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ConstantCalculus-CircleConstant-0.03
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/ ConstantCalculus::CircleConstant

NAME

ConstantCalculus::CircleConstant - Perl extension for the high accuracy calculation of circle constants using big numbers.

VERSION

Version 0.02

SYNOPSIS

  # Load the Perl module.
  use ConstantCalculus::CircleConstant;

Calculate the value of the circle constant Pi:

  # Declare the variable for Pi.
  my $pi = undef;

  # Set the number of places.
  my $places = 100;

  # Use the calculation method for Pi.
  $pi = pi_chudnovsky($places);
  print $pi . "\n";

  # Use the calculation method for Pi.
  $pi = pi_borwein25($places);
  print $pi . "\n";

  # Use the calculation algorithm for Pi directly.
  $pi = chudnovsky([
      Places => 100,     # Set the number of places to 100
      Terms => 9,        # Set the number of terms to 9 
      Precision => 115,  # Set the precision to 115
      Trim => 115,       # Trim the value of Pi to 115 places      
      Type => "pi"       # Calculate the value of Pi
  ]);
  print $pi . "\n";

Calculate the value of the circle constant Tau:

  # Declare the variable for Tau.
  my $tau = undef;

  # Set the number of places.
  my $places = 100;

  # Use the calculation method for Tau.
  my $tau = tau_chudnovsky($places);
  print $tau . "\n";

  # Use the calculation method for Tau.
  my $tau = tau_borwein25($places);
  print $tau . "\n";

  # Use the calculation algorithm for Pi directly.
  $tau = chudnovsky([
      Places => 100,     # Set the number of places to 100
      Terms => 9,        # Set the number of terms to 9 
      Precision => 115,  # Set the precision to 115
      Trim => 115,       # Trim the value of Tau to 115 places      
      Type => "tau"      # Calculate the value of Tau
  ]);
  print $tau . "\n";

Use BBP for calculation with respect to Pi:

  # Print the nth hexadecimal digit of Pi.
  my $nth = 1;
  print bbp_digit($nth, 14);  # Use precision 14

  # Print the nth hexadecimal digit of Pi upwards.
  my $nth = 1;
  print bbp_digits($nth, 32, 128);  # Output 32 digits and use precision 128

  # Print Pi using BBP.
  my $places = 1;
  print bbp_algorithm($places, 14); # Use precision 14

Use predefined values of Pi and Tau.

    # Print Pi with 1000 decimal places.
    print $PI_DEC . "\n";

    # Print Pi with 1000 hexadecimal places.
    print $PI_HEX . "\n";

The square brackets in the above subroutine calls means that the named arguments within the brackets are optional. Every named argument will be preset in the subroutine if not defined.

REQUIREMENT

CPAN bignum

DESCRIPTION

The circle constant is a mathematical constant. There are two variants of the circle constant. Most common is the use of Pi (π) for the circle constant. More uncommon is the use of Tau (τ) for the circle constant. The relation between them is τ = 2 π. There can be found other denotations for the well known name Pi. The names are Archimedes's constant or Ludolph's constant. The circle constant is used in formulas across mathematics and physics.

The circle constant is the ratio of the circumference C of a circle to its diameter d, which is π = C/d or τ = 2 C/d. It is an irrational number, which means that it cannot be expressed exactly as a ratio of two integers. In consequence of this, the decimal representation of a circle constant never ends in there decimal places having a infinite number of places, nor enters a permanently repeating pattern in the decimal places. The circle constant is also a not algebraic transcendental number.

Over the centuries scientists developed formulas for approximating the circle constant Pi. Chudnovsky's formula is one of them. A algorithm based on Chudnovsky's formula can be used to calculate an approximation for Pi and also for Tau. The advantage of the Chudnovsky formula is the good convergence. In contradiction the convergence of the Leibniz formula is quit bad.

