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.

翻译：我们考虑了Jonathan和Peter Borweins对美元和基本功能的高精度计算的一些贡献。 特别是参考他们的书《 Pi 和 AGM 》 (Wiley,1987年)。 这里的“ AGM”是高尔斯和图例的算数几何平均值。 由于AGM是四倍结合的, 它可以与快速乘法相结合, 给出 $ 和 位元 的快速算法, 以及更广义的基本函数。 这些算法以几乎线性的时间 $O( M)( n)\ log n) 运行, 其中, $M( n) 是美元乘以倍增。 我们概述了在 Pi 和 AGM 中给出的一些结果和算法, 并提出了一些相关( 但新) 的结果。 特别是, 我们改进了某些四倍和四倍一致的算法的算法的快速算法, 比如高尔斯- Legendre 算法。 我们显示, 它们相当于Borwe\ rigalalalalalalalalation $。