from scipy.optimize import minimize_scalar


def f(x, a = 350, b = 2, f0 = 110):
    return sum([a*(x[i]**2 - x[i+1])**2 + b*(x[i] - 1)**2 for i in range(0, len(x)-1)]) + f0
   

def next_step(x, step, index):
    x[index] += step
    return x


def next_value(h):
        return f(next_step(x, step, index))


def search(f, x0, e):
    x = x0
    while True:
        for i in range(len(x)):
            x_old = x
            x = next_step(x, minimize_scalar(next_value).x, i) 
        if sum(x - x_old) * 1.0/len(x) < e:
            break
    return x


def pattern_search(f, x0, e, l):
    while True:
        x1, x2, x3, x4 = x0, exploring_search(f, x0), x1 + l*(x2-x1), exploring_search(f, x3)     
        if np.linalg.norm((x2 - x4))/len(x0) < e:
            break
        else:
            x0 = x2
    return x2, f(x2)
    

def gradient(x, a = 350, b = 2):
    return np.array([2*a*x[0], a + b*x[1]*2 - 2*b])
    

def gradient_descent(x, nm, eps, max_step):
	def reverce(a): 
		return f(x - a*gradient(x))
    result = minimize_scalar(reverce(a), bounds = (-max_step, max_step), method = nm)
    new_x = x - result.x*gradient(x)
    if f(x) - f(new_x) < eps:
        return (f(x), x)
    return gradient_descent(new_x, nm)