assigned an "iteration number" to each tree, which basically told them which step of the calculation the tree belonged to. If you added up all the trees with iteration number 1, you got your first guess. Add up trees with iteration number 2, and you get a better guess, and so on. This process was known to recreate the famous Newton-Raphson method, the standard "hiker in the fog" approach.

The New Twist: The Function $\phi$
DeFranco's big idea is to change the rules of how we count these trees. Instead of a fixed rule, he introduces a custom function called $\phi$ (phi). This function acts like a traffic controller for the trees. It looks at a tree and decides how many "steps" it should count as, based on how many branches the tree has and how deep those branches go.

He defines a specific rule:

If a tree is just a single dot, it counts as step 0.
If a tree is a cluster of smaller trees, he looks at the "iteration number" of each sub-tree. He finds the highest number among them (let's call it the "max").
Then, he counts how many sub-trees share that highest number (the "multiplicity").
Finally, he uses his custom function $\phi$ to decide the final step number for the whole tree.

The paper proves that if you organize your trees using this new $\phi$ rule, the sum of the weights of all trees up to a certain step number ( $N$ ) follows a very specific pattern. This pattern leads to a new formula for finding the root.

The Main Discovery: The Recurrence Relation
The core of the paper is a mathematical proof showing that these sums, which DeFranco calls $Z_N$ , satisfy a specific recurrence relation. In plain English, this means that the new guess ( $c_N$ ) can be calculated using a combination of the previous few guesses ( $c_{N-1}, c_{N-2}, \dots$ ) and the values of the function and its derivatives at those points.

The formula looks complicated, but the logic is elegant:
$c_N = c_{N-1} - \frac{\text{A Complex Mix of Past Steps}}{f'(c_{N-1})}$

The "Complex Mix" in the numerator is where the magic happens. It combines the function's value at a point $L$ steps back ( $f(c_{N-L})$ ) with a series of corrections based on how many trees had the same "max" iteration number. The function $\phi$ controls exactly which past steps get included in this mix.

Why This Matters
The paper suggests that by choosing different functions for $\phi$ , you can create different "variants" of the Newton-Raphson method.

If you choose a very simple $\phi$ (where it only counts trees with one branch as step 0 and everything else as step 1), you get back the classic Newton-Raphson method.
But if you choose a more complex $\phi$ , you get a multi-point method. This means your new guess isn't just based on the immediate past; it's based on a whole history of previous guesses, weighted by the tree structure.

The author proves that this method works for any function that is differentiable enough times (specifically, $L$ times). The paper explicitly states that this is a proven mathematical result for formal series (infinite sums of variables), and it suggests that this structure can be applied to actual functions to create new algorithms.

What the Paper Doesn't Say
It is important to note what this paper doesn't do. It doesn't claim that this method is faster or better than existing methods for every problem. It doesn't provide a list of which $\phi$ function is the "best" one to use. It also doesn't show computer simulations proving it works on real-world data. Instead, it lays the mathematical groundwork, proving that these multi-point methods can be organized this way and that they follow a consistent, predictable pattern.

The Future of the Hunt
DeFranco ends by suggesting two exciting paths for future work. First, he wants to find explicit formulas that show exactly what these new guesses look like in terms of the actual zeros of the function. Second, he wonders if this tree-based organization can be used for other types of mathematical "zeros" that involve negative numbers or different kinds of trees.

In short, this paper doesn't just give you a new tool; it gives you a new way to build tools. By treating the steps of a calculation like a family tree and organizing them with a custom function $\phi$ , it opens the door to a whole new family of mathematical methods for finding hidden numbers. It's a reminder that even in a field as old as calculus, there are still new ways to arrange the pieces of the puzzle.

Technical Summary: Multi-point Variants of the Newton-Raphson-Simpson Method

Problem Statement
The paper addresses the construction of multi-point variants of the Newton-Raphson-Simpson method for finding roots of functions. While previous works (specifically Traub [5] and Džunić et al. [4]) have explored general multi-point methods, this paper seeks to derive a specific recurrence relation arising from the combinatorial organization of "formal zeros." The core problem is to generalize the iteration number assignment for rooted plane trees—previously defined in the author's prior work [1, 2, 3]—using an arbitrary function $\phi$ , and to determine how this generalization affects the resulting iterative series and the associated root-finding algorithm.

