The following formula can be used to calculate the pi value

The following formula can be used to calculate the pi value. Set a = 1, b = 1/sqrt(2), t = 1/4 and x = 1 Repeat the following commands until the difference between a and b is within some desired accuracy: y = a a = (a + b)/2 b = sqrt(b*y) t = t - x*(y - a)^2 x = 2*x From the resulting values of a, b and t, an estimate of pi is Pi_est = ((a + b)^2)/(4*t) How many repeats are needed to estimate pi to an accuracy of 1e-8? Write a function that ask for the accuracy (as an input) and output the number of iteration used to estimate the pi value according to that accuracy.

Solution

function [n,pi_es]=pi_est(TOL)
a=1;
b=1/sqrt(2);
t=1/4;
x=1;n=0;
while(abs(a-b)>TOL)
    n=n+1;
    y=a;
    a=(a+b)/2;
    b=sqrt(b.*y);
    t=t-x.*((y-a).^2);
    x=2.*x;
end
pi_es=((a+b).^2)./(4.*t);

>> [k,p]=pi_est(1e-8)

k =

     3

p =

    3.1416