I'm trying to understand exactly what WorkingPrecision, AccuracyGoal and PrecisionGoal mean for the result of NDSolve.

I presume WorkingPrecision simply means the number of decimal places used internally by Mathematica at various points throughout the calculation on its scratchpad, and therefore essentially gives upper limit to what the accuracy/precision of final result can be.

Now I understand Accuracy/Precision somewhat in the lab sense (Accuracy is how close you are to the true value, Precision is how repeatable the value you get is in some sense; or to use the dartboard analogy-if you're near the bullseye that's accurate-if you hit the the outskirts in the same place 100 times that's precise but not accurate), but not sure I know how these correlate to the Mathematica concepts...

If I set AccuracyGoal->3, PrecisionGoal->4 in NDSolve, what does that say about the function I get spat out? It looks like the definition on the help pages is that AccuracyGoal of 3 would mean 3 significant figures are correct, whereas PrecisionGoal of 4 would give 4 digits after the decimal are correct...e.g if the answer spat out is 89.7895. What does it mean though in this case to say3 sig figs are correct, but 4 digits after decimal place are also correct? seems inconsistent.(just a rule of thumb?)

The help pages also state:

With AccuracyGoal->a and PrecisionGoal->p, Mathematica attempts to make the >numerical error in a result of size be less than 10^-a+|x|10^-p

does this mean if I did have AccuracyGoal->3, PrecisionGoal->4 and NDSolve spat out 89.7895 the numerical error would be 10^-3+89.7895*10^4=0.0997895? so my answer is really 89.7895+/-0.0997895 ? or is there a different definition of numerical error here?

thanks for any clarifications