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
Quote:
I can confirm the result given by Valentin to the first 30 digits of precision, using a highprecision 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:
/ 10
 x^{2}*sin(x^{3}).dx
/0
To some 750+ decimal places, my quadrature routine produces:
0.14587364123643233630725025779820134374806272608726769409905827138475545678
91873252258156604930609502083829281765371738264413094048636339249999429802062379
57961603563346425312582138935758511981714815713913326990354637466454659373462989
23856880365641819094067543287366102194594287500142135303138814301671482938831794
27974049757460366605942774949622910049520201352146447612765559727085467322830992
54379599727222565251775009728250768063382128194525269251261178725736672752044626
44859088056137118266467542972367110856139562847936891992188412495430195619961309
13352677329362058906113949895974412408123236712255902834497437414519914257696481
63439748933708957007997706981128617570867623590746630295371101538752591300416917
574665836311221415826565867926678856523022042718456...
which is demonstrably correct to all digits shown. On the other hand, the naive approach using the HP71B and asking for full precision gives:
>INTEGRAL(0,10,0,IVAR^2*SIN(IVAR^3))
.145873640643
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
.145873641554
You might want to try assorted calculators and your own highprecision quadrature to see how they cope with this simple integral.
Best regards from V.
Edited: 14 May 2012, 11:44 p.m.
