fminbnd fails to converge with exitflag=1

I'm trying to use fminbnd to find the extrema of a function, but I'm having an issue with the tolerance. Here is a simple example:

>> [x,fval,exitflag]=fminbnd(@cos,3,4,optimset('TolX',1e-12))

x = 3.1416
fval = -1
exitflag = 1

>> x-pi

ans = 4.3792e-10

The exitflag indicates that fminbnd converged within the specified tolerance. So why is x-pi>TolX???

Answer 1

[Anton Semechko]

because TolX is a measure of change in x between two successive iterations and not an indication of how close it is to a true minimum, which is generally not known

[John]

Thanks, Anton. Good point...

The real problem actually comes when there is no local min on x = [x1,x2]. For example:

>>   [x,fval,exitflag]=fminbnd(@cos,0,1,optimset('TolX',1e-12))

x = 1.0000
fval = 0.5403
exitflag = 1

    >> x-1

ans = -2.4233e-08

When this is no local min, the documentation says fminbnd will return an interior point x1<x<x2 which is no more than 2*TolX from x1 or x2.

[Anton Semechko]

Sorry, can't help you here. You got to take this issue up with the MathWorks folks.

[Alan Weiss]

Thanks for the excellent question. You found an error in the documentation. I will address this error soon.

In fact, if you look at the code for fminbnd by executing "edit fminbnd", you see that there is a tolerance of TolX and also a value sqrt(eps) that comes into play. The true final tolerance relates to

 sqrt(eps)*x + TolX/3

where x is the current point. Now sqrt(eps) ~ 1e-8, so if you choose a TolX any smaller than that, and x ~ 1, then the value of TolX has almost no effect.

Alan Weiss

MATLAB mathematical toolbox documentation