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</p> " data-medium-file="https://i1.wp.com/dogdogfish.com/wp-content/uploads/2014/05/rubbish.png?fit=300%2C226" data-large-file="https://i1.wp.com/dogdogfish.com/wp-content/uploads/2014/05/rubbish.png?fit=676%2C509" class="size-medium wp-image-169" src="https://i1.wp.com/dogdogfish.com/wp-content/uploads/2014/05/rubbish.png?resize=300%2C226" alt="A Circle in a Square" srcset="https://i1.wp.com/dogdogfish.com/wp-content/uploads/2014/05/rubbish.png?w=812 812w, https://i1.wp.com/dogdogfish.com/wp-content/uploads/2014/05/rubbish.png?resize=300%2C226 300w" sizes="(max-width: 300px) 100vw, 300px" data-recalc-dims="1" />

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 #!/usr/bin/python    import numpy as np  import math  import matplotlib.pyplot as plt    """  Calculate pi using Monte-Carlo Simulation  """    """  First - the maths:  A circle has area Pi*r^2  A square wholly enclosing above circle has area 4r^2  If we randomly generate points in that square we'd expect the ratio of points in the square/points in the circle to equal the area of the square divided by the circle.  By that logic n_in_sq/n_in_cir = 4/Pi and so Pi = (4 * n_in_cir)/n_in_sq  """    class pi_calculator(object):        def __init__(self, radius, iterations):          self.radius = radius          self.iterations = iterations          self.square = square(radius)          self.circle = circle(radius)        def scatter_points(self):          for _ in range(self.iterations):              point_x, point_y = ((2*self.radius) * np.random.random_sample(2)) - self.radius              self.square.increment_point_count(point_x, point_y)              self.circle.increment_point_count(point_x, point_y)        def return_pi(self):          return (4.0*self.circle.return_point_count())/self.square.return_point_count()        def calculate_accuracy(self, calc_pi):          absolute_error = math.pi - calc_pi          percent_error = 100*(math.pi - calc_pi)/math.pi          return (absolute_error, percent_error)        def return_iterations(self):          return self.iterations    class square(object):        def __init__(self, radius):          self.radius = radius          self.lower_x = -radius          self.lower_y = -radius          self.upper_x = radius          self.upper_y = radius          self.point_count = 0          def in_square(self, point_x, point_y):          return (self.upper_x > point_x > self.lower_x) and (self.upper_y > point_y > self.lower_y)          def increment_point_count(self, point_x, point_y, increment = 1):          if self.in_square(point_x, point_y):              self.point_count += increment        def return_point_count(self):          return self.point_count    class circle(object):        def __init__(self, radius):          self.radius = radius          self.point_count = 0        def in_circle(self, point_x, point_y):          return point_x**2 + point_y**2 < self.radius**2        def increment_point_count(self, point_x, point_y, increment=1):          if self.in_circle(point_x, point_y):              self.point_count += increment        def return_point_count(self):          return self.point_count      if __name__ == '__main__':      axis_values = []      pi_values = []      absolute_error_values = []      percent_error_values = []      for _ in range(1,3000,30):          pi_calc = pi_calculator(1, _)          pi_calc.scatter_points()          print "Number of iterations: %d    Accuracy: %.5f" % (pi_calc.return_iterations(), math.fabs(pi_calc.calculate_accuracy(pi_calc.return_pi())[0]))          axis_values.append(_)          pi_values.append(pi_calc.return_pi())          absolute_error_values.append(math.fabs(pi_calc.calculate_accuracy(pi_calc.return_pi())[0]))          percent_error_values.append(math.fabs(pi_calc.calculate_accuracy(pi_calc.return_pi())[1]))        improvement_per_iteration = [absolute_error_values[index] - absolute_error_values[index-1] for index, value in enumerate(absolute_error_values) if index > 0]      fig = plt.figure()      fig.suptitle('Calculating Pi - Monte Carlo Method')      ax1 = fig.add_subplot(2,2,1)      ax2 = fig.add_subplot(2,2,2)      ax3 = fig.add_subplot(2,2,3)      ax4 = fig.add_subplot(2,2,4)      plt.subplots_adjust(wspace=0.3, hspace=0.3)      ax1.set_xticklabels([str(entry) for entry in axis_values[::len(axis_values)/5]], rotation=30, fontsize='small')      ax1.set_xlabel('Iterations')      ax1.set_ylabel('Calculated value of Pi')      ax1.plot(pi_values, 'k')      ax1.plot([math.pi for entry in axis_values], 'r')      ax2.set_ylabel('Absolute error')      ax2.set_xticklabels([str(entry) for entry in axis_values[::len(axis_values)/5]], rotation=30, fontsize='small')      ax2.set_xlabel('Iterations')      ax2.plot(absolute_error_values, 'k', label="Total Error")      ax3.set_ylabel('Absolute percentage error (%)')      ax3.set_xticklabels([str(entry) for entry in axis_values[::len(axis_values)/5]], rotation=30, fontsize='small')      ax3.set_xlabel('Iterations')      ax3.plot(percent_error_values, 'k', label="Percent Error")      ax4.set_ylabel('Absolute improvement per iteration')      ax4.set_xticklabels([str(entry) for entry in axis_values[::len(axis_values)/5]], rotation=30, fontsize='small')      ax4.set_xlabel('Iterations')      ax4.plot(improvement_per_iteration, 'k', label="Absolute change")      plt.savefig('pi_calculation.png')      plt.show()