HP71B Integral Questions

[Wes Loewer]

(02-06-2020 11:16 PM)Albert Chan Wrote:  A simple test showed that HP71B INTEGRAL do extrapolations from trapezoids, not mid-points rule.

>10 DISP INTEGRAL(-1, 1, 1E-5, 1/SQRT(1-IVAR^2)), IBOUND
>RUN
3.14156045534              -3.1415397954E-5

this failure to converge (65535 sample points !) is due to missing end points evaluation.

The 50g also chokes on this integral:
FIX 3: result = 3.13952114413, IERR = 3.13821310172E-3, N = 1023
FIX 4: result = 3.14133371247, IERR = 3.14116984018E-4, N = 8191
FIX 5: bails out after N = 10259

It's very well possible that I have simply believed incorrectly for over a dozen years that the Romberg method hp used involved midpoints rather than trapezoids. I went back and reread the Kahan's original article where he writes, "The simplest method is the midpoint rule, whose nodes all lie in the middles of panels all of the same width" along with the statement that "no sample need be drawn from either end of the interval of integration." I guess I jumped to the conclusion that midpoints were used.

I sure would love to see the source code on this.