For example, if the black ball only just enters into a pocket after being hit by the cue ball, a very small error in calculating the trajectory of the cue ball could result in a much larger error in the post-collision trajectory of the black ball - which in turn might allow it to just ricochet off the side of the pocket - which in turn could result in half a dozen balls winding up half a meter away from where they should be. What is meant here (I believe) is that a comparison of reality versus the simulation would be very sensitive to small numerical errors - such as might result from normal arithmetic precision issues. "This particular example also turns out to be numerically unstable: a small error in any calculation will cause catastrophic changes in the final position of the billiard balls." So what is meant here is not 'instability' in the sense of the numbers blowing up to infinity or something - although many practical physics engines do exhibit these kinds of instabilities. John Nagle 02:50, 10 June 2007 (UTC) Reply The actual line of the article says this: But it would violate WP:OR to put that in the article. Overall, it turned out to be slower than using an RK4 integrator and tiny time steps. Getting the solution to converge can require very small steps in a high-dimensional nonlinear space and very frequent recomputation of the Jacobian. The problem is that the integration becomes more stable, but solving the implicit system when it is highly stiff turns out to be difficult. "If you used an implicit method, for example, it would be unconditionally stable." Actually, as one of the few people to ever try ragdoll physics with an implicit integrator, I can report that it doesn't help much. Moreover, one would probably use an adaptive time stepper because the problem would become very stiff once collisions start to take place. Certain explicit algorithms would be unconditionally unstable, but some of course would be stable provided the time step is sufficiently small. How can you say that the numerical algorithm is unstable when you haven't even presented the numerical algorithm? If you used an implicit method, for example, it would be unconditionally stable. I've been doing some work on the ragdoll physics article and related subjects.
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