The subject-line query says it all.

I was fiddling around with numeric integration, and the first integral I usually attempt is the one discussed in Kahan's 1980 paper on the 34C integration (on the MoHPC DVD). Namely, we are talking the integral of the following between w=0 and w=1:

2*w^2/((w-1)*(w+1)) - w/ln(w)

The WP34S emulator is fast but it seems to top out at 7 accurate significant digits even if one sets the display to the max possible (i.e., ALL 11, FIX 11, or SCI 11).

My HP15C LE is pretty fast and gets more like 8 or 9 digits even if I use something as low as SCI 6 or SCI 7. My 35S is much slower but, still, does better than the best attainable with the 34s.

This is admittedly a challenging integral (Kahan presents it as a potentially pathological example in the aforementioned paper), but I was wondering if the programmers here limited the number of function computations in the Romberg Integration or whatever other algorithm was used.

I note that Thomas Okken, via Hugh Steers, uses Romberg Integration in Free42. Free42 gets all 12 displayed digits correct for this integral, and quickly at that.

I am very interested in numerical integration and wondering if anyone could tell me about the innards of the algorithm.

Les

P.S. I am working with 2.2, not the version 3 beta. Indeed, I don't know where to get the latter.

*Edited: 23 Nov 2011, 4:01 a.m. *