class RooIntegralMorph: public RooAbsCachedPdf

 Class RooIntegralMorph is an implementation of the histogram interpolation
 technique described by Alex Read in 'NIM A 425 (1999) 357-369 'Linear interpolation of histograms'
 for continuous functions rather than histograms. The interpolation method, in short,
 works as follows.

   - Given a p.d.f f1(x) with c.d.f F1(x) and p.d.f f2(x) with c.d.f F2(x)

   - One finds takes a value 'y' of both c.d.fs and determines the corresponding x
     values x(1,2) at which F(1,2)(x)==y.

   - The value of the interpolated p.d.f fbar(x) is then calculated as
     fbar(alpha*x1+(1-alpha)*x2) = f1(x1)*f2(x2) / ( alpha*f2(x2) + (1-alpha)*f1(x1) ) ;

 From a technical point of view class RooIntegralMorph is a p.d.f that takes
 two input p.d.fs f1(x,p) an f2(x,q) and an interpolation parameter to
 make a p.d.f fbar(x,p,q,alpha). The shapes f1 and f2 are always taken
 to be end the end-points of the parameter alpha, regardless of what
 the those numeric values are.

 Since the value of fbar(x) cannot be easily calculated for a given value
 of x, class RooIntegralMorph is an implementation of RooAbsCachedPdf and
 calculates the shape of the interpolated p.d.f. fbar(x) for all values
 of x for a given value of alpha,p,q and caches these values in a histogram
 (as implemented by RooAbsCachedPdf). The binning granularity of the cache
 can be controlled by the binning named "cache" on the RooRealVar representing
 the observable x. The fbar sampling algorithm is based on a recursive division
 mechanism with a built-in precision cutoff: First an initial sampling in
 64 equally spaced bins is made. Then the value of fbar is calculated in
 the center of each gap. If the calculated value deviates too much from
 the value obtained by linear interpolation from the edge bins, gap
 is recursively divided. This strategy makes it possible to define a very
 fine cache sampling (e.g. 1000 or 10000) bins without incurring a
 corresponding CPU penalty.

 Note on numeric stability of the algorithm. Since the algorithm relies
 on a numeric inversion of cumulative distributions functions, some precision
 may be lost at the 'edges' of the same (i.e. at regions in x where the
 c.d.f. value is close to zero or one). The general sampling strategy is
 to start with 64 equally spaces samples in the range y=(0.01-0.99).
 Then the y ranges are pushed outward by reducing y (or the distance of y to 1.0)
 by a factor of sqrt(10) iteratively up to the point where the corresponding
 x value no longer changes significantly. For p.d.f.s with very flat tails
 such as Gaussians some part of the tail may be lost due to limitations
 in numeric precision in the CDF inversion step.

 An effect related to the above limitation in numeric precision should
 be anticipated when floating the alpha parameter in a fit. If a p.d.f
 with such flat tails is fitted, it is likely that the dataset contains
 events in the flat tail region. If the alpha parameter is varied, the
 likelihood contribution from such events may exhibit discontinuities
 in alpha, causing discontinuities in the summed likelihood as well
 that will cause convergence problems in MINUIT. To mitigate this effect
 one can use the setCacheAlpha() method to instruct RooIntegralMorph
 to construct a two-dimensional cache for its output values in both
 x and alpha. If linear interpolation is requested on the resulting
 output histogram, the resulting interpolation of the p.d.f in the
 alpha dimension will smooth out the discontinities in the tail regions
 result in a continuous likelihood distribution that can be fitted.
 An added advantage of the cacheAlpha option is that if parameters
 p,q of f1,f2 are fixed, the cached values in RooIntegralMorph are
 valid for the entire fit session and do not need to be recalculated
 for each change in alpha, which may result an considerable increase
 in calculation speed.