Fractional Part  A Difficult Integral

06292014, 11:36 AM
Post: #1




Fractional Part  A Difficult Integral
Here some results for integrating FP from zero to 5.8:
Maple V Release 3.0 evalf(int(frac(x), x=0..5.8)) 2.820001887 Maxima 5.23.2 quad_qags(xfloor(x), x, 0, 5.8) [2.720000001263673,1.5265947617137954*10^ 8,3171,0] PARI 2.4.2 intnum(x=0,5.8,frac(x)) 2.770657207613055496607250297 WolframAlpha 2011528 integrate(frac(x),x,0,5.8) 2.82 CASIO fx9860GII (& fx5800P) ∫_0^5.8▒〖Frac X〗 dx 2.72 HP 42S GaussLagrange 16 point: Divisions: 1 2.32 2 2.71441362769 3 2.95110930932 4 2.84122268726 5 2.72066778956 6 2.8405122545 HP 50G Inbuilt integration programme: ∫_0^5.8▒〖FP(X)〗 dX FIX 6 2.820116 in 1,232.15 seconds GaussLobatto 4 point formula with 7 and 13 point Kronrod extensions: FIX 2 2.80 3 2.819 4 2.8199 5 2.82000 6 2.820000 in 12.34 seconds SHARP EL9650 ∫_0^5.8▒〖fpart X〗 dxX 2.419454779 TI84 Plus 2.53MP fnInt(fPart(X),X,0,5.8) 2.720014333 TI86 (tol = 1E5) fnInt(fPart x,x,0,5.8) 2.71992436738 TI89 (& voyage 200 & 92(plus)) ∫_0^5.8▒〖fPart (x)〗 dx 2.72000000036 

06292014, 11:55 AM
Post: #2




RE: Fractional Part  A Difficult Integral
(06292014 11:36 AM)Gerald H Wrote: Here some results for integrating FP from zero to 5.8: Another example (apart from WolframAlpha) where a pencil and paper approach is better. Greetings, Massimo +×÷ ↔ left is right and right is wrong 

06292014, 05:18 PM
Post: #3




RE: Fractional Part  A Difficult Integral  
06292014, 06:17 PM
Post: #4




RE: Fractional Part  A Difficult Integral
Change the upper limit to 6.4 and try evaluating with the builtin integrator on any HP calculator that has one.


06292014, 06:47 PM
Post: #5




RE: Fractional Part  A Difficult Integral
(06292014 06:17 PM)kakima Wrote: Change the upper limit to 6.4 and try evaluating with the builtin integrator on any HP calculator that has one. Link for the lazy: Numerical Integration on the 35S. This is another example where most HPcalculators fail. Cheers Thomas 

06292014, 07:57 PM
Post: #6




RE: Fractional Part  A Difficult Integral  
06292014, 09:40 PM
(This post was last modified: 06292014 10:36 PM by pito.)
Post: #7




RE: Fractional Part  A Difficult Integral
(06292014 06:47 PM)Thomas Klemm Wrote:(06292014 06:17 PM)kakima Wrote: Change the upper limit to 6.4 and try evaluating with the builtin integrator on any HP calculator that has one. What about to use a simple Monte Carlo method as a "preprocessor" to get a "rough estimate", ie.: Code: NIntegrate[x  Floor[x], {x, 0, 6.4}, returns always nice results around 3.08. So we can see whether the specific "precise" integration method with some "difficult" integrands has failed or not.. PS: another example from above posts: Code: NIntegrate[ 

06302014, 01:17 PM
Post: #8




RE: Fractional Part  A Difficult Integral
Numerical integration of things like Frac(x)dx do not gain by using highorder rules (Simpson's, Gauss, etc.) because the first derivative is not continuous.
Monte Carlo (and quasi Monte Carlo) does no worse than its usual performance (which isn't that good anyway) as the variance (or variation) is small. One QMC example is to use the points (2j1)/2N for j=1,N as an Npoint integration formula. Another is to use Frac(N*Sqrt(2)) as a sequence to integrate these types of functions. The last sequence is easy to compute; just add Frac(Sqrt(2)) at each step and reduce below 1. 

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