Thus, within 1 or 2 refinement iterations, the 1-step errors can grow by many orders of magnitude, resulting in failed refinements. QT used a Bulirsch-Stoer integrator by Press et al. However, if the boundary conditions are the same, then the solutions should be the same. I have not pursued this question at all, but it seems an interesting one. One way to test it is to try to find a shadow for using a different initial guess for , although I have not tried this.
Similar comments apply to .
However, there are at least 3 arguments against this dependence being a problem. Further study of this possible problem may be helpful. If it does turn out to be a problem, there are many possible fixes. First, we could attempt to shadow only between points that are known to have accurate approximations to the stable and unstable directions. One measure of this accuracy is to start with two different arbitrary unit vectors at time 0 and evolve them forward using equation 1.4 .