I converted this to Post because it is just a statement of fact and suggestion for improvement.

Vote Up: I agree that the icon should be modified. I suggest some flourish (perhap waves with an arrow) that connotes "flow" in one corner.

I think the easiest thing to do is to replace dot (.) by underscore (_) in your file names, except for the file-type extensions if there are any. Although you may not be having major problems yet, this seems like a "bug waiting to happen".

@MaPal93 Maple's internal file format .m isn't human readable, so don't convert it to a PDF. In order to upload it here, change the name to .txt.

@Muhammad_Qozeem The green arrow is in the MaplePrimes editor, not in Maple itself.

Do you mean the mean and standard deviation of the underlying normal random variable, or of the lognormal r.v. itself? I think that the former would be more usual.

@Kitonum I think that your implicitplot3d command is missing a multiplication sign after the y.

@acer It's even more efficient if you avoid direct indexing of the rtable in rtable_scanblock's 1st argument and instead use the 2nd argument to literally specify a "block" (i.e., submatrix, slice, subarray, etc.) for it to "scan":

(m,n):= (99, 10^5):
M:= LinearAlgebra:-RandomMatrix((m,n), density= 0.3):
L1:= CodeTools:-Usage([seq](rtable_scanblock(M, [i, ..], ':-NonZeros'), i= 1..m)):
memory used=9.03KiB, alloc change=0 bytes,
cpu time=234.00ms, real time=231.00ms, gc time=0ns

L2:= CodeTools:-Usage([seq](rtable_scanblock(M[i, ..], [..], ':-NonZeros'), i= 1..m)):
memory used=75.56MiB, alloc change=0 bytes,
cpu time=687.00ms, real time=511.00ms, gc time=265.62ms

evalb(L1=L2);
                              true

What I give here is only a "hunch", i.e., a vague intuitive feeling not backed by direct evidence. I suspect that the solution that you want is a singular solution (like I wrote to you in another thread). That is, I suspect that there is some valuation of some parameters that will yield a solution that cannot be obtained by applying the same valuation to a general solution. The parameter valuations that lead to singular solutions can often be guessed by using valuations that would produce zeros in denominators in the general solution.

@sursumCorda The timing difference that you show is due almost entirely to the different methods of random-number generation: In your fast case, the random numbers are generated all at once in a matrix; in your slow do-loop case, they are generated one pair per iteration. In the example below, I've taken your do-loop case and modified it so that the random numbers are generated in a matrix.
 

Download iRandomPointVector3.mw

@vs140580 Why not just compute the 500x500 (symmetric) correlation matrix? It's much smaller than the 1000x125250 matrix that you're asking for.

Just to clarify: If starting with 500 coluimns, you want to append new columns of all pairwise products for a total of 125250 columns?

@sursumCorda Okay, I understand what you were trying to highlight: The 1+ is to draw the attention of the human reader to a mathematical fact; you weren't intending for it to make any difference to Maple. If there were some situation where Maple treated 1+20230321 and 20230322 differently, that would be a much more serious bug. (I'm not saying that your reported bug isn't serious. I think it's serious.)

Vote Up. Your first algorithm had an innovative use of symbolic polynomial algebra, and this one has a very innovative use of matrix algebra.

What aspect, if any, of this bug are you trying to highlight by the weirdly specified coefficient (1 + ...)?

@dharr Vote Up. It's a great and surprisingly short (codewise) algorithm with an innovative use of symbolics. I haven't tried it yet, but from a quick read-through on my phone, I'd guess that the storage requirement is at least O(n^(k+2)), where k is the final output (and note that necessarily k >= n). Perhaps the procedure can be refactored to reduce this. Two things that might help:

1. Carefully control the number of times expand is used.
2. Send the result of any particular expand to the garbage collector as soon as it's known that it won't be needed later.

(Your code may already do both of those things; I won't know until I actually run it.)

Note that Maple only stores a single copy of any polynomial; all other references, including as parenthesized subexpressions of other polynomials, are just pointers to that master copy. It's useful to keep this in mind when attempting to reduce storage requirements.