@MapleEnthusiast 
1. The "how" answer is useless. What linear system?
Suppose e.g. that we start with a stochastic matrix S having some symbolic elements. If P is an arbitrary nonsingular matrix also with some symbolic elements then P^(-1).S.P has the spectral radius <= 1 but it will be very hard to prove it without knowing S and P.

2. A Neumann series ( = Sum(A^k, k=0..infinity)) cannot converge if A's spectral radius is >= 1.

@MapleEnthusiast The Gershgorin theorem is too simple to hope for a proof. It is useless even to prove the result for standard stochastic matrices. It would be more important to explain how (and why) the matrix M was obtained.

@Carl Love An interesting (and strange) fact:

restart;
Digits:= 15:
Sin:= proc(x::realcons)
    Digits:= 2;
    evalf(sin(x))
end proc
:
evalf(Int(Sin, 0..1*Pi/2, method= _d01ajc));
#                        1.00000000000000

So, sin has been computed with 2 digits precision and the result is "perfect".

Compare with:

restart;
Digits:= 15:
Sin:= proc(x::realcons)
    Digits:= 20;
    evalf(round(1000*sin(x))/1000)
end proc
:
evalf(Int(Sin, 0..Pi/2, method= _d01ajc));
#Error, (in evalf/int) NE_QUAD_MAX_SUBDIV:
#  The maximum number of subdivisions has
#  been reached: max_num_subint = 200

evalf(Int(Sin, 0..Pi/2, method= _d01ajc, epsilon=1e-4)); 
                        1.00000605573377

@Carl Love It is not a very good idea to do so. It will not work for nonlinear equations, and even for linear ones, some solutions could be lost. E.g.  2*x=4  mod 16.

@mthkvv The situation is more complicated. Some authors take different branches, such that the result is complex for nu>1, 0<k<1 (see e.g. Oldham K., Myland J., Spanier J. - An Atlas of Functions, Springer, 2009).
Unfortunately, Maple's documentation does not contain the details.

@acer @Carl Love  thank you both. It would be nice to know whether there was a good reason for this construct, or it's just for fun. In the latter case, let's hope that at least option encrypted will not be used.

Edit. For combinat:-numbpart more similar constructs are used:

kkk := [op(7, eval(combinat:-numbpart))]:
showstat(kkk[4][NumberOfPartitions]);

revealing some undocumented options.

@acer  Thank you, this is very interesting. My impression was that the lexical table is used only for internal purposes.
I wonder why a procedure would be placed here.
Your example would be greatly appreciated.

@Carl Love  The procedure combinat:-partition simply calls partition1(n,m) . This is not an export, local or library proc and I was not able to find it. Do you know something about it?

@Carl Love  Yes, you are right. Yesterday it was late and I had not the time to test it.
Here is a version a bit faster.

NumPartStrict1:=proc(n, m)
  option remember;
  local k;
  if m=0 then return `if`(n=0,1,0) fi;
  if m*(m+1)<2*n then return 0 fi;
  if m>=n then return 1 + thisproc(n,n-1) fi;
  add(thisproc(n-m+k-1,m-k),k=1..m)
end:

CodeTools:-Usage(NumPartStrict1(2000,100));
memory used=159.09MiB, alloc change=76.01MiB, cpu time=2.77s, real time=2.76s, gc time=93.75ms

                  344032826568833636909684517

Please provide a more detailed statement of the problem.

@Aliocha  It's of course possible to use LineInt, but then you must enter the correct complex components for the vector (otherwise the result is always real). As you see, using int is simpler.

The problem is delicate. In Maple there is no "complex diff". The function abs is nowhere complex-differentiable, not even at real points. The abs(1,z) is a compromise between real and complex differentiation. When using diff with abs in a nonreal context, there will be always problems.

@mzivari It's just a simple change of variables, taking into account the constraints.

@mzivari  Yes, it is nonnegative; not easy to prove though (some work is needed for this).
f >= 0 (in the domain) iff  fuv >=0  for u,v >=0,  where:

f := (x, y, z) -> 1 - x^4 + ln(x^4) + z^4 - y^4 - y^4*(ln(z^4) - ln(y^4)):
fuv:=f(sqrt(1+u), sqrt(1+u+v), sqrt(1+2*u+v)); # >=0 for u,v>=0

plot3d(fuv, u=0..10, v=0..10); # graphically should be clear

@chrisc The integral could exist in some distributional sense. If you know how this distribution is defined, maybe Maple could be used to retrive the desired result.