博温兄弟派和AGM
The Borwein brothers, Pi and the AGM
摘要：
我们考虑Jonathan和Peter Borweins对$π$和初等函数的高精度计算的一些贡献，特别参考他们的书“Pi and the AGM”（Wiley，1987）。这里“AGM”是高斯和勒让德的算术-几何平均。因为AGM二次收敛，所以它可以与快速乘法算法相结合，给出$π$的$n$位计算的快速算法，更一般地，给出初等函数的快速算法.这些算法以几乎线性的时间$O(M(n)\log n)$运行，其中$M(n)$是$n$位乘法的时间。我们概述了在Pi和AGM中给出的一些结果和算法，并提出了一些相关的（但新的）结果。特别是，我们改进了一些二次和四次收敛的算法$π$，如高斯-勒让德算法公布的误差界。我们表明，一个迭代的Borwein-Borwein四次算法的$π$是等价的两个迭代的高斯-勒让德二次算法的$π$，在这个意义上，他们产生完全相同的序列的近似$π$，如果使用精确的算术。
英文摘要：
We consider some of Jonathan and Peter Borweins' contributions to the high-precision computation of $π$ 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 $π$, 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 $π$, such as the Gauss-Legendre algorithm. We show that an iteration of the Borwein-Borwein quartic algorithm for $π$ is equivalent to two iterations of the Gauss-Legendre quadratic algorithm for $π$, in the sense that they produce exactly the same sequence of approximations to $π$ if performed using exact arithmetic.
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