The challenge in providing an algorithm for the circle constant is that all decimal places must be correct in terms of the formula. Based on the desired decimal place number or precision, the number of places must be correct. The provided algorithm takes care of this. At the moment the result of the implemented algorithm was checked against the known decimal places of Pi up to 10000 places.

APPROXIMATIONS OF PI AND TAU

Fractions such as 22/7 or 355/113 can be used to approximate the circle constant Pi, whereas 44/7 or 710/113 represent the approximation of the circle constant Tau.

At the moment Chudnovsky's formula is fully implemented in the module to calculate Pi as well as Tau. The algorithm from Borwein from 1989 is implemented for experimental purposes. The most popular formula for calculating the circle constant is the Leibniz formula.

IMPLEMENTATION

To be able to deal with large numbers pre decimal point and after the decimal point as needed, the Perl module bignum is used. The main subroutine argument is the number of places, which should be calculated for the circle constant.

If terms and precision is not given, both are estimated from the given number of places. This will result in a value of Pi, which is accurate to the requested places. If places, terms and/or precision is given, the behaviour of the algorithm can be studied with respect to terms and/or precision.

The number of iterations is calculated using the knowledge, that each iteration should result in e.g. 14 new digits after the decimal point. So the value for the calculation of the number of terms is set to e.g. 14. To make sure that reverse as less as possible digits are changed, the number of terms to calculated is uneven. So the sign of the term to add is negative after the decimal point.

To prevent rounding errors in the last digit, the precision is a factor of e.g. 14 higher than the requested number of places. The correct number of places is realised by truncating the calculated and possibly rounded value of Pi to the requested number of places.

EXAMPLES

Calculation of Pi

Example 1

  # Load the Perl module.
  use ConstantCalculus::CircleConstant;

  # Declare the variable for Pi.
  my $pi = undef;

  # Set the number of places.
  my $places = 100;  

  # Calculate Pi.
  $pi = chudnovsky($places);
  print $pi . "\n";

Example 2

  # Load the Perl module.
  use ConstantCalculus::CircleConstant;

  # Declare the variable for Pi.
  my $pi = undef;

  # Set the number of places.
  my $places = 100;  

  # Set the number of terms.
  my $terms = 50;  

  # Set the precision.
  my $precision = 115;  

  # Set the value for trim.
  my $trim = 115;  

  # Calculate Pi.
  $tau = chudnovsky(
      Places => $places, 
      Terms => $terms,      
      Precision => $precision,
      Trim => $trim, 
      Type => "pi"  
  );
  print $pi . "\n";

Example 3

  # Load the Perl module.
  use ConstantCalculus::CircleConstant;

  # Create an alias for the module method.
  *pi = \&pi_borwein25;

  # Declare the variable for Pi.
  my $pi = undef;

  # Set the number of places.
  my $places = 100;  

  # Calculate and print the value Pi.
  $pi = pi($places);
  print $pi . "\n";

Example 4

  # Print decimal Pi with 100 places.
  my $pi = $PI_DEC;
  $pi = substr($pi, 0, 102); 
  print $pi . "\n";

Example 5

  # Print hecadecimal Pi with 100 places.
  my $pi = $PI_HEX;
  $pi = substr($pi, 0, 102); 
  print $pi . "\n";

Calculation of Tau

Example 1

  # Load the Perl module.
  use ConstantCalculus::CircleConstant;

  # Declare the variable for Tau.
  my $tau = undef;

  # Set the number of places.
  my $places = 100;  

  # Calculate Tau.
  $tau = chudnovsky($places);
  print $tau . "\n";

Example 2

  # Load the Perl module.
  use ConstantCalculus::CircleConstant;

  # Declare the variable for Tau.
  my $tau = undef;

  # Set the number of places.
  my $places = 100;  

  # Set the number of terms.
  my $terms = 50;  

  # Set the precision.
  my $precision = 115;  

  # Set the value for trim.
  my $trim = 115;  

  # Calculate Tau.
  $tau = chudnovsky(
      Places => $places, 
      Terms => $terms,      
      Precision => $precision,
      Trim => $trim, 
      Type => "pi"  
  );
  print $tau . "\n";

