The Mathematica Journal


Sanjar M. Abrarov, Rehan Siddiqui, Rajinder K. Jagpal, Brendan M. Quine
Published May 24, 2021


Lehmer defined a measure

where the may be either integers or rational numbers in a Machin-like formula for . When the are integers, Lehmers measure can be used to determine the computational efficiency of the given Machin-like formula for . However, because the computations are complicated, it is unclear if Lehmers measure applies when one or more of the are rational. In this article, we develop a new algorithm for a two-term Machin-like formula for as an example of the unconditional applicability of Lehmers measure. This approach does not involve any irrational numbers and may allow calculating rapidly by the NewtonRaphson iteration method for the tangent function. Read More »

A Tutorial

Peyton Cook
Published March 3, 2021
This article is intended to help students understand the concept of a coverage probability involving confidence intervals. Mathematica is used as a language for describing an algorithm to compute the coverage probability for a simple confidence interval based on the binomial distribution. Then, higher-level functions are used to compute probabilities of expressions in order to obtain coverage probabilities. Several examples are presented: two confidence intervals for a population proportion based on the binomial distribution, an asymptotic confidence interval for the mean of the Poisson distribution, and an asymptotic confidence interval for a population proportion based on the negative binomial distribution. Read More »