``It is in vaine to goe about to make the shadowe straite, if the bodie
whiche giveth the shadowe bee crooked.''

-- Stefano Guazzo, *Civile Conversation.* (1574)

*
Shadowing is a branch of chaotic systems theory that tries
to show that, even in the face of the exponential magnification of small
errors, numerical solutions have some validity. It does this by trying to
show that, for any particular computed solution (the ``noisy''
solution), there exists a true solution with slightly different
initial conditions that stays uniformly close to the computed
solution. If such a solution exists, it is called a true
shadow of the computed solution.
An approximation to true shadowing is numerical shadowing,
whereby an iterative refinement algorithm is applied to a noisy
solution to produce a nearby solution with less noise. If this
iterative process converges to a solution with noise close to the
machine precision, the resulting solution is called a numerical
shadow. Numerical shadowing is very compute intensive,
because it requires the storage and manipulation of the full
phase-space trajectory of the system,
at much higher precision than the original computation.
*

- Introduction
- The refinement procedure of GHYS
- A new shadowing procedure: SLES
- A wider perspective: Why shadowing?

Sun Dec 29 23:43:59 EST 1996

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