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J-F Garnier · 02-03-2020, 03:00 PM

(02-03-2020 01:58 PM)Albert Chan Wrote: 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 ...

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.

I'm interested in documenting the Math ROM algorithms, can you please post the Excel formulae you used to reconstruct the INTEGRAL results?

From the HP Journal article, July 1984 p31 on the Math ROM "A Closer Look at INTEGRAL":
"INTEGRAL proceeds by computing a sequence of weighted sums [...]. These sums are accumulated in an extended-precision, eight-level Romberg scheme.[...] These iterations produce a sequence of approximate integrals I(k,j), where k=0,1,... and j=0,1,...,min(k,7). Here, I(k,0) is the direct weighted sum of 2^k samples integration points, and I(k,j) is the Romberg extrapolation given by I(k,j) = (4^j.I(k,j-1) - I(k-1,j-1)) / (4^j-1)."

Is the 8-level Romberg scheme close to the Richardson Extrapolations you used?

About IBOUND:
IBOUND is an estimation of the absolute error of the computed integral. Some details on how it is computed are also mentioned in the article cited above. IBOUND is usually close to the user-provided target precision P times the integral estimation.

J-F