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A root finding method with arbitrary order of convergence

This paper presents a polynomial-based fixed-point iteration method that allows for the calculation of the $M$ -th root of a positive number $a$ with an arbitrary order of convergence.

Original authors: Alois Schiessl

Published 2026-02-10

📖 3 min read🧠 Deep dive

CC BY 4.0

Original authors: Alois Schiessl

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 "Super-Powered Magnifying Glass": A Simple Guide to Schiessl’s Root-Finding Method

Imagine you are standing in a massive, dark forest, and you know there is a single, glowing treasure chest hidden somewhere on the ground. You don't have a map, but you have a compass that tells you if you are getting warmer or colder.

In mathematics, finding a "root" (like a square root) is exactly like trying to find that treasure chest. Most mathematicians use a tool called Newton’s Method. It’s like walking toward the treasure by taking steps based on the slope of the ground. It works great, but it has a speed limit: every time you take a step, you roughly double the amount of information you have.

Alois Schiessl’s paper introduces a way to break that speed limit.

1. The Problem: The Speed Limit of Math

Most traditional methods are like a runner who doubles their speed every minute. It’s fast, but if you want to find a treasure buried under a mountain of precision (like a million decimal places), even a fast runner might take too long.

Schiessl’s method is like a teleporting runner. Instead of just doubling their speed, they can choose to triple it, quadruple it, or even make it ten times faster. He calls this "Arbitrary Order of Convergence."

2. The Secret Sauce: The "Magic Polynomial"

How does he do this? Instead of just looking at the slope of the ground (the derivative), he uses something called a Polynomial.

Think of a polynomial as a custom-molded stencil.

A standard method uses a simple, straight stencil (a line).
Schiessl’s method allows you to design a highly complex, curvy stencil that fits the shape of the "treasure" almost perfectly.

By choosing a specific number (which he calls $P$ ), you are essentially deciding how "curvy" and "detailed" your stencil is. If you choose a higher $P$ , your stencil becomes more sophisticated, allowing you to leap much closer to the treasure in a single bound.

3. The "Turbo Button" (The Math Part)

In the paper, Schiessl defines a special function $F(x)$ . In everyday terms, this function is a mathematical slingshot.

When you pull the rubber band back (choose your $P$ value), you are loading the slingshot with energy.

If $P=1$ , it’s a standard slingshot (quadratic convergence—doubling your accuracy).
If $P=3$ , it’s a high-powered cannon (quartic convergence—quadrupling your accuracy).

The paper proves mathematically that this "slingshot" will always hit the target (the root) and that you can make it as powerful as you want.

4. The Proof in the Pudding: One Million Digits

To prove his method wasn't just theoretical nonsense, Schiessl put it to the ultimate test: Calculating the square root of 2 to one million decimal places.

Using his "high-powered cannon" ( $P=3$ ), the computer didn't just crawl toward the answer; it exploded toward it. In just nine steps, the accuracy jumped from a tiny fraction to a million digits. He did this in just three seconds on a standard home computer.

Summary: Why does this matter?

If you are a scientist or an engineer working with massive amounts of data or ultra-precise physics (like calculating the trajectory of a spacecraft or the behavior of subatomic particles), you need answers that are both perfectly accurate and incredibly fast.

Schiessl has provided a "tuning knob" for math. Instead of being stuck with the standard speed of Newton's Method, mathematicians can now turn the knob to "Turbo," "Super-Turbo," or "Hyper-Drive," depending on how much precision they need.

Technical Summary: A Root Finding Method with Arbitrary Order of Convergence

Author: Alois Schiessl
Date: February 10, 2026

1. Problem Statement

The paper addresses the fundamental numerical problem of calculating the $M$ -th root of a positive real number $a$ (denoted as $\sqrt[M]{a}$ ). While traditional methods like Newton’s Method are widely used, they are limited to a fixed quadratic order of convergence (order 2), meaning the number of correct digits approximately doubles with each iteration. The author seeks a method that is restricted to polynomial evaluations—which are computationally efficient—but allows for an arbitrary order of convergence $(P+1)$ by adjusting a parameter $P$ .

2. Methodology

The proposed method shifts the focus from the standard function $f(x) = x^M - a$ to a specific fixed-point iteration based on the function $f(t) = 1 - \frac{t^M}{a}$ .

The Fixed-Point Function:
The author defines a specialized fixed-point function $F(x)$ using a definite integral:
$F(x) = \left[ \prod_{\ell=1}^{P} \left( 1 + \frac{1}{\ell \cdot M} \right) \right] x \int_{0}^{x} \left( 1 - \frac{t^M}{a} \right)^P dt$

Key Mathematical Steps:

Polynomial Reduction: By applying the Binomial Theorem to the integrand and performing term-by-term integration, the author proves that $F(x)$ is a polynomial of degree $(P \cdot M + 1)$ with only $(P + 1)$ terms.
Coefficient Pre-calculation: The coefficients $c_k$ of the polynomial are derived such that they remain constant throughout the iteration process, allowing for efficient computation via Horner’s scheme.
Convergence Analysis: Using the Banach Fixed-Point Theorem and Taylor series expansion, the author proves that:
1. $\sqrt[M]{a}$ is a fixed point of $F(x)$ .
2. The first $P$ derivatives of $F(x)$ at the fixed point are zero.
3. The $(P+1)$ -th derivative is non-zero.
4. Consequently, the sequence $x_{n+1} = F(x_n)$ converges to $\sqrt[M]{a}$ with an order of convergence exactly equal to $P+1$ .

3. Key Contributions

Arbitrary Convergence Order: Unlike Newton's method, the user can choose $P$ to achieve any desired order of convergence (e.g., cubic, quartic, quintic, etc.).
Polynomial-Only Approach: The method avoids complex transcendental functions during the iteration phase, reducing the problem to a simple polynomial evaluation.
Efficiency via Sparsity: Despite the high degree of the polynomial, the number of non-zero terms is kept low $(P+1)$ , making it computationally lightweight for high-precision arithmetic.

4. Results and Computational Validation

The author validates the method through two primary computational experiments:

Example 1 (Quadratic Convergence): Calculating $\sqrt[3]{10}$ using $P=1$ (order 2). The results demonstrated that the number of correct digits doubled with each step, as expected.
Example 2 (High-Precision Quartic Convergence): Calculating $\sqrt{2}$ $2$ to one million decimal digits using $P=3$ $P = 3$ (order 4).
- Performance: The calculation was completed in approximately 3 seconds on a standard consumer-grade PC (AMD Ryzen 5 5600G, 32GB RAM) using MAPLE software.
- Verification: The error magnitude (the number of leading zeros in the difference between iterations) approximately quadrupled at each step, confirming the quartic convergence order.

5. Significance

This paper presents a highly scalable algorithm for high-precision root finding. Its significance lies in its predictability and flexibility: for applications in computational mathematics or cryptography where extreme precision is required, this method allows a developer to balance the trade-off between the complexity of the polynomial (determined by $P$ ) and the speed of convergence. The ability to reach a million digits of precision in seconds makes it a potent alternative to standard iterative solvers.