@Kitonum An easy generalization to any number of dimensions, with the input and output both being vectors:

PV:= proc(v::Vector) 
local x:= `tools/genglobal`(:-_x), pv:= Vector(upperbound(v), symbol= x);
    eval(pv, solve(v.pv, {seq(pv)}))
end proc: 

Or, a simpler version, allowing the user to specify the free global stem:

PV:= proc(v::Vector, free::name= :-_X) 
local pv:= Vector(upperbound(v), symbol= free);
    eval(pv, solve(v.pv, {seq(pv)}))
end proc: 

@vv Thanks for finding that. Here's a correction:

f:= z-> z/abs(z)^2:  r:= t-> exp(I*t):
IntegrationTools:-Change(Int(f(z), z= r(-%Pi)..r(%Pi)), z= r(t));
value(subsindets(%, ln(exp(anything)), simplify, symbolic));

The above returns 0. Changing f to z-> 1/z, it returns 2*I*Pi.

@Christopher2222 I agree with your policy regarding voting. For almost any legible and legitimate math/Maple Question, if the author has reputation < 11, I give it a vote up. The 11 is the cut-off below which the Question gets the red spam flag.

I'm curious: Is "Maple Companion User" a generic username by which Questions get posted directly from the Maple Companion app? If so, I think it's a bad method: There should be a better way to distinguish users.

I'll say again: The product pull-down list for Questions needs to include Maple Companion.

Of the numerous ways to correct your procedure, this is my favorite:

b:= proc(x::nonnegative)
local n:= trunc(x);
    procname(n):= `if`(n < 2, n, (thisproc(n-1) + thisproc(n-2))/2)
end proc:

And multiple matched pairs of (column) vectors can be plotted with 

plot([<X1|Y1>, <X2|Y2>])

There's no limit on how many pairs can be put in the [...].

@nm To upload an animation:

1. Right click on it for the context menu.
2. Export As => GIF
3. Use the green uparrow to to upload it just like you would upload a worksheet.

@Rouben Rostamian  Oh, I forgot to say that the procedure is meant to be run in a separate worksheet, kept in the background.

I don't understand what you mean by "power series of epsilon". I do understand power series of t, power series of x, or perhaps even a joint x and t power series. 

I think that using a Maple procedure such as below is a lot easier than watching an external file monitor such as fswatch.

CheckForNewData:= proc(
    fn::string, #filename
    {
        start::posint:= FileTools:-ModificationTime(fn),
        sleep::positive:= 2 #seconds
    }
)
    while FileTools:-ModificationTime(fn) <= start do 
        Threads:-Sleep(sleep)
    od;
    return "File touched."
end proc
:

@mmcdara "Reading between the lines" of the Question, I suspect that the OP is waiting for new data to appear in a file so that it can be processed by Maple. I don't think that they are actually interested in a file comparison such as is provided by diff or similar OS commands.

@Teep You should add this procedure to my code, right after AssignToClusters:

#Returns the partition of the point indices.
PartitionIntoClusters:= proc(U::Matrix)
local CL:= AssignToClusters(U), k;
    [seq([ListTools:-SearchAll(k,CL)], k= 1..max(CL))]
end proc
:

Running this on the same example as before, I get

CP:= PartitionIntoClusters(U);
[[4, 5, 6, 10, 14, 15, 19, 20], [1, 2, 3, 7, 8, 9, 11, 12, 13, 16, 17, 18],
[21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]]

You may get a different partition due to randomization of the initial configuration done in my algorithm.

The partitioning of the original matrix of points can then be done by

map2(index, XM, CP);

I should've mentioned this at the time: The continuous red curve in my plot (along the top of the histogram) is not meant to be a proper probability function. It's just a simple connecting of the dots of the discrete function. Sorry if this caused any confusion. 

My guess is that there was a temporary slowdown in the Maplesoft license server. I recall a few times in the past week where startup seemed slow, but the problem seems to have passed. Does the problem persist for you? Here's an experiment: Compare startup times with and without an Internet connection. 

That's a rather banal Post. Were you just testing the MaplePrimes posting system?