Newton.sage

def Newton(g,x0,n):
    x = var('x')
    f(x) = g(x)
    N = x0
    for i in range(n):
        N = N - f(N)/f.diff(x)(N)
    return N

def Newton2(F,G,x0,y0,n):
    x = var('x')
    y = var('y')
    f(x,y) = F(x,y)
    g(x,y) = G(x,y)
    N = vector([x0,y0])
    for i in range(n):
        A = jacobian((f,g),(x,y))(N[0],N[1])
        N = N - A^(-1)*vector([f(N[0],N[1]),g(N[0],N[1])])
    return N