We are working on the solution of large non-linear PDE (say the Navier-Stokes equation) which we solve using Newton's method with an analytical formulation of the jacobian. For very large systems, we are interested in optimally choosing the tolerance of our iterative linear solver to maximise our use of the Newton's method quadratic convergence rate, while ensuring that we do not oversolve each linear system. I have read that a good approach to achieve this is the Eisenstat-Walker method which is described in the following paper: https://epubs.siam.org/doi/10.1137/0917003
However, even after reading through the paper, I am unable to clearly identify the algorithm to implement. My math knowledge on this type of analysis is limited.
So my understanding of this is difficult. In the case of the newton method, $s_k$ should always be equal to one if we do a full step? How then do we chose the initial tolerance? And I am unsure on to chose $\theta$ within the given interval?
I really am unsure how to translate the results of this article, which are very interesting for our applications, into a clear algorithm...