import math

import cmath

def dft(p):

	"""Quadratic formula from the definition."""

	N = len(p)

	w = cmath.exp(-2*math.pi*1j/N)  # primitive N-th square root of 1

	x = [0]*N

	y = [0]*N

	for k in range(N):

		x[k] = w**k

		for n in range(N):

			y[k] += p[n] * x[k]**n

	return y

def compute_inverse(N1, N2):

	for i in range(N2):

		if N1*i % N2 == 1:

			return i

	raise ValueError("Failed to find an inverse to {0} mod {1}.".format(N1, N2))

def prime_fft(p, divisors):

	"""https://en.wikipedia.org/wiki/Prime-factor_FFT_algorithm"""

	if len(divisors) == 1:

		return dft(p)

	N = len(p)

	N1 = divisors[0]

	N2 = N//N1

	N1_inv = compute_inverse(N1, N2)

	N2_inv = compute_inverse(N2, N1)

	ys = []

	for n1 in range(N1):  # compute rows

		p_ = [p[(n2*N1+n1*N2) % N] for n2 in range(N2)]

		ys.append(prime_fft(p_, divisors[1:]))

	for k2 in range(N2):  # compute cols

		p_ = [row[k2] for row in ys]

		y_ = dft(p_)

		for (yi, row) in zip(y_, ys):  # update col

			row[k2] = yi

	# remap and output

	res = [0]*N

	for k1 in range(N1):

		for k2 in range(N2):

			res[(k1*N2*N2_inv+k2*N1*N1_inv) % N] = ys[k1][k2]

	return res

# arr = [9,8,5,3,8,4,9,7,0,9,5,6,6,2,4]

# print(dft(arr))

# print()

# print(prime_fft(arr,[3,5]))

"""

def fft(x):

    N = len(x)

    if N <= 1: return x

    even = fft(x[0::2])

    odd = fft(x[1::2])

    T = [cmath.exp(-2j*math.pi*k/N)*odd[k] for k in range(N//2)]

    return [even[k] + T[k] for k in range(N//2)] + \

           [even[k] - T[k] for k in range(N//2)]

def fft_mixed(x, r1, r2):

	A = dft(x)

	for k0 in range(0, r2):

		pass

def bit_reverse(x, width):

	y = 0

	for i in range(width):

		y <<= 1

		y |= x&1

		x >>= 1

	return y

def bit_reverse_seq(x):

	n = len(x)

	width = round(math.log2(n))

	return [x[bit_reverse(i, width)] for i in range(n)]

def fft2(x):

	y = bit_reverse_seq(x)

	n = len(x)

	omegas = [(-2*math.pi*1j/n)**i for i in range(0, n)]

	b = 1

	while b<n:

		for j in range(0, n, 2*b):

			for k in range(0, b):

				alpha = omegas[n*k//(2*b)]

				u = y[j+k]

				t = alpha*y[j+k+b]

				y[j+k] = u+t

				y[j+k+b] = u-t

		b *= 2

	return y

"""

# GNU GPLv3, see LICENSE

from unittest import TestCase

from ..fft import *

class TestFFT(TestCase):

	def test_dft(self):

		self.assertEqual(dft([0]), [0+0j])

		self.assertEqual(dft([1]), [1+0j])

		self.assertEqual(dft([2]), [2+0j])

		all(self.assertAlmostEqual(a, b) for (a, b) in zip(dft([3, 1]), [4+0j, 2+0j]))

		all(self.assertAlmostEqual(a, b) for (a, b) in zip(dft([3, 1, 4]), [8+0j, 0.5+2.59807621j, 0.5-2.59807621j]))

		all(self.assertAlmostEqual(a, b) for (a, b) in zip(dft([3, 1, 4, 1]), [9+0j, -1+0j, 5+0j, -1+0j]))

		all(self.assertAlmostEqual(a, b) for (a, b) in zip(

			dft([3, 1, 4, 1, 5]),

			[14+0j, 0.80901699+2.04087031j, -0.30901699+5.20431056j, -0.30901699-5.20431056j, 0.80901699-2.04087031j]

		))