论文标题
Lehmer的无条件适用于$π$的两项类似机械的公式
Unconditional applicability of Lehmer's measure to the two-term Machin-like formula for $π$
论文作者
论文摘要
Lehmer定义了一个度量$$μ= \ sum \ limits_ {j = 1}^j \ frac {1} {\ log_ {10} \ left（\ left |β_j\ loft |β_j\ right | \ right）}，$β_j$在$β_j$中可能是全体数字或$β_j$的$β_j$，或$β_j$是$ palional in Machin like $ $ $ upe $ cule $。当$β_J$是整数时，Lehmer的度量可用于确定给定的机器样公式的计算效率。但是，由于计算很复杂，因此尚不清楚当$β_j$中的一个或多个合理时，莱默的度量是否适用。在本文中，我们开发了一种针对$π$的两届机械式公式的新算法，以此作为莱默度量无条件适用性的一个例子。这种方法不涉及任何非理性数字，并且可以通过牛顿$ - $ raphson迭代方法来迅速计算$π$。
Lehmer defined a measure $$ μ=\sum\limits_{j=1}^J\frac{1}{\log_{10}\left(\left|β_j\right|\right)}, $$ where the $β_j$ may be either integers or rational numbers in a Machin-like formula for $π$. When the $β_j$ are integers, Lehmer's measure can be used to determine the computational efficiency of the given Machin-like formula for $π$. However, because the computations are complicated, it is unclear if Lehmer's measure applies when one or more of the $β_j$ are rational. In this article, we develop a new algorithm for a two-term Machin-like formula for $π$ as an example of the unconditional applicability of Lehmer's measure. This approach does not involve any irrational numbers and may allow calculating $π$ rapidly by the Newton$-$Raphson iteration method for the tangent function.
