@dpaddy 

If in the integral of f*dg  one of the functions  f,g  is C^1 (at least piecewice) then the integral can be reduced to a Riemann one
and it is not difficult to write a Maple program to do this.
(But be aware of the fact that there exists (pathological) functions for which Maple fails in computing the Riemann integral.)

Otherwise, you have to study theoretically the problem, and if the integral exists try to reduce it to the previous case (if possible).
Note that the Stieltjes integral is not very simple in the general case. Even if both f,g are continuous, it may not exist.

@AmusingYeti 

Any 3d plot!

plot3d(x^2-y^2,x=-1..1, y=-1..1);
#Here is the print screen; I have posted such images here long time ago.
(I have to  use Maxima for vector 3d plots!)

@AmusingYeti 

It must be added that unfortunately the .eps export for 3d graphics is terrible (unusable), at least in Windows.

@Carl Love 

g(x)=floor(x):

inf(x*dg(x), x=0..5) =

x*g(x)|(x=5) - x*g(x)|(x=0)  - int(g(x), x=0..5) =

25 - 10 =

15

Now, using Maple:

Int(x*D(g)(x), x=0..5):
IntegrationTools[Parts](%,x):
eval(%, g=floor):
value(%);
                               15

# Wrong result if trying directly!
int(x*D(floor)(x), x=0..5);
                               0

@Carl Love 

The Stieltjes integral  Int_[0,5]  x d(floor(x))    cannot be computed by Maple. It must be transformed by hand.

@Carl Love 

Thank you for the interesting info.

Best regards,
V.

@Joe Riel 

Thank you, I found it.
Unfortunately it seems that for an interested user it will be harder and harder to know what is Maple actually doing. It used to be so nice to see the Maple code when something went wrong or, on the contrary, suprisingly well!

@Tyna 

V is supposed to be a vector.

V^+  converts this vector from row to column (or vice-versa).

A and B were filled with some values (by RandomMatrix) in order to obtain numeric results
(you did not provide the numeric matrices).

If V is a matrix (instead of a vector), the objective function is not a scalar! You cannot divide two matrices or minimize a matrix!
[Actually this would be possible in vector optimization, but then a lot of details must be considered.]

So, you should post the whole problem (or a smaller version) if you want a numeric solution.
But it has to make sense :-)

@Joe Riel 

Yes, I forgot about _rest. I have used myself initially args[3..-1] but it was slower.

(But actually the timing seems to be rather inconsistent here.)

@Carl Love 

BTW, I always had the impression that op(L) is faster than L[] for lists, but it seems that the latter is a bit faster.

P.S. Do you have an example where a local for the variable in seq is mandatory? (probably only for multiple kernels [or threads?] in some circumstances).

P.P.S. (related to this thread).  I wanted to see the code for combinat:-choose. When called with 1 argument L, it returns Choose1(L).

I was not able to identify Choose1 (via opaquemodules etc) . Do you have any idea about it?
The life was easier without modules and objects :-)

@Carl Love 

Yes, thank you, I have corrected.

@Carl Love 

But for this problem you don't need arbitrary-length arithmetic. A simple routine to add two lists containing the digits is enough and it will be faster; the reversing is automatic in this approach.

@Carl Love 

For such problems needing computer speed and memory rather than the symbolic power of Maple,
a simple C program seems to be more suitable.

The database of exact solutions for Einstein's equations seems to be (almost?) 100% done.
It would be interesting an estimate (%) for the material contained in Abramowitz and Stegun/NIST
(or maybe Gradshteyn & Ryzhik). 

.

What kind of decomposition do you hope for this example? A factorization of the denominator is needed and this implies the complicated roots of a cubic (supposing that the indeterminates are x,y).