@Markiyan Hirnyk 

Of course you do, it's not something new.

@Markiyan Hirnyk 

They include the cases when both sides are undefined (but actually solutions in the Riemann sphere).
If you want to exclude this cases, simply take integer _Z
with  - _Z <> W + 2*W^2,  for integer W.

@one man 

For a cone the geodesic distance is straightforward, so the exact parametric equation of the spiral is obvious.

@Carl Love 

Looking at showstat(`mod/Factors`)  we can see that a table s is created and then
factors := convert(s,'multiset');
As shown in my answer (and reply), this will (could) produce a random order.

To test this, I'd suggest to modify `convert/multiset` inserting indexorder and see whether the random behavior persists.

By coloring the cube with color f(x,y,z) you will have information on f only on the boundary of the cube.
That is, only f(0,y,z), f(1,y,z),...,f(x,y,1) are visualized.
You can't "see" e,g. the value f(0.5, 0.6, 0.7).
So, you will see the max/min only if they are on the boundary.

@Carl Love 

Sorry for the confusion, but I thought that the reference to Eigenvectors is obvious because of the presence of lambda.

@Carl Love 

Eigenvectors produces a random order for the eigenvalues because after solving the characteristic equation with the local variable lambda, indices is called.
You are right about indexorder. The procedure which calls indices without indexorder is `convert/multiset`. If we change it inserting indexorder, the random behavior disappears.

@Markiyan Hirnyk 

OK, it's enough for me too.
P.S. I wish you good luck in obtaining small Groebner bases using the cited articles.

@Markiyan Hirnyk 

I was not saying that the paper is not serious. But their approach is only for (some) numeric problems. The Groebner bases are used for much more!

@Markiyan Hirnyk 

I did not read them because as I have added (probably you did not see because it was during your reply) I am interested in general Groebner bases which are inherently huge.

Edit. OK, you made me curious and I have looked over [1]. As I have anticipated, they approximate a Groebner basis using Newton's method. That is not what most people want!

@Markiyan Hirnyk 

OK, but if for 70 equations we need supercomputers then ...

Edit. I was (and am) interested in general Groebner bases, not in some versions designed for numerics. Here the situation is unfortunately clear: the Grobner basis is unique and very often is HUGE.

@Markiyan Hirnyk 

The Groebner basis technique is excellent in theory and works for small size problems. I like it very much.
Unfortunately for even moderately large problems it cannot be used. The compexity could be double exponential O(a^(b^n))   !!

@Carl Love 

Yes, irem is a little faster than mods, but if p and nops(S) are huge,
taking (at least from time to time) the irem of the partial product
will speed up the computations.

@acer 
Probably a more instructive example would be:

progbar:= proc(p::realcons)
local x;
if p>100 or p<0 then DocumentTools:-Tabulate( [[]] ): return NULL fi:
DocumentTools:-Tabulate([ sprintf("%a %c", p, "%"),
                          plot(100,x=0..p, view=[0..100,0..100],
                                  filled,axes=box, tickmarks=[[],[]],labels=["",""], size=[450,50]) ],
                          exterior=none, interior=none, widthmode=pixels, width=450 );
end:

s:=0;
"starting ...";
for i to 10000 do
  s:=s + evalf(1/i);
  if i mod 100 = 0 then  progbar(i/100) fi;
od:
progbar(101):
's'=s;

@dpaddy 

If g is continuous and C^1 piecewise continuous then you can use IntegrationTools[Split].
If g is given as piecewise, Maple should be able do compute directly int(f*D(g), a..b);

If g is C^1 piecewise continuous but not (globally) continuous you must also consider the jumps.