3 ms·

> And notice this had nothing to do with the system being ill-conditioned; it was purely due to FP round-off making x just a tiny bit smaller than h. This does

> And notice this had nothing to do with the system being ill-conditioned; it was purely due to FP round-off making x just a tiny bit smaller than h. This doesn't sound right: the comparison function x(t) >= h is extremely ill-conditioned at the point t where x(t)=h, having condition number Inf.

mehrdadn 9y ago
That is not the system being ill-conditioned. The system would be well-conditioned. That's just a comparison that is assumed to hold true inside the simulator because it's trying to advance its time to that of the next event. It's internal to the simulation algorithm.
- conistonwater 9y ago
  I think it is: your code ends up taking different paths based on the result of a comparison, which is ill-conditioned. The underlying system might be well-conditioned, but the question you are asking about it is not.
  - mehrdadn 9y ago
    I'm not disputing whether there is an ill-conditioning present here; I am saying that it is not the system that is ill-conditioned or otherwise somehow particularly pathological, and that this is obviously not due to a modeling error. Please consider re-reading my comments since my point is something other than what you are disputing -- you seem to be disputing a point that I was never making.
    - conistonwater 9y ago
      It is not the system that is ill-conditioned or pathological, nor is it modelling error, I agree. It is the comparison x(t) >= h that is ill-conditioned, and you'd be wrong to rely on its result (much like catastrophic cancellation, it is a bad idea to compare approximately equal numbers with errors in them, no matter where the errors come from—the errors could come from measurements, and the same logic would apply). We might have different ideas about what in your example is specified as given (with full precision) and what is specified approximately? But I do not think I misunderstood anything you wrote, please consider re-reading my comments too.

mehrdadn 9y ago

That is not the system being ill-conditioned. The system would be well-conditioned. That's just a comparison that is assumed to hold true inside the simulator because it's trying to advance its time to that of the next event. It's internal to the simulation algorithm.

conistonwater 9y ago
I think it is: your code ends up taking different paths based on the result of a comparison, which is ill-conditioned. The underlying system might be well-conditioned, but the question you are asking about it is not.
- mehrdadn 9y ago
  I'm not disputing whether there is an ill-conditioning present here; I am saying that it is not the system that is ill-conditioned or otherwise somehow particularly pathological, and that this is obviously not due to a modeling error. Please consider re-reading my comments since my point is something other than what you are disputing -- you seem to be disputing a point that I was never making.
  - conistonwater 9y ago
    It is not the system that is ill-conditioned or pathological, nor is it modelling error, I agree. It is the comparison x(t) >= h that is ill-conditioned, and you'd be wrong to rely on its result (much like catastrophic cancellation, it is a bad idea to compare approximately equal numbers with errors in them, no matter where the errors come from—the errors could come from measurements, and the same logic would apply). We might have different ideas about what in your example is specified as given (with full precision) and what is specified approximately? But I do not think I misunderstood anything you wrote, please consider re-reading my comments too.