[R] integrate function

[Richard Condit]

After finding an error in R's integrate function, I spent some time testing 
various alternative numerical integration routines I took from the 
Numerical Recipes book by Press et al. Here is a brief summary of what I found.

As near as I can tell, no method for numerical integration is foolproof. I 
believe that no matter the method, functions could be invented to defeat 
it. In particular, as Ole Christensen pointed out, if a function has 
positive values only in a very narrow range, then numerical routines cannot 
be guaranteed to find it.

R's integrate function works better more often than the alternatives I 
tested. It succeeded with some very tricky functions that foiled the other 
methods. But it did fail in some circumstances.

My only advice about the R integrate function would be to point out in 
documentation that the value returned as "abs error" is not in fact the 
error. There is no way I found of calculating the deviation between a 
numerical estimate of an integral and the true integral. Abs error is 
simply the change in the estimate between the last 2 iterations of the 
numerical routine. In many cases, this should be a rough approximation of 
the discrepancy between estimate and true integral, but if integrate fails, 
abs error fails too. I would suggest abs error not even be returned with 
the estimate of the integral, since it doesn't add information (since 
abs.tol is set in the function).

I can provide more information about the various methods and their errors, 
if anyone is interested.


Rick



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