Jacobi-Newton iteration

To solve Eq. (4.86) we employ the Jacobi-Newton iteration technique, which proceeds iteratively in an alternating sequence of local and global minimization steps. Let p be the local density at lattice site i in the A th local and the /th global minimization step. A local estimate for the corresponding minimum value of fi is obtained via Newton s method (see Eq. (D.6)] that... 

There are iterative methods (e.g., Jacobi, Gauss-Seidel, Newton) whose purpose is simply to provide solutions for the steady-state equations, others (e.g., Euler and its improved versions) aim to give trajectories. Cycling will be felt as a disagreeable iteration artifact in the first case, as an indication of a probably cyclic trajectory in the second case. The relation between the behavior in a simple iteration method (e.g., Jacobi) and the real trajectory is interesting, if not simple. Consider, for instance, a simple negative loop comprising three inhibitory elements ... 

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