Time-lag in Derivative Convergence Time-lag in Derivative Convergence for Fixed Point Iterations
(2004)
In an earlier study it was proven and experimentally confirmed on a 2D Euler code
that fixed point iterations can be differentiated to yield first and second order derivatives of
implicit functions that are defined by state equations. It was also asserted that the resulting
approximations for reduced gradients and Hessians converge with the same R-factor as the
underlying fixed point iteration.
A closer look reveals now that nevertheless these derivative values lag behind the functions
values in that the ratios of the corresponding errors grow proportional to the iteration counter
or its square towards infinity. This rather subtle effect is caused mathematically by the
occurrence of nontrivial Jordan blocks associated with degenerate eigenvalues. We elaborate
the theory and report its confirmation through numerical experiments.