On Wednesday, February 1, 2017 at 7:26:17 AM UTC+8, Richard Fateman wrote:   
   > There are a few points that I think are worth mentioning.   
   >   
   > 1. The Risch/ etc/ methods determine algebraic anti-derivatives.   
   > It is a bit of a stretch sometimes to identify them with functions   
   > with certain properties. It is typically possible to multiply or add   
   > or otherwise muck around with these functions by, for example, use   
   > of delta or step functions. Or adding strange constants.   
   >    The results of a subsequent differentiation need not be affected.   
      
   For rational functions integration, there exists algorithms that   
   produce continuous anti-derivatives.  It's fairly simple, I see no   
   reason not to implement it.  As for other kinds of integration,   
   I don't think there's an algorithm that always works, but I think   
   heuristics can solve some of them.   
      
   >    b. If a single variable, you are presumably being presented with a   
   > function that is computable and continuous [yeah, well some version of   
   > continuous for some version of integration]  within the limits of   
   > integration (yes, I know you might not know the limits.  But you really   
   > have a lot of nerve asking for the integral of a function you are not   
   > willing to define adequately between the limits of interest.)   
   > Continuing on this -- a continuous function of the kind being discussed   
   > here can almost always be integrated by numerical methods.  That is   
   > given f(x), excellent methods exist to produce a program F(a,b)   
   > which returns the integral of f from a to b  (where a and b are   
   > numerical constants).   
   >   Arbitrarily high precision methods are available, as are error   
   > bounds.  Chance of being fooled very low.   
      
   There are problems with numeric methods too, for example   
   if the integrand includes variables, then numeric integration   
   must compute over and over again for these variables.   
   Another one is precision, zero is zero, no matter how many   
   digits are computed, you can't know if it's really zero or just   
   a small number.   
      
   > 3. You might argue  (certainly I have) that symbolic results display   
   > more information than (say) a table of numbers for various a,b.  Yet   
   > if the result is going to be run through a plotting program, not so   
   > convincing.  A partial symbolic result can be obtained by doing a   
   > numerical-partial-fraction decomposition of some expressions --   
   > any denominator that looks like a polynomial in one variable with   
   > coefficients that can be resolved to floating-point numbers can be   
   > factored into a product of linear factors, thereby requiring only   
   > integration of  / (x-r)^n.   
      
   If the integrand contains variables, then a symbolic result will   
   better than a plot (you can't draw a 4D function).   
      
   > I'm not saying that Bronstein's work, or the programmers attempting to   
   > solve the problems in this thread are "wrong", just that the context and   
   > relevance to (say) scientific computation is easy to misconstrue.   
   >   
   > Cheers.   
   > RJF   
      
   I think symbolic integration is also full of fun.   
      
   Cheers.   
      
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