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How does Mathematica perform numerical calculations?

Posted 9 years ago

I have asked about precision and accuracy settings in the past and my most recent post got no replies. So, I am going to ask the question differently this time.

I have a function that I need to numerically integrate. It is a function with two parametric parameters. In performing the integration varying the parameters two things happen.

At some threshold of one parameter, the result seems to be similar to a bifurcation. The results abruptly take a step to a new value and as I narrow the domain to "zero in" on the jump point, it does not go away even as I increase the precision and accuracy requirements to extreme levels until it simply never runs to completion. I have never worked with functions that exhibited a bifurcation. How does Mathmatica deal with these, which leads to the next question?

As I increase the accuracy and precision varying the second parameter, the Plot goes from smooth to broken up. As the accuracy and precision is increased further and using the Table and ListPlot functions, the calculated points begin to look like data with a noise component. In other words, the plotted points look as if there is some statistical component and the occasional outlier seems to be consistent with the RMS limits given the vertical scale used in the plot. Also, with enough plotted points, the "smooth" curve from the less precise calculations seems to be a regression fit to the higher precision data points.

I am puzzled by what I may be looking at. It seems as though my function is "funky" to begin with but the transitions in the way the integral is numerically calculated may be introducing unwanted noise into the results. If these descriptions make sense to anyone, I would appreciate some discussion.

POSTED BY: Luther Nayhm
5 Replies

I suggest you post the calculation you're doing. It's easier to help with a specific question than a very general one.

POSTED BY: Frank Kampas

What I wrote here probably has something to offer in the way of explaining jumps involving error estimates. In brief, a refinement might or might not be made depending on which side of a test you are on (just passing, or just failing), and that can give rise to a jump.

Also, as noted in another response, absent a concrete example it is nearly impossible to speculate (and that applies to more than one question today).

POSTED BY: Daniel Lichtblau
Posted 9 years ago

Yes, thanks. It does, but it opened up another set of questions. The issue was not just the smooth curve, it was what the higher precision and accuracy curves were, what they mean if anything and whether they are telling me things that the smooth curve was not. This point is a bit clearer in the attachment above in the last figure of the set of four.

I would have replied to the previous post you commented on, but that thread was closed for "drifting" off topic. Thanks for getting back on this.

POSTED BY: Luther Nayhm
Posted 9 years ago

Yes, for sure. The attached worksheet also contains a dialog as to what I am trying to accomplish and what my questions are relating to the plots I am generating.

Attachments:
POSTED BY: Luther Nayhm

I suspect the jumps are artifacts of error estimates deciding either to refine or not.

POSTED BY: Daniel Lichtblau
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