add FFT algorithm to be merged

Fast_Fourier_Transform.py

@@ -0,0 +1,56 @@
+#  Recursive Fast Fourier Transform (FFT) applied
+# to polynomial multiplication. The basic algorithm proceeds
+# as followes:
+# 1. Add n higher-order zero coefficients to A(x) and B(x)
+# 2. Evaluate A(x) and B(x) using FFT for 2n points
+# 3. Pointwise multiplication of point-value forms
+# 4. Interpolate C(x) using FFT to compute inverse DFT
+# FFT can perform Discrete Transform and Inverse Discrete
+# Transform (IDFT) in average O(n log n) time.
+
+from math import sin,cos,pi
+
+# Recursive function of FFT
+def fft(a):
+
+	n = len(a)
+
+	# if input contains just one element
+	if n == 1:
+		return [a[0]]
+
+	# For storing n complex nth roots of unity
+	theta = -2*pi/n
+	w = list( complex(cos(theta*i), sin(theta*i)) for i in range(n) )
+	
+	# Separe coefficients
+	Aeven = a[0::2]
+	Aodd = a[1::2]
+
+	# Recursive call for even indexed coefficients
+	Yeven = fft(Aeven)
+
+	# Recursive call for odd indexed coefficients
+	Yodd = fft(Aodd)
+
+	# for storing values of y0, y1, y2, ..., yn-1.
+	Y = [0]*n
+	
+	middle = n//2
+	for k in range(n//2):
+		w_yodd_k = w[k] * Yodd[k]
+		yeven_k = Yeven[k]
+		
+		Y[k]		 = yeven_k + w_yodd_k
+		Y[k + middle] = yeven_k - w_yodd_k
+	
+	return Y
+
+
+# Driver code
+if __name__ == '__main__':
+
+	a = [1, 2, 3, 4]
+	b = fft(a)
+	for B in b:
+		print(B)