# Easy Proof of Three Recursive $\pi$-Algorithms -- Einfacher Beweis dreier rekursiver $\pi$-Algorithmen

## TL;DR

This paper provides elementary algebra and calculus proofs demonstrating that Borwein's quartic algorithm and the Brent-Salamin algorithm both converge to pi, with the former being more iteration-efficient and having quartic convergence.

## Contribution

It offers new elementary algebra and calculus proofs establishing the equivalence, convergence, and convergence rates of Borwein's and Brent-Salamin pi algorithms.

## Key findings

- Borwein's quartic algorithm matches Brent-Salamin's output with fewer iterations.
- Brent-Salamin algorithm converges to pi using integral calculus.
- Borwein's algorithm exhibits quartic convergence, faster than quadratic.

## Abstract

This paper consists of three independent parts: First we use only elementary algebra to prove that the quartic algorithm of the Borwein brothers has exactly the same output as the Brent-Salamin algorithm, but that the latter needs twice as many iterations. Second we use integral calculus to prove that the Brent-Salamin algorithm approximates $\pi$. Combining these results proves that the Borwein brothers' quartic algorithm also approximates $\pi$. Third, we prove the quadratic convergence of the Brent-Salamin algorithm, which also proves the quartic convergence of Borwein's algorithm. -- -- Dieses Paper besteht aus drei unabh\"angigen Teilen: Erstens beweisen wir mit elementarer Algebra, dass der Borwein-Algorithmus vierter Ordnung die gleichen Ergebnisse liefert wie der Brent-Salamin-Algorithmus, wobei letzterer doppelt so viele Iterationen ben\"otigt. Zweitens beweisen wir mit Integralrechnung, dass der Brent-Salamin-Algorithmus gegen $\pi$ konvergiert. Hieraus folgt, dass der Borwein-Algorithmus vierter Ordnung ebenfalls gegen $\pi$ konvergiert. Drittens beweisen wir die quadratische Konvergenz des Brent-Salamin-Algorithmus und somit auch die quartische Konvergenz des Borwein-Algorithmus.