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

Lorenz Milla

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

Lorenz Milla

Number Theory · Submitted 9 Jul 2019

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Combining these results proves that the Borwein brothers' quartic algorithm also approximates $\pi$.

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arXiv ID1907.04110 (version 3)

Pith IDDP6ARUWF, Pith's permanent record of this paper and its review

CitationLorenz Milla. "Easy Proof of Three Recursive $\pi$-Algorithms -- Einfacher Beweis dreier rekursiver $\pi$-Algorithmen." arXiv:1907.04110 (2019). https://pith.science/paper/DP6ARUWF

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,780 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.13 · human participants 0.01 · controlled comparison 0.09 · randomized assignment 0.01 · states quantitative result 0.84 · compares with prior results 0.87 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; show every role.

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.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	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.	contribution 0.81	0.91	0.17	asserted 1.00
s1	Second we use integral calculus to prove that the Brent-Salamin algorithm approximates $\pi$.	contribution 0.56	0.91	0.06	asserted 1.00
s2	Combining these results proves that the Borwein brothers' quartic algorithm also approximates $\pi$.	result 0.72	0.87	0.44	asserted 1.00
s3	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.	contribution 0.60	0.90	0.20	asserted 1.00
s4	Zweitens beweisen wir mit Integralrechnung, dass der Brent-Salamin-Algorithmus gegen $\pi$ konvergiert.	contribution 0.68	0.92	0.06	asserted 1.00
s5	Hieraus folgt, dass der Borwein-Algorithmus vierter Ordnung ebenfalls gegen $\pi$ konvergiert.	result 0.71	0.82	0.24	asserted 1.00
s6	Drittens beweisen wir die quadratische Konvergenz des Brent-Salamin-Algorithmus und somit auch die quartische Konvergenz des Borwein-Algorithmus.	contribution 0.51	0.93	0.28	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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arXiv listing abstract v3

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S sections
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original 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.

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1 arXiv paper cite this in 1 sentence

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 1 paper

We show that an iteration of the Borwein-Borwein quartic algorithm for $\pi$ is equivalent to two iterations of the Gauss-Legendre quadratic algorithm for $\pi$, in the sense that they produce exactly the same sequence of approximations to $\pi$ if performed using exact arithmetic.

The Borwein brothers, Pi and the AGM · 0.88

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