Comparative analysis of Algorithms to Calculate π

Presented by

Rohit Kumar

Trivia

• The Greek letter π, often spelled out pi in text, was adopted for the number from the Greek word for perimeter "περίμετρος", probably by William Jones in 1706 .

• According to the Bible, π = 3!

• “Pi Day” is celebrated on March 14th (3-14) which is also the Birthday of Albert Einstein.

• Current World Record : Chou Lu of China recited 67,890 decimal places of pi on 20 November 2005.

• The value of pi has been calculated to more than a trillion (1012) digits .

What is π ?

• Pi is the Ratio of any circle’s circumference to its diameter in Euclidean Geometry.

• Numerical value of π : 3.14159 26535 89793 23846 26433 83279…

• Pi is an irrational no and hence its decimal expansion never ends or repeats.

Calculating πBrief History

• Earliest known approximations(around 1900 BC) : Babylonia - 25/8 Egypt - 256/81 India - 339/108 ≈ 3.139 • Around 480, the Chinese mathematician Zu

Chongzhi gave the approximation π = 355/113 ≈ 3.141592.

• Around 1400 Madhava (India) calculated Pi correct to 11 decimal places using infinite series.

π in computer age

• Calculating billions of digits is possible using infinite series.

• The digits of π behave as Pseudo-Random numbers and hence are of interest for Cryptographic Applications.

• The Random number produced by π are of high ‘quality’.However the conjecture that π produces random numbers has still not been proved !!

1. Madhava of Sangamagrama

• Calculates the value of π correct up to 14 decimal places in 31 iterations.

Leibniz Formula :

•Calculates the value of π correct upto 3 decimal places in 10,000 iterations

Euler’s Formula :

• Calculates the value of π correct upto 3 decimal places in 10,000 iterations.

John Machin :

• Calculates the value of π correct up to 14 decimal places in 11 iterations.

Ramanujan Formula :

• Computes the value of correct up to 16 decimal places in 2 iterations.

•Adds 8 decimal places in every iteration.

Bailey–Borwein–Plouffe formula :

• Calculates the value of π correct up to 15 decimal places in 11 iterations,

• Can be used to calculate any arbitrary digit without calculating the previous digits.

Continued Fraction :

• Calculates the value of π correct upto 10 decimal places in 10,000 iterations

Infinite Product :

• Calculates the value of π correct upto 4 decimal places in 10,000 iterations