...solutions.

There is a subspace of measure zero, called the stable subspace, in which small perturbations do not result in vastly different solutions. But since it has measure zero, the probability of random numerical errors being restricted to this subspace is zero.

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...time

The e-folding time is , where is the Lyapunov exponent.

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...particles.

This is in stark contrast to N-body simulations of our solar system, in which case we are interested in the precise evolution of the particular solar system we live in, not a hypothetical one very similar to it.

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...i+1.

  In other words, let

be the first-order ODE. (Often written as y' = f(t,y) in ODE texts.) Note that is the solution of equation 1.0 using as the initial condition and integrating h to time i+1. Then is the Jacobian of equation 1.0. The Jacobian measures how changes if is changed by a small amount. The resolvent is the integral of J(t) along the path , and tells us how a small perturbation from at time gets mapped to a perturbation from at timestep . That is, is the solution of the so-called variational equation

where I is the identity matrix. (The reason the arguments to R seem reversed is for notational convenience: they satisfy the identity , and so a perturbation at time gets mapped to a perturbation at time by the matrix-matrix and matrix-vector multiplication .) Finally, the linear map in the GHYS refinement procedure is [13].

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...exist.

``Proof'': For any system, even a chaotic one, given any true orbit of fixed length in time, a small enough perturbation in the initial conditions in any direction produces a small change in the final conditions, although this perturbation must be exponentially smaller for chaotic systems than for non-chaotic systems. (If the perturbation is restricted to the stable subspace, then obviously a similar solution will be obtained.) Thus given any true orbit that -shadows a noisy orbit, we can find infinitely many other true orbits nearby that also -shadow the noisy orbit. However, it may be that all the true orbits are packed into a space unresolved by the machine precision.

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...SLES

SLES is pronounced ``sleeze'', because this seems like such a sleezy, naïve method.

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...orbit,

This formulation is not complete, because we do not yet know how to include the boundary conditions in g. See further work for more details.

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...backwards.

This oversight is not obvious from the description in their appendix B, it appears only in their code.

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...other.

If the force is ``softened'', then this problem goes away.

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...meaningfully.

It is trivial to show that the geometric mean of a set of numbers is the arithmetic mean of their logarithms, to a suitable base.

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...RUS.

For example, with M=25 moving particles, a RUS of length S=128 requires about 100 megabytes of memory, and this space scales as . ((25 particles 6 dimensions per particle) 8 bytes per double precision number 4 (2 resolvents, 2 sets of basis vectors each covering half the space) 128 steps).

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...small.

This neglects to mention the error of the resolvent itself, which depends on . However, the resolvent error doesn't seem to matter much in comparison to the error introduced by the size of the perturbation .

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...errors

The error in the error, so to speak.

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...recomputed.

It is interesting to note in passing that the larger the shadow steps, the more obvious it is that refinement converges to a solution of the accurate integrator, not to a true solution. For, if the shadow step errors were small enough, and the machine precision allowed it, we would be able to ``shadow'' an entire noisy orbit using a single huge shadow step.

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...procedure.

More iterations, but each iteration takes less time. There is clearly a trade-off here that is probably problem dependent.

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...result.

This is probably because it did too much work in the less accurate case, so that the solution it computed had at least 2 more accurate digits than were requested. Then, in the more accurate case, it managed somehow to more accurately judge its errors, and avoided extra work. A difference of is less likely to cause this to happen.

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...Insertion

``But the one thing we should absolutely never do is to ignore or renounce the observational facts because we cannot explain them.''
- Halton Arp

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...(0,1).

Astronomers will note that this does not correspond to any realistic astronomical system; however, it seems unlikely that the precise particle distribution will effect shadowing results. This is supported by the close correspondence of my results with those of QT, even though they used a more realistic ``Plummer'' distribution.

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...spots.

If the orbit has a trouble spot, then Constant Resolvent and Re-use RUS from previous successful shadow will have no effect, because the RUS will be re-computed in an attempt to find a shadow. The other optimizations - Cheaper Accurate Integrator, Large Shadow Steps, and Backwards Resolvent by Inverting Forward Resolvent - still offer significant performance improvement.

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...spots.

Although I have never seen a clear definition of ``trouble spot'', it seems that other practitioners of shadowing, eg.,  GHYS and QT, use the term to refer to any trajectory on which refinement encounters a point on the trajectory where the 1-step errors do not decrease with refinement, although they do remain small and bounded.

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...up.

Although it is completely by chance that it took exactly the same time to 3 digits.

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...0..S

0..S means the orbit from the beginning of shadow step 0 to the end of shadow step S.

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...process

See the discussion on stellar kinematics in [25], particularly pages 431-438.

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...encounters.

There is some disagreement that these two factors exist. There seems to be an ongoing debate between Kandrup and Smith [19, 20, 21], who argue that the growth of errors is due both to the global potential and to collisional effects, while Goodman, Heggie, and Hut [10] argue that collisional encounters are the only process for the magnification of errors.

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...shadow.

Think of it like this: F is produced by perturbing m's orbit to have small 1-step errors under the influence of some arbitrary, mysterious, external forces produced from a potential that is a function only of time. This potential comes from the other particles, each of whom moves as if it was sliding along a rigid, massless wire winding through space. The motion of each particle along its wire - its motion in C - is a function of time, and only time.

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...valid.

Note that the measure of 1-step error, and thus the measure of a quality of any refinement procedure, is still the full phase-space 1-step error, which is cheap to compute. In this case, the savings per refinement is that we need to compute M resolvents per timestep, rather than one resolvent.

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...is.

This idea, and the next, on related areas, are based on ideas from Scott Tremaine.

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...researchers.

Note, however, that the practicality of attempting to integrate the solar system back several billion years seems dubious. For, if it is already known that the system is chaotic, then there are an almost unlimited number of small perturbations that we cannot know about - small undiscovered comets, or comets that passed through our system once from deep space billions of years ago, which we cannot possibly account for; or stars in our galaxy that perturbed the planets over the past few billion years.

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Access count (updated once a day) since 1 Jan 1997: 8503