Curve tolerance between the calculated points

In order to plot a curve in MATLAB a set of points is calculated based on some formula. The function FPLOT also supports plotting with some tolerance. Its help says the following about this functionality:

The FPLOT function begins with a minimum step of size (XMAX-XMIN)*TOL.
The step size is subsequently doubled whenever the relative error
between the linearly predicted value and the actual function value is
less than TOL.

So, if I plot a curve based on some expression and with some predefined tolerance TOL, is the error of the line segment approximation between any two calculated points always smaller than TOL?

Sorry, this is not obvious for me.

A confident yes/no answer would suffice, but it's also better to explain the reason and/or give an estimate for the maximal error of approximation of a curve with linear segments if a tolerance algorithm like in FPLOT is used.

[Matt J]

You should provide a link to that excerpt in the online documentation. I can't find that explanation of the TOL parameter either in "help fplot" or "doc fplot".

However, I don't see how any general plotting tool could determine the error between a line segment approximation and a continuum of points on the true continuous curve. That would require symbolic analysis at the very least. I imagine perhaps that the line segment approximation between two points is extrapolated to obtain a predicted value at neighboring points. The error spoken of here might be the error between that extrapolation and the true value of the curve.

[Ilya]

Sorry, the excerpt is from the function file fplot.m. I can delete it (or a forum administrator can do this..) if such excerpts are not allowed. I really was not aware if this were not allowed.

[John D'Errico]

To be honest, I was not impressed with the help for fplot. I found it quite a bit confusing as to the purpose of the code, the requirements on the function, and all this even for a highly experienced MATLAB user like me.

[Matt J]

I can delete it (or a forum administrator can do this..) if such excerpts are not allowed.

No, I meant you should provide a link to it so that we can know where to find it and read its full text. It wasn't in the help doc...

Answer 1

[John D'Errico]

No. You cannot know the maximum error of approximation between parts of the curve that were not sampled. I can always give you any simple (and very well behaved) curve, plus a delta function in some spot that will not have been sampled. The error of approximation can then be arbitrarily large, and fplot will never know it.

Of course, the curve I suggested as a counter-example will not be continuous or even differentiable. However, if you want that property and still see fplot "fail" to see that spike, then just add a Gaussian instead of a delta function to the well-behaved function, but choose a Gaussian with an extremely small width. As such, it behaves like a delta function, but is still theoretically infinitely differentiable.

Thus you cannot assure that the error will ALWAYS be smaller than tol, for any supplied tolerance. This is a fundamental issue with virtually any computational algorithm when applied to a completely general function. You can always choose some (locally degenerate) function that will cause it to "fail" to achieve a given tolerance.

[Ilya]

This sounds very sound. Thanks a lot. And what if the function is several times differentiable (e.g. sin(nx), cos(mx) or a linear combination of sin(nx), cos(mx)) and there is no "noise" (e.g. n and m in sin(nx), cos(mx) are less than some small integer, say 5 or 10). Can some bounds be guaranteed for this case? Or, at least, do some bounds on the error of "piecewise-linear" approximation exist in this case if a tolerance algorithm like in FPLOT is used?

[John D'Errico]

Again, consider my counter-case, wherein a linear function plus a narrow Gaussian will result in arbitrarily large errors. The composite function is infinitely differentiable, yet will allow no bounds on the error.

If you choose some set of infinitely differentiable functions (i.e., your trig example) but where the derivatives are LIMITED IN MAGNITUDE ove the domain of interest, then yes you can probably ensure that the tolerance is met.

[Ilya]

Thanks a lot, this is now a quite complete information about the topic.