Example 3

  # Load the Perl module.
  use ConstantCalculus::CircleConstant;

  # Create an alias for the module method.
  *tau = \&tau_borwein25;

  # Declare the variable for Pi.
  my $tau = undef;

  # Set the number of places.
  my $places = 100;  

  # Calculate and print the value Pi.
  $tau = tau($places);
  print $tau . "\n";

MODULE METHODS

Main Methods

chudnovsky_algorithm()

Implementation of the Chudnovsky formula.

borwein25_algorithm()

Implementation of the Borwein 25 formula.

pi_borwein25()

Calculate Pi with the Borwein 25 algorithm.

tau_borwein25()

Calculate Tau with the Borwein 25 algorithm.

pi_chudnovsky()

Calculate Pi with the Chudnovsky algorithm.

tau_chudnovsky()

Calculate Tau with the Chudnovsky algorithm.

bbp_algorithm()

Apply the BBP algorithm.

bbp_digits()

Calculate as much as possible hexadecimal digits.

bbp_digit()

Calculate one hexadecimal digit.

S()

Calculate the S terms of the BBP algorithm

Other Methods

factorial()

The subroutine calculates the factorial of given number.

sqrtroot()

The subroutine calculates the square root of given number.

modexp()

The subroutine returns the result of a modular exponentiation. A modular exponentiation is an exponentiation applied over a modulus. The result of the modular exponentiation is the remainder when an integer b (base) is exponentiated by e (exponent) and divided by a positive integer m (modulus).

estimate_terms()

Estimates the terms or iterations to get the correct number of place.

truncate_places()

Truncate the number of places to a given value.

MODULE EXPORT

Generic export

  • chudnovsky_algorithm

  • borwein25_algorithm

  • pi_borwein25

  • tau_borwein25

  • pi_chudnovsky

  • tau_chudnovsky

  • bbp_algorithm

  • bbp_digits

  • bbp_digit

  • S

  • $PI_DEC

  • $PI_HEX

Export on demand

  • factorial

  • sqrtroot

  • modexp

  • estimate_terms

  • truncate_places

LIMITATIONS

Limitations are not known yet.

BUGS

Bugs are not known yet.

NOTES

The implemented chudnovsky algorithm is used in a representation where the well known terms are optimised for calculation.

It seems that the Perl builtin function sqrt() cannot used in general for determining the value of Pi or Tau with respect to some calculation formulas.

Chudnovsky's formula is working using the Perl builtin function sqrt(). In contradiction Borwein25's formula fails in calculation using the Perl builtin function sqrt().

Using a coded user defined subroutine for calculating of a square root, Borwein25's could be used for the calculation.

OPEN ISSUE

Further calculations with higher precision are outstanding to check the accuracy of the correctness of the last digits of the calculated circle constant.

SEE ALSO

Programming informations

Mathematical informations

  • Sources w.r.t. the circle constant Pi

  • Sources w.r.t. the circle constant Tau

  • Resources about the Leibniz formula

  • Resources about the Chudnovsky formula

  • Resources about Borwein's formulas

Bibliography

  • David H. Bailey, The BBP Algorithm for Pi, September 17, 2006

  • David H. Bailey, A catalogue of mathematical formulas involving π, with analysis, December 10, 2021

  • David H. Bailey, Jonathan M. Borwein, Peter B. Borwein and Simon Plouffe, The Quest for Pi, June 25, 1996

AUTHOR

Dr. Peter Netz, <ztenretep@cpan.org>

COPYRIGHT AND LICENSE

Copyright (C) 2022 by Dr. Peter Netz

This library is free software; you can redistribute it and/or modify it under the same terms of The MIT License. For more details, see the full text of the license in the attached file LICENSE in the main module folder. This program is distributed in the hope that it will be useful, but without any warranty; without even the implied warranty of merchantability or fitness for a particular purpose.