Methodology
The methodology proceeds through a combinatorial framework involving formal series and rooted plane trees:

Formal Zero Definition: The author defines a formal series $Z$ representing a "formal zero" of a polynomial $f(z) = \sum a_i z^i$ . This series is constructed as a sum over the set of rooted plane trees, where each tree $T$ is assigned a weight $w(T)$ based on the coefficients of $f(z)$ .
Generalized Iteration Number: In prior work, the iteration number $iter(T)$ was defined such that trees of iteration number $N$ $N$ corresponded to the $N$ $N$ -th term of the Newton-Raphson-Simpson recurrence. This paper generalizes this concept via Definition 4. For a function $\phi: \mathbb{Z}_{\ge 1} \to \mathbb{Z}_{\ge 0}$ $ϕ : Z_{\geq 1} \to Z_{\geq 0}$ , the iteration number $iter_\phi(T)$ $i t e r_{ϕ} (T)$ is defined recursively:
- For a single vertex $T=()$ , $iter_\phi(T) = 0$ .
- For a tree $T = (T_1, \dots, T_l)$ , let $n = \max(\{iter_\phi(T_i)\})$ and $r$ be the count of subtrees achieving this maximum. Then $iter_\phi(T) = n + \phi(r)$ .
Partial Sums and Generating Functions: The author defines $Z_N$ as the partial sum of weights $w(T)$ for all trees where $iter_\phi(T) \le N$ .
Derivation of Recurrence: By analyzing the structure of trees with iteration number $\le N$ , the author derives a recurrence relation for $Z_N$ (Lemma 1 and Lemma 2). Under specific constraints on $\phi$ (specifically that its image is $[0, L]$ and $\phi^{-1}(\{0\}) = \{1\}$ ), they simplify this relation to obtain a closed-form recurrence involving derivatives of $f$ .

Key Contributions and Results
The primary contribution is Theorem 1, which establishes a recurrence relation for the partial sums $Z_N$ under the condition that $\phi$ maps to $[0, L]$ and $\phi^{-1}(\{0\}) = \{1\}$ . The theorem states:
$Z_N = Z_{N-1} - \frac{f(Z_{N-L}) + \sum_{k=1}^{L-1} \sum_{r \in \phi^{-1}([0,k])} \frac{(Z_{N-k} - Z_{N-k-1})^r f^{(r)}(Z_{N-k-1})}{r!}}{f'(Z_{N-1})}$

Based on this theoretical result, the paper introduces Definition 6, the max-phi method. This is a multi-point iterative algorithm for an arbitrary $L$ -differentiable function $f: K \to K$ (where $K$ is a field). The $N$ -th iterate $c_N$ is defined by:
$c_N = c_{N-1} - \frac{f(c_{N-L}) + \sum_{k=1}^{L-1} \sum_{r \in \phi^{-1}([0,k])} \frac{(c_{N-k} - c_{N-k-1})^r f^{(r)}(c_{N-k-1})}{r!}}{f'(c_{N-1})}$

The paper notes that when $L=1$ , the only valid $\phi$ yields the standard Newton-Raphson-Simpson method. Furthermore, Remark 1 highlights that if $\phi$ satisfies specific monotonicity conditions ( $\phi^{-1}([0, k]) = [1, \lambda_k]$ ), the terms in the summation correspond to Taylor polynomial approximations of $f$ .

Significance and Claims
The paper claims that the main result (Theorem 1) provides a natural combinatorial derivation for a new class of multi-term root-finding methods. The significance lies in the ability to generate these methods systematically by choosing different functions $\phi$ , thereby organizing the terms of the formal zero series in a way that dictates the multi-point structure of the iteration.

The author maintains a modest scope, focusing strictly on the formal series derivation and the definition of the method. They do not present numerical experiments, convergence proofs for specific functions, or experimental applications. Instead, they identify two specific directions for Further work:

Obtaining explicit formulas for the max-phi iterates $Z_N$ in terms of the actual zeros of $f(z)$ .
Applying the max-phi framework to other formal zeros expressible as sums over rooted plane trees with negative vertex degrees.

The Great Mathematical Hunt for Hidden Numbers

The Tree-Map Method: A New Way to Find Math's Hidden Keys

Technical Summary: Multi-point Variants of the Newton-Raphson-Simpson Method

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