Recall that in Calculus I you saw that, given an initial guess x_o, you could approximate the root of f(x) = 0 by replacing f(x) by its first-order Taylor approximation centered at x_o and then solving the approximation for its root. This procedure is called Newton's method for functions of one variable. It turns out that Newton's method works for systems of multivariate equations as well. Consider solving the system of nonlinear equations

f (x, y) = 0

g (x, y) = 0

Suppose that the solution to the system of equations is (x_i, y_i). Newton's method is derived by replacing f (x₁, y₁) and g(x₁, y₁) in the system above by their first-order Taylor approximations centered at some initial guess (x₀, y₀), and then solving the resulting system of equations for (x_i, y_i). If the improved guess (x₁, y_i) is not sufficiently close to the root the method is then repeated using (x_i, y_i) as the initial guess to obtain then next iterate (x₂, y₂).

(a) Explain what the method is doing geometrically.

(b) Show that in general, one step of Newton's method on the above system results in the equations

x₁ = x₀ + {(gf_y - fg_y) / (f_xg_y - f_yg_x)},    y₁ = _y0 + {(fg_x - gf_x) / (f_xg_y - f_yg_x)},

where all functions and partial derivatives are evaluated at the initial guess (x_o, y_o).

(c) Perform two iterations of Newton's method, starting with the initial guess (x_o, y_o) = (2, 1), for the system with f (x, y) = y (1 - x²) and g (x , y) = 2 - xy. What is the exact solution to the system?

Are your approximations getting closer and closer to the true solution?