HP71B Integral Questions

[Albert Chan]

(02-05-2020 06:20 PM)J-F Garnier Wrote:  The IBOUND value is still not clear for me, both on the HP71 and HP75. With same number of samples and same extrapolation, why is the IBOUND value different and depends on the user-supplied target accuracy P? For instance in the cases above, IBOUND is exactly divided by 10 when P is changed from 1E-4 to 1E-5 - same samples, same extrapolation, same INTEGRAL result.

I think P is maximum relative error allowed.

Let E = | rough expected integral result |     → Maximum absolute error ≈ IBOUND = E × P

Then, it just walk across the romberg numbers, using the difference as a guide, pick the correct gap.

(02-03-2020 01:58 PM)Albert Chan Wrote:  Extending rows of 128 intervals, extrapolate to its limits, we have:

62522713769             Δ
62563113307 40399538
62563049732     -63575
62563050740      1007
62563050648        -91
62563050583          -65 <-\
62563050564          -19 <-- extrapolations does not improve estimate, actually made it worse
62563050559            -5 <-/

Say, E = 62563e6

For P=1E-5, 63575 < (IBOUND = 625630) < 40399538     → return 62563050740
For P=1E-6, 1007 < (IBOUND = 62563) < 63575              → return 62563050648

This implied we need at least 4 romberg numbers, or minimum of 7 sample points.