Quadruple Integral Calculation

[admin]

Is anybody interested in quadruple integrals?

[admin]

People, who needs to numerically evaluate quadruple, quintuple, sextuple, and septuple integrals, write to me.

[admin]

I was interested how engineers and scientists measure volumes of rockets and etc. I had search Google for volume calculation procedure, found Wikipedia article which offers for measuring volume of irregular 3D objects by fluid displacement. O-la-la, no wonder that rockets with expensive satellites fall regularly. Now I am interested how engineers and scientists measure hyper-volumes of 4D objects.

[admin]

Quadruple Integral Calculator Real15 is ready and available for trying http://www.tvalx.com/products/QuadrupleI...eal15.html .

[admin]

Earlier I asked if my Triple Integral Calculator Level 2 is the only triple integral calculator in the world. After that Wolfram-Alfa widget Triple Integral Calculator appeared on the web. Let's see what will happen with quadruple integral calculator. For today Google shows only my forum:

[admin]

Half a year past from announcement of Quadruple Integral Calculator but there is no answer from major math software producers. Our Quadruple Integral Calculator Real 15 stands alone like a rock. It's phenomenal. I invite public to discuss this. Google shows only one exetools forum where quadruple integral calculator is mentioned. Someone user says "Why use 20 different calculators for such tasks? Why not just use MATLAB (for numeric stuff) and Mathematica (for symbolic stuff)? That would be way better than using a bunch of ugly, unprofessional mini-calculators!" Our answer is : Why use MATLAB and Mathematica for calculating quadruple integrals if they cannot do it. Use just one Quadruple Integral Real 15.

[admin]

I have discovered on the Google page Softpedia review of Quadruple Integral Calculator Real 15. Thank you Softpedia! On screen shots however you used the same functions for low and upper limits. That's why you got zero for result. Limits should be different. For example -cos(x) for lower limit and cos(x) for upper limit.

[admin]

Someone in Paris tried
0;1;0;2*Math.PI;0;2*Math.PI;(1+Pow(Pow(x, 2)+Pow(y, 2)-2*x*y*Cos(w), 0.5))/Pow(Pow(x, 2)+Pow(y, 2)+1-2*x*y*Cos(w)-2*x*Cos(z)+2*y*Cos(z+w), 0.5);
probably source formula was (1+(x^2+y^2+2*x*y*cos(w))^0.5) / (x^2+y^2+1-2*x*y*cos(w)-2*x*cos(z)+2*y*cos(z+w))^0.5
It's a serious shot. I cannot see what limits were for x. I got 0.718... for x in [0.2, 0.21]. Then got message "calculation was aborted by user" for [0.2, 0.3]. That means that inner calculations acceded max level, 10 by default. After that I tried to exclude endpoints and got 7.028... for x in [0.2, 0.3] and 0.01;1-0.01;0.01;2*Math.PI-0.01;0.01;2*Math.PI-0.01;(1+Pow(Pow(x, 2)+Pow(y, 2)+2*x*y*Cos(w), 0.5)) /Pow( (Pow(x, 2)+Pow(y, 2)+1-2*x*y*Cos(w)-2*x*Cos(z)+2*y*Cos(z+w)), 0.5); for other limits with working uncertainty 0.01 and max level 15. Finally I got 61.73 for x in [0.1, 0.9] in one minute. It seems that higher accuracy requires high max level and long time, probably hours.
Trying to use Triple Integral Calculator Real 15 with w=0 I got divergence at x=0 . It hinted me that the quadruple integral diverges at zeroes of the denominator. Even if the integral converges, the algorithm requires max level 30 for x in [0.00001, 0.001] and working uncertainty 10^-5 . So the quadruple integral requires max level no less than 30 for such precision. It's hard to estimate how long it could be.

[admin]

Quadruple Integral Calculator Real 15 for now works only on national versions of Windows 7 with dot as decimal separator. I plan to adapt it soon to versions with comma as decimal separator.
If you need it now then use the decimal separator native to your version of Windows in Working Uncertainty and x-range. For example 1,5E-5 1E-6, or 0,999 for German, French and others with comma as decimal separator. But all numerals in formulas should be strictly in C# format. That is : no group separators are allowed and dot is decimal separator.
There is also a bug in Second Integral formulas (incorrect copy/paste).

[admin]

Example of calculation in Windows 7 with comma as decimal separator:

source functions:
function f(x,y,z) = (1-x*x*y*y*z*z*w*w)/((1-x*x)*(1-y*y)*(1-z*z)*(1-w*w)*(1+x*x*y*y*z*z*w*w))^(3/2)
upper contour for y = 1-1E-3
lower contour for y = 0
upper surface for z = 1-1E-3
lower surface for z = 0
upper hypersurface for w = 1-1E-3
lower hypersurface for w = 0

inner functions:
function f(x,y,z) = (1-x*x*y*y*z*z*w*w)/Pow((1-x*x)*(1-y*y)*(1-z*z)*(1-w*w)*(1+x*x*y*y*z*z*w*w), 3/2)
upper contour for y = 1-1E-3
lower contour for y = 0
upper surface for z = 1-1E-3
lower surface for z = 0
upper hypersurface for w = 1-1E-3
lower hypersurface for w = 0

upper limit for x = 0,999
lower limit for x = 0
integral = 129,463174716663
estimated uncertainty = 3,32369495981544E-07
working uncertainty = 1E-3