Gauss-Legendre Algorithm


Gauss-Legendre algorithm

1. Initial value setting; 

    a = 1    b = 1 / SqRt(2)    t = 1/4    x = 1

2. Repeat the following statements until the difference of a and b is within the desired accuracy; 

    y = a

    a = (a+b) / 2

    b = SqRt(b*y)

    t = t − x * (y−a)²

    x = 2 * x

3. Pi is approximated with a, b and t as; 

    π = ((a+b)²) / (4*t)

The algorithm has second order convergent nature. Then if you want to calculate up to n digits, iteration count of the order log2(n) is sufficient. E.g. 19 times for 1 million decimal digits, 31 times for 3.2 billon decimal digits.

Note: The text is from the program SuperPi.Exe. The equations were converted to standard character set by Harry Smith.

Return to Computing Pi
Return to Harry's Home Page

This page accessed times since October 20, 2004.
Page created by: [email protected]
Changes last made on Sunday, 17-Jul-05 11:03:44 PDT
Hosted by www.Geocities.ws

1