[Paper Review] Old and new algorithms for pi

Richard P. Brent|arXiv (Cornell University)|Mar 12, 2013

History and Theory of Mathematics3 citations

TL;DR

This paper corrects misconceptions in Prof. Adlaj's critique of the Brent-Salamin algorithm for computing π, arguing that while Gauss and Legendre laid foundational groundwork, the algorithm's efficiency is contingent on modern computational technology. It emphasizes that pre-computer-era mathematicians like Euler, Legendre, and Gauss would not have valued such algorithms for π, as they were impractical without fast multiplication and electronic computation, and highlights that elementary functions can be computed as efficiently as π using the arithmetic-geometric mean.

ABSTRACT

This is a letter to the editor concerning Semjon Adlaj's article "An eloquent formula for the perimeter of an ellipse", AMS Notices 59, 8 (2012), 1094-1099.

Motivation & Objective

• To clarify the historical attribution of the Brent-Salamin algorithm and correct the suggestion of naming it after Gauss and Euler.
• To argue that the algorithm's significance is contingent on modern computational technology, not inherent mathematical superiority.
• To demonstrate that pre-computer-era mathematicians like Gauss, Legendre, and Euler would not have appreciated such algorithms due to the lack of fast multiplication and electronic computation.
• To show that the arithmetic-geometric mean enables efficient evaluation of all elementary functions, not just π, up to a constant factor.
• To challenge the notion that computing π as a single constant is a meaningful or historically relevant computational goal.

Proposed method

• Analyzes historical sources, including Gauss’s unpublished notebook entry from May 1809, to assess whether Gauss viewed his identity as an algorithm for π.
• Compares the computational efficiency of the Brent-Salamin algorithm with classical Machin-like formulas based on arctangent series and binary splitting.
• Uses the Schönhage-Strassen algorithm for fast multiplication as a benchmark for modern computational feasibility.
• Applies the theory of arithmetic-geometric mean (AGM) to show that all elementary functions, including π, e^π, and π/e, can be computed with comparable speed.
• Evaluates the relevance of terms like 'Gauss-Legendre' and 'Gauss-Euler' via Google search frequency to argue against misleading nomenclature.
• Contrasts the computational context of 19th-century mathematics with modern algorithmic efficiency, emphasizing technological dependence.

Experimental results

Research questions

• RQ1 Why is the Brent-Salamin algorithm not historically accurate to name after Gauss and Euler, despite their foundational contributions?
• RQ2 How did the absence of fast multiplication algorithms affect the practicality of early π-computing methods?
• RQ3 What role did Gauss’s 1809 notebook entry play in the development of the Brent-Salamin algorithm, and how did Gauss interpret it?
• RQ4 Why would Euler, Legendre, and Gauss have found the Brent-Salamin algorithm unremarkable, given their historical context?
• RQ5 Can the arithmetic-geometric mean be used to compute all elementary functions with the same efficiency as π, and what does this imply for the significance of π-computation?

Key findings

• The Brent-Salamin algorithm is not a practical method for computing π without modern fast multiplication algorithms like Schönhage-Strassen.
• Machin-like formulas using arctangent series remain competitive with the Brent-Salamin algorithm when combined with binary splitting and fast multiplication.
• Gauss’s 1809 notebook entry likely represented an identity involving elliptic integrals, not an algorithm for computing π, as π appears only in the denominator.
• The term 'Gauss-Legendre' is commonly used for quadrature, not π algorithms, and 'Gauss-Euler' is mostly irrelevant in this context.
• The arithmetic-geometric mean enables the evaluation of all elementary functions, including π, e^π, and π/e, with the same asymptotic efficiency, up to a constant factor.
• The focus on computing π as a single constant is arguably misplaced, as the same computational power yields a broader class of constants.