Code to estimate pi

see above image.

Please help for both calculations.

thanks

[Image Analyst]

John, somehow the formula image got deleted. Can you put it back?

[Rena Berman]

(Answers dev) Restored edit

Answer 1

[Star Strider]

Use the MATLAB vector capabilities:

N = 1E+8;                                                           % Number Of Terms
pie1 = sqrt(sum(6./[1:N].^2))
pie2 = sqrt(8 + sum(16./((2*[1:N] - 1).^2 .* (2*[1:N] + 1).^2)))

pie1 =
           3.1415926440655

pie2 = 
          3.14159265358975

Answer 2

[Image Analyst]

Here's code that gets to pi even faster. Dubbed the world's ugliest formula by mathematicians, it's so accurate after one iteration that MATLAB can't even detect a difference after just one term! Very interesting article if you want to read it. If someone can show me how to get the true difference, which is less than 1E-16, I'd appreciate it. By the way, most of the mathematicians agreed on which equations were beautiful and which were ugly, with Euler's identity, 1+eiπ=0, consistently rated the most attractive equation in the lot.

% Computes Ramanujan's formula for pi.
% http://www.scientificamerican.com/article/equations-are-art-inside-a-mathematicians-brain/
% On the bottom of the heap of beautiful formulas, mathematicians consistently rated 
% Srinivasa Ramanujan's infinite series for estimating pi as the most ugly.

% Problem: convergence it so fast that double precision is not
% precise enough to show the convergence curve.

clc;    % Clear the command window.
close all;  % Close all figures.
clear;  % Erase all existing variables. Or clearvars if you want.
workspace;  % Make sure the workspace panel is showing.
format long g;
format compact;
fontSize = 24;

% Compute the sum for a bunch of different number of terms
% in the series to see the shape of the error curve
% as it converges towards the true value of pi.
for kMax = 1 : 10 
  thisSum = 0;
  for k = 0 : kMax
    numerator = factorial(4 * k) * (1103 + 26390*k);
    denominator = factorial(k)^4 * 396^(4*k);
    thisTerm = numerator / denominator;
    thisSum = thisSum + thisTerm;
  end
  theConstant = 2 * sqrt(2) / 9801;
  oneOverPi = theConstant * thisSum;
  estimatedPi(kMax) = 1 / oneOverPi
  theError(kMax) = estimatedPi(kMax) - pi
end

% Plot the value of estimated pi.
subplot(2, 1, 1);
plot(estimatedPi, 'b*-', 'LineWidth', 2);
grid on;
title('Estimated \pi', 'FontSize', fontSize);
xlabel('Number of Terms in the Sum', 'FontSize', fontSize);

% Plot the value of the error (difference between true pi and estimated pi).
subplot(2, 1, 2);
plot(theError, 'b*-', 'LineWidth', 2);
grid on;
title('Error', 'FontSize', fontSize);
xlabel('Number of Terms in the Sum', 'FontSize', fontSize);

% Set up figure properties:
% Enlarge figure to full screen.
set(gcf, 'Units', 'Normalized', 'OuterPosition', [0 0 1 1]);
% Get rid of tool bar and pulldown menus that are along top of figure.
set(gcf, 'Toolbar', 'none', 'Menu', 'none');
% Give a name to the title bar.
set(gcf, 'Name', 'Estimation of pi', 'NumberTitle', 'Off')