The Borwein brothers, Pi and the AGM

TL;DR

This paper reviews Borwein brothers' work on high-precision pi computation using the AGM, introduces new results, and compares different algorithms' convergence and efficiency.

Contribution

It provides improved error bounds for pi algorithms and establishes equivalence between Borwein-Borwein quartic and Gauss-Legendre quadratic methods.

Findings

Improved error bounds for pi algorithms.

Equivalence of Borwein-Borwein quartic and Gauss-Legendre quadratic algorithms.

Algorithms run in almost linear time with respect to bit-length.

Abstract

We consider some of Jonathan and Peter Borweins' contributions to the high-precision computation of $\pi$ and the elementary functions, with particular reference to their book "Pi and the AGM" (Wiley, 1987). Here "AGM" is the arithmetic-geometric mean of Gauss and Legendre. Because the AGM converges quadratically, it can be combined with fast multiplication algorithms to give fast algorithms for the $n$-bit computation of $\pi$, and more generally the elementary functions. These algorithms run in almost linear time $O(M(n)\log n)$, where $M(n)$ is the time for $n$-bit multiplication. We outline some of the results and algorithms given in Pi and the AGM, and present some related (but new) results. In particular, we improve the published error bounds for some quadratically and quartically convergent algorithms for $\pi$, such as the Gauss-Legendre algorithm. 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.