def brents(f, a_k, b_k):
	mflag = True
	a_k_1 = None
	b_k_1 = None
	b_k_2 = None

	m_k = (a_k + b_k)/2

	if f(a_k) == f(b_k_1) or f(b_k) == f(b_k_1):
		s_k = a_k * f(b_k) - b_k*f(a_k)/(f(b_k) - f(a_k))
	else:
		s_k = (a_k * f(b_k) * f(b_k_1) / ((f(a_k)-f(b_k))*(f(a_k)-f(b_k)))
			+ b_k * f(a_k) * f(b_k_1) / ((f(b_k)-f(a_k))*(f(b_k)-f(b_k_1)))
			+ b_k_1 * f(a_k) * f(b_k) / ((f(b_k_1)-f(a_k))*(f(b_k_1)-f(b_k)))
			)

	tp = (3*a_k + b_k)/4

	# test if in interval (no guarentee a_k>b_k, so I did it this way)
	if (s_k < tp and s_k < b_k) or (sk > tp and sk > b_k):
		c_k = m_k
		mflag = 1
	elif mflag and abs(s_k-b_k) > abs(b_k-b_k_1)/2:
		c_k = m_k
		mflag = 1
	elif not mflag and abs(s_k-b_k)>abs(b_k_1-b_k_2)/2
		c_k = m_k
		mflag = 1
	else:
		c_k = s_k
		mflag = 0

	a_k_1 = a_k
	b_k_1 = b_k
	b_k_2 = b_k_1

	if f(a_k)*f(c_k) < 0:
		a_k, b_k = a_k, c_k

	if f(c_k)*f(c_k) < 0:
		a_k, b_k = c_k, b_k

	if abs(f(a_k)) < abs(f(b)):
		a_k, b_k = b_k, a_k