I haven't really dug into the source of eliminate, but it looks like it is trying to preserve the variables in the lhs of the original equations as much as possible in the output equations.

It looks like you can trick it into given the form you want by having it eliminate the squares of the original equations:

elm:=eliminate({x^2=(cos(v) *cos(u))^2, y^2=(cos(v)*sin(u))^2, z^2=sin(v)^2},{u,v});

The downside to this, is that it gives 4 answers with different substitions (but with the same output "eliminated" equation) which makes the output slightly annoying to process. In this case, this works:

op(map2(op,2, {elm}));

John May
Mathematical Software
Maplesoft

Actually, the solutions returned in Maple 9.5 are not complex, they just happen to be written in terms of complex numbers.  If you want to get the solutions written without complex numbers you can do something like:

sol:=[solve(x^3-8*x^2+12*x-4,x)];
evalf[50](sol); # Imaginary parts show up, but they are numerically 0.
sol:=simplify(map(evalc,sol));
evalf(sol); # no imaginary parts

In Maple 11, the fact that RealDomains package has a hard time figuring out that all the roots are real and it only returns the ones that it can tell for sure are actually real. This is a weakness in the RealDomains package (it doesn't use evalc).

John May
Mathematical Software
Maplesoft

If you change your '.'s to '*', it should work fine (as long as you change eq1 to eqn1 in the call to solve).

eqn1 := x + y + z = 0;
eqn2:= a*x + b*y + c*z=0;
eqn3:= b*c*x+c*a*y+a*b*z=1;

solve( {eqn1,eqn2,eqn3},{x,y,z} );

The easy answer to the question:

So the question is, how do I get ReducedRowEchelonForm to stop doing this?

is probably this:

with(LinearAlgebra):
A:=<<27.6,3100,250>|<30.2,6400,360>|<162,23610,1623>>;
evalf(ReducedRowEchelonForm(convert(A,rational)));

I don't think you can avoid at least some discussion of the difference between exact numbers and floating point numbers, however.  That is, to Maple (and most computer software), 27.6 is very different from 276/10.

John May
Mathematical Software
Maplesoft

You can convert the piecewise function into a list.  Look at ?convert,pwlist

In this list every other entry will be a function, and the others will be the bounds.  Depending on whether the domain of the piecewise function is bounded, the first entry of the list may be a function or a bound.  Assuming the first entry is a polynomial, you can do the following:

#f is your original piecewise function
ftmp:=convert(f,pwlist);
flist:=[seq](ftmp[i], i=1..nops(ftmp), 2); #may need to start at i=3 instead
# flist is a list of all the polynomials.

John May
Mathematical Software
Maplesoft

solve() isn't really the right tool for this, but it should be able to find the same solution that fsolve does: P = 0.5772706212 (instead of finding nothing).

Solve works on equations with floats by running convert/rational on them, then running evalf on the output.  Part of what trips this process up is the numerical error in the argument to tan() in the input expression.    If the input expression is written only in terms of cosine:

.2666845389e-11*Pi^2/(.1450695744e-10*Pi+.1281254481e-13*Pi*cos(.5637758661e-1 *P^(1/2)))=P

then solve works.

John May
Mathematical Software
Maplesoft

If you are only interested in computing over the real numbers, you can use: with(RealDomain): at the beginning of your Maple session.  This package redefines a number of Maple commands, including solve, to ignore complex answers.

John May
Mathematical Software
Maplesoft

- For a general equation or system of equations, the fsolve command computes a single real root.

My documentation (Maple 9) says: "if m and n are integers then irem returns r such that m = n*q + r, abs(r) < abs(n) and m*r >= 0"