Newton’s Method for Real Equations

Much of the complexity which we have seen in Newton’s method for complex polynomials is known to be closely linked to the underlying complex analytic structure. Thus, it appears to be an interesting question to ask what the situation is like for systems of real equations. Note, however, that a complex analytic map ℂ∋x(x) can be regarded as a function of two real variables in a canonical way. viz. ƒ(x) — (ƒ1(x1, x2), ƒ2(x1, x2)) such that the Cauchy-Riemann differential equations are satisfied: (7.1) $$ |\frac{{\partial f_1 }} {{\partial x_1 }} = \frac{{\partial f_2 }} {{\partial x_2 }},\frac{{\partial f_1 }} {{\partial x_2 }} = - \frac{{\partial f_2 }} {{\partial x_1 }} $$