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A fixed-point iteration method for the number Pi with arbitrary odd order of convergence based on the sine function

This paper introduces a fixed-point iteration method based on the sine function that achieves an arbitrary odd order of convergence $(2P+1)$ for the high-precision computation of $\pi$ .

Original authors: Alois Schiessl

Published 2026-03-18

📖 5 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

Imagine you are trying to hit a bullseye on a dartboard, but the bullseye is the number Pi ( $\pi$ ), and you are throwing darts in the dark. Most methods of calculating Pi are like throwing darts one by one, hoping you get a little closer each time. Some methods are like using a laser sight that corrects your aim significantly with every throw.

This paper, written by Alois Schiessl for "Pi Day 2026," introduces a super-powered laser sight. It's a mathematical recipe that doesn't just get you closer to Pi; it zooms in on the target with such extreme speed that the number of correct digits you get multiplies wildly with every single step.

Here is the breakdown of how this "magic trick" works, using simple analogies.

1. The Goal: Finding the Perfect Spot

Pi is an irrational number, meaning it goes on forever without repeating. To calculate it, mathematicians use Fixed-Point Iteration.

The Analogy: Imagine you are walking toward a door (Pi). You take a step, look at where you are, and take another step based on that new position. If your steps are calculated correctly, you will eventually stop right in front of the door.
The Problem: Most walking methods are slow. You might take 100 steps to get close.
The Solution: This paper offers a method where, if you are already somewhat close, one step gets you 9 times closer (or even 27 times closer, or 81 times closer, depending on how you set it up).

2. The Secret Ingredient: The "Sine" Wave

The method relies on the sine function ( $\sin(x)$ ), which you might remember from high school math as a wavy line.

The Analogy: Think of the sine function as a magic compass. When you are standing exactly at Pi, the compass points straight up (the value is 0). If you are slightly off, the compass tilts and tells you exactly how far and in which direction you need to move to get back to the center.
The author builds a complex formula using this compass. He takes the standard sine wave and adds a "correction layer" on top of it.

3. The "Dial" (The Variable $P$ )

The most exciting part of this paper is the arbitrary order of convergence.

The Analogy: Imagine a video game difficulty setting.
- Level 1 ( $P=1$ ): You get 3x more accuracy with every step. (Cubic convergence).
- Level 2 ( $P=2$ ): You get 5x more accuracy.
- Level 4 ( $P=4$ ): You get 9x more accuracy.
The author proves that you can turn this dial to any odd number you want. If you want to calculate Pi to a million digits, you can set the dial to a high number, and the math will explode in accuracy incredibly fast.

4. How the Math Works (The "Self-Correcting" Loop)

The formula looks scary in the paper, but the logic is simple:
$New Position = Old Position + (Correction Factor \times \text{Sine Wave})$

Start: You guess a number close to Pi (like 3.14).
Calculate: You plug that number into the formula. The formula uses the sine function to figure out the "error" (how far off you are).
Correct: It adds a massive correction to your guess.
Repeat: You take the new, better guess and do it again.

Because the formula is designed so perfectly, the "error" doesn't just shrink; it gets squared, cubed, or raised to the 9th power with every single step.

5. The Real-World Test: The Million-Digit Challenge

The author didn't just write theory; he tested it on a computer.

The Challenge: Calculate one million digits of Pi.
The Old Way: This would usually take thousands of steps and massive computing power.
The New Way:
- He started with a guess that was already accurate to 18 digits.
- He set the "Dial" to Level 4 (9x convergence).
- Step 1: Accuracy jumps to ~160 digits.
- Step 2: Accuracy jumps to ~1,400 digits.
- Step 3: Accuracy jumps to ~13,000 digits.
- Step 4: Accuracy jumps to ~118,000 digits.
- Step 5: Accuracy jumps to ~1,000,000 digits.
- Step 6: He confirms the result.

Result: In just 6 steps, he calculated Pi to over a million digits. It's like walking across a room and, with every step, doubling the distance you cover until you've crossed the entire universe.

Why is this cool?

Usually, to get more precision, you have to do more work (more steps, more complex math). This method is self-correcting. It realizes, "Oh, I'm getting close, so I'll switch to a super-powerful mode automatically."

The author even suggests that if you have infinite time and memory, you could calculate Pi to infinite precision using this method. It turns the calculation of Pi from a slow, grinding march into a rocket ship launch.

Summary

This paper presents a new, incredibly fast way to calculate Pi. By using a clever combination of the sine function and a customizable "power level," the method allows the number of correct digits to multiply exponentially with every single calculation. It's a mathematical shortcut that turns a slow walk into a teleportation beam.

1. Problem Statement

The paper addresses the challenge of computing the mathematical constant $\pi$ to high precision. While many algorithms exist (e.g., Chudnovsky algorithm, Machin-like formulas), this work proposes a novel fixed-point iteration method capable of achieving an arbitrary odd order of convergence. The goal is to construct a simple iterative scheme based on the sine function that allows for rapid convergence to $\pi$ without requiring complex series expansions or modular arithmetic at every step, provided a sufficiently close initial guess is available.

