https://github.com/abhrankan-chakrabarti/pi-calculation-with-catalan-numbers

This repository contains a Python script that calculates the digits of π (pi) using a novel algorithm based on Catalan numbers and fixed-point arithmetic. The algorithm efficiently computes the digits of π by leveraging the properties of Catalan numbers, making it a unique and effective method for high-precision calculations.
https://github.com/abhrankan-chakrabarti/pi-calculation-with-catalan-numbers

arbitrary-precision catalan-numbers computational-mathematics fixed-point-arithmetic gmpy2 high-precision high-precision-calculation math mathematical-algorithms numerical-methods pi pi-calculation pi-digits-calculation python

Last synced: 4 months ago
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This repository contains a Python script that calculates the digits of π (pi) using a novel algorithm based on Catalan numbers and fixed-point arithmetic. The algorithm efficiently computes the digits of π by leveraging the properties of Catalan numbers, making it a unique and effective method for high-precision calculations.

• Host: GitHub
• URL: https://github.com/abhrankan-chakrabarti/pi-calculation-with-catalan-numbers
• Owner: Abhrankan-Chakrabarti
• License: mit
• Created: 2024-06-21T17:07:25.000Z (over 1 year ago)
• Default Branch: main
• Last Pushed: 2024-06-22T22:30:33.000Z (over 1 year ago)
• Last Synced: 2025-05-21T16:15:15.704Z (5 months ago)
• Topics: arbitrary-precision, catalan-numbers, computational-mathematics, fixed-point-arithmetic, gmpy2, high-precision, high-precision-calculation, math, mathematical-algorithms, numerical-methods, pi, pi-calculation, pi-digits-calculation, python
• Language: Python
• Homepage: https://abhrankan-chakrabarti.github.io/pi-calculation-with-catalan-numbers/
• Size: 39.1 KB
• Stars: 2
• Watchers: 1
• Forks: 0
• Open Issues: 0
• Metadata Files:
  • Readme: README.md
  • License: LICENSE

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README

          
# Catalan π Algorithm

This repository contains an implementation of a newly discovered algorithm for calculating the digits of π using Catalan numbers, derived from ![6 \arcsin(0.5)](https://latex.codecogs.com/svg.latex?6%20%5Carcsin%280.5%29). The algorithm uses high-precision arithmetic and optimized series summation to achieve efficient computation.

## Introduction

Calculating the digits of π has been a topic of interest for mathematicians and computer scientists alike. This project introduces a novel algorithm for π calculation that leverages Catalan numbers and fixed-point arithmetic.

## Algorithm

The series representation for π used in this algorithm is derived from the inverse sine function:

![\pi = 6 \arcsin(0.5)](https://latex.codecogs.com/svg.latex?%5Cpi%20%3D%206%20%5Carcsin%280.5%29)

By expanding ![\arcsin(x)](https://latex.codecogs.com/svg.latex?%5Carcsin%28x%29) as a series, we get:

![series](https://latex.codecogs.com/svg.latex?%5Carcsin%28x%29%20%3D%20%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%7D%20%5Cfrac%7B%282n%29%21%7D%7B4%5En%20%28n%21%29%5E2%20%282n%20%2B%201%29%7D%20x%5E%7B2n%20%2B%201%7D)

Substituting ![x = 0.5](https://latex.codecogs.com/svg.latex?x%20%3D%200.5) and multiplying by 6, we derive the series:

![series](https://latex.codecogs.com/svg.latex?%5Cpi%20%3D%203%20%2B%206%20%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D%20%5Cfrac%7B%282n%20-%201%29%20%5Ccdot%20C_%7Bn-1%7D%7D%7B%282n%20%2B%201%29%20%5Ccdot%2016%5En%7D)

Where ![C_n](https://latex.codecogs.com/svg.latex?C_n) is the nth Catalan number.

## Implementation

The implementation of the algorithm in Python can be found in the `pi_gmpy.py` file.

## Usage

To use the script, run the following command in your terminal:

```sh

python pi_gmpy.py

```

You will be prompted to enter the number of digits of π you want to calculate.

## Example

```sh

$ python pi_gmpy.py

How many digits of π? : 100

π = 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679...∞

```

## License

This project is licensed under the MIT License - see the [LICENSE](LICENSE) file for details.

## Contact

For any questions or feedback, feel free to reach out via GitHub or email.