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An iterative method for computing $π$ by argument reduction of the tangent function

Sanjar M. Abrarov, Rehan Siddiqui, Rajinder Kumar Jagpal, Brendan M. Quine

Abstract

In this work, we develop a new iterative method for computing the digits of $π$ by argument reduction of the tangent function. This method combines a modified version of the iterative formula for $π$ with squared convergence that we proposed in a previous work and a leading arctangent term from the Machin-like formula. The computational test we performed shows that algorithmic implementation can provide more than $17$ digits of $π$ per increment. Mathematica codes, showing the convergence rate for computing the digits of $π$, are presented.

An iterative method for computing $π$ by argument reduction of the tangent function

Abstract

In this work, we develop a new iterative method for computing the digits of by argument reduction of the tangent function. This method combines a modified version of the iterative formula for with squared convergence that we proposed in a previous work and a leading arctangent term from the Machin-like formula. The computational test we performed shows that algorithmic implementation can provide more than digits of per increment. Mathematica codes, showing the convergence rate for computing the digits of , are presented.
Paper Structure (11 sections, 128 equations)

This paper contains 11 sections, 128 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Machin-Like Formulas
  4. Tangent Function
  5. Arctangent Function
  6. Results and Discussion
  7. Modified Iteration
  8. Methodology
  9. Mathematica Codes and Description
  10. Rational Numbers
  11. Conclusions