2. Methodology

The Core Function

The author defines a fixed-point function $S(x)$ parameterized by an integer $P \in \mathbb{N}$ . The function is constructed by truncating the power series expansion of $\arcsin(t)$ and substituting $t = \sin(x)$ .

The function is defined as:
$S(x) = x + \sum_{k=1}^{P} \left( \prod_{\ell=1}^{k-1} \frac{2\ell-1}{2\ell} \right) \frac{\sin(x)^{2k-1}}{2k-1}$

The iteration rule is:
$x_{n+1} = S(x_n)$

Theoretical Foundation

The method relies on the Taylor series of $\arcsin(t)$ :
$\arcsin(t) = \sum_{k=1}^{\infty} \left( \prod_{\ell=1}^{k-1} \frac{2\ell-1}{2\ell} \right) \frac{t^{2k-1}}{2k-1}$
By setting $t = \sin(x)$ , the infinite series theoretically equals $x$ (for $x$ near $\pi$ where $\sin(x)$ is small). The paper truncates this series at $P$ terms to create a finite polynomial in $\sin(x)$ added to $x$ .

Mathematical Proof

The paper provides a rigorous proof of three key properties:

Fixed Point: $S(\pi) = \pi$ because $\sin(\pi) = 0$ , causing all summation terms to vanish.
Vanishing Derivatives: The author proves that the first $2P$ derivatives of $S(x)$ at $x=\pi$ are zero ( $S^{(k)}(\pi) = 0$ for $1 \le k \le 2P$ ). This is achieved through careful differentiation and the application of the Leibniz rule on the term $\sin(x)^{2P-1}$ .
Convergence Order: The $(2P+1)$ -th derivative at $\pi$ is non-zero. Specifically:
$S^{(2P+1)}(\pi) = \left( \prod_{\ell=1}^{P} (2\ell-1) \right)^2$
According to the theory of fixed-point iteration, if the first $m-1$ derivatives at the fixed point are zero and the $m$ -th is non-zero, the convergence order is $m$ . Thus, the order of convergence is exactly $2P + 1$ .

Optimization Strategies

For practical high-precision computation, the author suggests three optimizations:

High-Precision Initial Guess: Start with an $x_0$ already accurate to $\approx 18$ digits.
Variable Precision Arithmetic: Since the method is self-correcting, early iterations do not require full precision. The working precision (number of digits) is increased geometrically ( $18 \times (2P+1)^n$ ) at each step to match the growing accuracy of the result, saving computational time.
Coefficient Pre-calculation: The coefficients of the polynomial are constant and can be pre-computed and stored.

3. Key Contributions

Arbitrary Odd Convergence: The paper introduces a unified framework where the convergence order can be tuned to any odd integer ( $3, 5, 7, \dots$ ) simply by increasing the parameter $P$ .
Simplicity of Implementation: Unlike some high-order methods that require complex rational approximations or multiple trigonometric evaluations, this method relies solely on the sine function and simple polynomial arithmetic.
Rigorous Derivation: The paper provides a complete analytical proof of the convergence order using Taylor expansions and the Leibniz rule, distinguishing it from heuristic numerical methods.
Practical Efficiency: The demonstration of calculating 1 million digits of $\pi$ in just 6 iterations using $P=4$ (9th-order convergence) highlights the method's extreme efficiency when combined with variable-precision arithmetic.

4. Results

The author performed computational tests using Maple 2025.2 on a standard consumer PC (AMD Ryzen 5 5600G).

Case 1 ( $P=1$ , 3rd Order):
- Starting from $x_0 = 3$ .
- After 7 iterations, the result achieved >900 correct decimal digits.
- The error reduction followed the cubic pattern ( $10^{-11} \to 10^{-34} \to 10^{-100}$ ).
Case 2 ( $P=4$ , 9th Order):
- Target: 1,000,000 digits.
- Starting from $x_0 \approx 3.14159...$ (18 digits).
- Used variable precision (starting at 162 digits, scaling up).
- Result: Achieved over 1,059,070 correct digits in just 6 iterations.
- The error exponent increased by a factor of roughly 9 at each step, confirming the theoretical 9th-order convergence.

5. Significance

This paper presents a mathematically elegant and computationally efficient method for high-precision $\pi$ calculation.

Theoretical Value: It demonstrates how truncating the $\arcsin$ series creates a powerful fixed-point iteration with controllable convergence order.
Practical Value: The method is highly suitable for arbitrary-precision arithmetic libraries. By combining a high-order scheme ( $P \ge 4$ ) with "doubling" (or in this case, multiplying by $2P+1$ ) precision strategies, it minimizes the cost of expensive high-precision sine evaluations.
Accessibility: The algorithm is relatively easy to implement compared to other ultra-fast algorithms (like the Chudnovsky algorithm which involves complex modular exponentiation), making it a strong candidate for educational purposes or specific high-precision applications where sine functions are readily available.

Note on Context: The paper is dated March 14, 2026 (Pi Day), suggesting it is a future-dated or hypothetical publication intended for the "Pi Day" celebration, though the mathematical content is presented as a rigorous derivation.