# 
# Fast discrete cosine transform algorithms (Python)
# 
# Copyright (c) 2020 Project Nayuki. (MIT License)
# https://www.nayuki.io/page/fast-discrete-cosine-transform-algorithms
# 
# Permission is hereby granted, free of charge, to any person obtaining a copy of
# this software and associated documentation files (the "Software"), to deal in
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# 

import math


# DCT type II, unscaled. Algorithm by Byeong Gi Lee, 1984.
# See: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.118.3056&rep=rep1&type=pdf#page=34
def transform(vector):
	n = len(vector)
	if n == 1:
		return list(vector)
	elif n == 0 or n % 2 != 0:
		raise ValueError()
	else:
		half = n // 2
		alpha = [(vector[i] + vector[-(i + 1)]) for i in range(half)]
		beta  = [(vector[i] - vector[-(i + 1)]) / (math.cos((i + 0.5) * math.pi / n) * 2.0)
			for i in range(half)]
		alpha = transform(alpha)
		beta  = transform(beta )
		result = []
		for i in range(half - 1):
			result.append(alpha[i])
			result.append(beta[i] + beta[i + 1])
		result.append(alpha[-1])
		result.append(beta [-1])
		return result


# DCT type III, unscaled. Algorithm by Byeong Gi Lee, 1984.
# See: https://www.nayuki.io/res/fast-discrete-cosine-transform-algorithms/lee-new-algo-discrete-cosine-transform.pdf
def inverse_transform(vector, root=True):
	if root:
		vector = list(vector)
		vector[0] /= 2
	n = len(vector)
	if n == 1:
		return vector
	elif n == 0 or n % 2 != 0:
		raise ValueError()
	else:
		half = n // 2
		alpha = [vector[0]]
		beta  = [vector[1]]
		for i in range(2, n, 2):
			alpha.append(vector[i])
			beta.append(vector[i - 1] + vector[i + 1])
		inverse_transform(alpha, False)
		inverse_transform(beta , False)
		for i in range(half):
			x = alpha[i]
			y = beta[i] / (math.cos((i + 0.5) * math.pi / n) * 2)
			vector[i] = x + y
			vector[-(i + 1)] = x - y
		return vector