HP71B Integral Questions

[Albert Chan]

(02-02-2020 03:31 PM)Albert Chan Wrote:  Why does IA(127) have different area, for P=1E-5 vs P=1E-6 ?
Both are using exactly the same 127 sample points.

Answering myself, first column = trapezoidal area of u-transformed FNA(X)

Code:

Intervals       Richardson Extrapolations
1             0
2      25715027    34286703                
4   13721555536 18286835705 19503672306         
8   50890703124 63280418987 66279991205 67022472458     
16  59914609212 62922577908 62898721836 62845050894 62828668848 
32  61913859850 62580276729 62557456650 62552039743 62550890679 ...
64  62401515156 62564066925 62562986272 62563074044 62563117315 ...
128 62522713769 62563113307 62563049732 62563050740 62563050648 ...

Note: u-transformed integrand returned 0 for integrand limit of -1 and 1, thus 1 trapezoid have zero area.

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 <-/

The numbers are done in Excel, which use IEEE-754 binary64.
Except for slight rounding errors, the bold numbers matched IA(127) for P=1E-5 and 1E-6.