Optimum step size control for Newton-Raphson solution of nonlinear vector equations

The problem of solving a non-linear vector equation of the form
x = f(\theta)
for a
\theta
which corresponds to a given
x
is attacked by Newton-Raphson. To keep the lengthy evaluations of partial derivatives at a minimum, each step is optimized to get the search as close to the solution as possible. Substantial savings in computation time are realized and solutions can be obtained efficiently even when the initial guess is not close to the ultimate answer.