Unconditional Applicability of Lehmer’s Measure to the Two-Term Machin-like Formula for π

S. Abrarov, R. Siddiqui, R. Jagpal, B. Quine
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引用次数: 4

Abstract

Lehmer defined a measure depending on numbers beta_i used in a Machin-like formula for pi. When the beta_i are integers, Lehmer's measure can be used to determine the computational efficiency of the given Machin-like formula for pi. However, because the computations are complicated, it is unclear if Lehmer's measure applies when one or more of the beta_i are rational. In this article, we develop a new algorithm for a two-term Machin-like formula for pi as an example of the unconditional applicability of Lehmer's measure. This approach does not involve any irrational numbers and may allow calculating pi rapidly by the Newton-Raphson iteration method for the tangent function.
Lehmer测度对π两项类机公式的无条件适用性
Lehmer根据类似Machin的π公式中使用的数字beta\i定义了一个测度。当beta\i是整数时,Lehmer测度可以用来确定给定的类Machin公式对pi的计算效率。然而,由于计算很复杂,当一个或多个beta_i是有理的时,尚不清楚Lehmer的测度是否适用。在本文中,我们为π的两项类Machin公式开发了一个新的算法,作为Lehmer测度无条件适用性的例子。这种方法不涉及任何无理数,并且可以通过切线函数的Newton-Raphson迭代方法快速计算pi。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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