|Re: Another tough integral|
Message #16 Posted by Valentin Albillo on 14 May 2012, 11:36 p.m.,
in response to message #15 by Steve Perkins
I can confirm the result given by Valentin to the first 30 digits of precision, using a high-precision method I programmed a few years back. Adaptive integration using double exponential quadrature.
My method was designed for tough integrals, but doesn't achieve results very quickly in high precision applications. It took about a minute to get (and confirm) 30 digits.
As always I admire Valentin's abilities and the tools he creates.
First of all, thank you very much for your appreciation and kind words.
It would be simple to check my 1000+ digit result using Mathematica, Maple, or PARI/GP for instance but I have none of them at hand right now, just my eReader with 894 books, 1200+ (mostly math) papers, 550+ audio files, and about two dozen assorted applications installed on it which, regrettably, don't include any of those.
However, I have no doubts whatsoever that my results are correct, as I've thoroughly tested the routine innumerable times using much more troublesome cases than Gerson's fairly mild integral, with reasonable success so far.
Just for instance, this simple, elementary integral is probably a harder nut than Gerson's if dealing with it straightforwardly:
To some 750+ decimal places, my quadrature routine produces:
which is demonstrably correct to all digits shown. On the other hand, the naive approach using the HP-71B and asking for full precision gives:
An interval approach fares somewhat better:
>S=0 @ FOR I=0 TO 9 @ S=S+INTEGRAL(I,I+1,0,IVAR^2*SIN(IVAR^3)) @ NEXT I @ S
You might want to try assorted calculators and your own high-precision quadrature to see how they cope with this simple integral.
Best regards from V.
Edited: 14 May 2012, 11:44 p.m.