Consider the following ordinary differential equation (ODE): \frac{d u}{d t}=g(u) To solve this numerically, you can use the backward Euler method, for some time step At > 0 \frac{u^{n+1}-u^{n}}{\Delta t}=g\left(u^{n+1}\right)

The numerical result from this process is the sequence uº, u', u², ..., which can be interpreted as an approximation to the exact solution sampled at times 0, delta t,2 delta t, . .. (a) If g(u) is a general nonlinear function and is differentiable, write down an iteration which determines un+1 from Newton's method. (b) The convergence of Newton's method depends on the choice of the initial guess. What would be a sensible choice for an initial guess?