clicliclic@freenet.de  wrote:   
   >   
   > This confirms that the Kauers heuristic is no panacea. It remains to   
   > be understood, though, why it can circumvent the problem of "multiple   
   > non-linear factors" for cube-root integrands, but not for 4th-root   
   > integrands. This anyway makes that problem look artificial rather than   
   > fundamental.   
      
   This can be explained in terms of divisors.  This would require   
   looking at Trager method and probably is too long for a post.   
      
   Just little explanation about "multiple non-linear factors":   
   this is really efficiency problem.  Trying simple solution   
   in current FriCAS implementation is likely to trigger very   
   long compute time, so FriCAS just gaves up.  But _some_   
   cases are much easier than other and I hope to improve this   
   part.   
      
   Now, concerning transformations, most transformation leave problem   
   in "general position".  But some special transformation may   
   put it in simpler form (in particular, for Kauers method having   
   pole at infinity makes things easier).  Kauers degree 3 examples   
   were in easier form, but FriCAS transformed them to harder   
   variant (FriCAS had to do this as it did not handle pole at   
   infinity).   
      
   > I suppose you do not apply the Kauers code iteratively as described in   
   > Section 6.3 of his ISSAC'08 article. The iteration is stated to have   
   > worked for all of his test integrands.   
      
   I do not know what he tested.  But he published results about   
   four curves.  Due to random coefficients particular examples   
   may hit or not some special features.  For given features all   
   examples work essentially the same, one just plugs in different   
   numbers and calculation goes on the same.  So question is   
   how many _really different_ test cases he had.  My estimate is that   
   in published set something between 20 and 50.  In particular his   
   choice of curves means that some features possible in general will   
   not appear in published tests.   
      
   More generaly, random testing is useful for finding bugs   
   (and it found several bugs in FriCAS).  But some important   
   featurs have very low probablity to appear in random tests,   
   unless probabilty distribition was specially biased.   
   So saying that something works in 99.9% of random cases is   
   does not say much.  It means "there is something in it",   
   rather than "the method almost always works".   
      
   --   
                                 Waldek Hebisch   
      
   --- SoupGate-Win32 v1.05   
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