numerical differentiation?

[Brian Gough]

One way to solve the general problem of writing a routine to operate
on a badly-behaved function is by providing an interval where the
function is well-behaved, and restricting evaluations to this
interval.  I do not claim any more insight than this ;-)

As you say, the typical symmetric derivative routine is sufficient for
most practical purposes. I am fully in favor of having a routine to do
that.  A separate routine to handle difficult cases might be useful.

Dave Morrison writes:
 > Hi Brian, Gerard,
 > 
 > I'm just not sure I see the point of requiring a domain to be defined
 > as a prerequisite for calculating a derivative.  Differentiation
 > depends on properties of a function in a neighborhood of the point at
 > which you want to know the derivative.  Sure, for integration you need
 > to know something about the function over the domain of integration,
 > but that's very different than the case for differentiation.  I've
 > looked and looked, but I can't find any implementation or discussion
 > of any algorithm which requires this.  The NAG routine D04AAF requires
 > you to specify an initial step size, as does the CERNLIB routine
 > DERIV, but neither requires you to specify a domain over which the
 > derivative is defined.  The discussion in Conte and de Boor says
 > nothing about domains.  The way to handle this is for the user to
 > choose between forward, backward or centered difference methods as
 > appropriate.
 > 
 > Besides, what domain would you specify?  Suppose I have a smooth
 > function with a discontinuous derivative, a cusp say, at each integer.
 > Now I want the derivative at x = 0.5.  Do I really have to tell the
 > routine that the derivative doesn't exist at x = 1,000,000?
 > 
 > Cheers,
 > Dave
 > 
 > Brian Gough wrote:
 > > 
 > > Regarding endpoint problems and singularities. In QUADPACK a similar
 > > situation occurs. It is handled by having two versions of the general
 > > integration function (QAG vs QAGP). One version assumes a well-behaved
 > > function which can be evaluated anywhere. The other version avoids
 > > using endpoints (in fact, it avoids a set of user-specified singular
 > > points).
 > > 
 > > For numerical derivatives it sounds as if you could have a similar
 > > setup.  One routine would be free to evaluate the function
 > > anywhere. Another routine would take a range as an argument and avoid
 > > the endpoints. In this way the routine can choose the type of
 > > derivative to use -- one-sided if the step-size would overshoot the
 > > end, but symmetric in the middle. Since the function will be varying
 > > the stepsize it may need to change its choice dynamically.
 > 
 > -- 
 > David Morrison  Brookhaven National Laboratory  phone: 631-344-5840
 >                 Physics Department, Bldg 510 C    fax: 631-344-3253
 > 		          Upton, NY 11973-5000  email: dave@bnl.gov
 > 
 >