Sigma in your sheet is a constant (or parameter or how you want to see it), otherwise you could not even plot, and it does not depend on R30.

Again: what should Sigma(R30) be?

Sigma in your sheet is a constant (or parameter or how you want to see it), otherwise you could not even plot, and it does not depend on R30.

Again: what should Sigma(R30) be?

For me it is hard to understand what you want, in your sheet you say "Sigma increases rapidly and the numerical problem of locating the solution gets tougher and tougher", but I think you mean that temp6c increases you want that zero. It is located at the peak in the log plot.

Of course you may have something different in mind, but it is not clear.

For me it is hard to understand what you want, in your sheet you say "Sigma increases rapidly and the numerical problem of locating the solution gets tougher and tougher", but I think you mean that temp6c increases you want that zero. It is located at the peak in the log plot.

Of course you may have something different in mind, but it is not clear.

I remembered that once at the GSL discussion list (which I do not longer follow) there
was a suggestion for a 'black-box method' for higher dimensional integrals, given by 
Steven G. Johnson, it was 2005 (have not checked for the concurrent state and the error
estimate seems to be not quite correct for the non-smooth nature here, but 'close'),
he is the one from 'fastest FFT'.

Using that it just takes some moments, ~ 5 seconds for dim=6, where Maple needs much time, 
(and beyond it will become costly, there will be a combinatorical increase ...), 5 decimals.

The appended source has to be compiled as C-code (not as C++, for that some notations
would have to be cleaned up).

May be a candidate for a private and compiled working horse.

int_multidim_exampl.zip

PS: when ever will MaplePrimes increase that nasty limitations for the length of uploaded file names ...

I remembered that once at the GSL discussion list (which I do not longer follow) there
was a suggestion for a 'black-box method' for higher dimensional integrals, given by 
Steven G. Johnson, it was 2005 (have not checked for the concurrent state and the error
estimate seems to be not quite correct for the non-smooth nature here, but 'close'),
he is the one from 'fastest FFT'.

Using that it just takes some moments, ~ 5 seconds for dim=6, where Maple needs much time, 
(and beyond it will become costly, there will be a combinatorical increase ...), 5 decimals.

The appended source has to be compiled as C-code (not as C++, for that some notations
would have to be cleaned up).

May be a candidate for a private and compiled working horse.

int_multidim_exampl.zip

PS: when ever will MaplePrimes increase that nasty limitations for the length of uploaded file names ...

@pagan 

Ok, but I meant 3 variables, sorry. For 2 the symmetry is 'evident':

  max(x,y); convert(%, piecewise,x);
  plot3d(%, x=0 .. 1, y=0..1, axes=boxed);

  piecewise(x<= y, y, ``);
  plot3d(%, x=0 .. 1, y=0..1, axes=boxed);

But can you sketch the idea why the symmetric group steps in? Ok, you
write 'don't know how to write it' - but the idea for it?

@pagan 

Ok, but I meant 3 variables, sorry. For 2 the symmetry is 'evident':

  max(x,y); convert(%, piecewise,x);
  plot3d(%, x=0 .. 1, y=0..1, axes=boxed);

  piecewise(x<= y, y, ``);
  plot3d(%, x=0 .. 1, y=0..1, axes=boxed);

But can you sketch the idea why the symmetric group steps in? Ok, you
write 'don't know how to write it' - but the idea for it?

@Markiyan Hirnyk Thx, and is more by accidence ...

For n=4 and n=5 I was able to verify your formula numerically, 4 Digits only (Maple 15)

For n=4 the symbollical approach works, for n=5 I lost patients on Maple and pagan's
version seems to be the way (though currently I can not yet fully understand it).

@Markiyan Hirnyk Thx, and is more by accidence ...

For n=4 and n=5 I was able to verify your formula numerically, 4 Digits only (Maple 15)

For n=4 the symbollical approach works, for n=5 I lost patients on Maple and pagan's
version seems to be the way (though currently I can not yet fully understand it).

Elegant! But not clear to me why it is true to use that 'triangle form'.

For dim = 2 I can follow, but ... would you mind to show dim=3?

Edit: the typo in the uploaded comment is no problem,
that's the correct pronunciation, hence phonetical correct :-)

Elegant! But not clear to me why it is true to use that 'triangle form'.

For dim = 2 I can follow, but ... would you mind to show dim=3?

Edit: the typo in the uploaded comment is no problem,
that's the correct pronunciation, hence phonetical correct :-)

@Markiyan Hirnyk 

Checking the results is numerically, the rest is symbolically.

My thoughts:

By symmetry reduce to the unit cubes, there are 2^dimension of those.
On them one has max(x^2,t^2) = max(x,t)^2 and max(a,b,c) = max(a, max(b,c)).

Now take assumptions, write as 'piecewise' and integrate iteratively.

Have not done that for higher dimensions, since the numerical check is painful,
but the 'recipe' should work.

@Markiyan Hirnyk 

Checking the results is numerically, the rest is symbolically.

My thoughts:

By symmetry reduce to the unit cubes, there are 2^dimension of those.
On them one has max(x^2,t^2) = max(x,t)^2 and max(a,b,c) = max(a, max(b,c)).

Now take assumptions, write as 'piecewise' and integrate iteratively.

Have not done that for higher dimensions, since the numerical check is painful,
but the 'recipe' should work.

@Markiyan Hirnyk

find it attached, it is based on a C routine due to Ooura, intdeo_hirnyk.mws

It is not a good shot here (depending too much on Digits)
and certainyl Pseudomodo's approach is far better (I often
forget to switch to the 'operator' form)