@Jesús Guillera 

I'm not a specialist of hypergeometric series, but if you're right then this is a Maple bug.

F(2)
                            infinity

But on the other side, if you "think that F(2) is convergent" why do you declare assume(abs(z)<1) ?


This communication (unfortunately in French) should put an end to this dicussion, mid of page 5 "Le rayon de convergence est évidemment l'unité"  which translate as "The convergence radius is obviously the unity" meaning the serie is convergent if abs(z) < 1 (abs(z) <= 1 ?).

An hypergeometric serie

 F := (z, a, b) ->sum(po(a,n)^2/po(b,n)^2*z^n/(n+1),n=1..infinity)

is convergent for any real z if at least one of a or b is a strictly negative integer (If I'm not mistaken, because the associated Pochammer symbol cancels out beyond a certain value of n).
But you are not in this situation with  a=1 and b=1/2 ...

See also Rivoal (still in french) from which this excerpt is taken:



The highlight text translate as "with sometimes a convergence on the unit circle"


So (once again I'm not a specialist), I advice you to verify if your hypergeometric serie converges for z > 1.

@C_R 

I vote up.

In fact I spent a little time on this question where I followed a slightly different path.
But I stpped digging after some time

@Kitonum 

Two reasons:

1. Firstly you assume that there exists some x such that 3628800 = x!  
2. Next your first command works only by chance:

   identify(fsolve(x!=3628800+1, x=0..infinity));
                             10.00000012

Putting aside the first point, something like this would be a little bit more satisfying:

restart;
Identify := proc(n)
  local i:
  i := identify(fsolve(x!=n, x=0..infinity));
  if i::integer then 
    print(n = ``(i)!);
  else
    i:= ifactors(n)[2]:
    if numelems(i)=1 then 
      print(n = i[1][1]^``(i[1][2]));
    else
      print("no identification found")
    end if
  end if
end proc:
Identify(3628800)
                   3628800 = factorial((10))
Identify(3628800*12)
                   "no identification found"
Identify(65536)
                                  (16)
                         65536 = 2    
Identify(3*4*5*6)
                   "no identification found"

What I had initially in mind was something like this

n := 3628800:
f := n+r:    # r being a "small" float number in absolute value
identify(f)  # expected output 10!

Propbably I wasn't clear enough in my question.

Nevertheless, thank you for your contribution.

@acer 

OuterProductMatrix and KroneckerProduct do the same thing here:

OuterProductMatrix(r,c)^%T, KroneckerProduct(c, r);

OuterProductMatrix(r,c), KroneckerProduct(r^+, c^+);

Is there an advantage in using one against the other (in fact I always use KP)?

TIA

@Debdp07 

Your question was "How to evaluate the right eigenvector ...".
In Wiki, parapragh"Left and right eigenvectors" you can read:





In LinearAlgebra:-Eigenvectors (see my first reply) it is written:


Tus according to Wiki, LinearAlgebra:-Eigenvectors  computes right eigenvectors.

Isn'it what you asked for?
I don't understand your reply, clarify if needed.

Help page says


Then E are the right eigenvectors

@Muhammad Usman 

As I'm naturally touchy, let's just say that I don't really appreciate your phrase "I am waiting for your positive response" which sounds to me like a do-my-job-and-do-it-well injuction.

That said, let's move on to the technical aspects and the reasons why "Here some lines are directed toward zero which is not needed".
There is no "needed" here: those lines are here because RK11 does not have the desired structure for surfdata, period.

In the end, your problem is this: you want to represent information and you've made the wrong choice of the Maple function. So match your needs to the possibilities offered by Maple and find a visualization that makes sense to you and that Maple accepts.

The attached file presents detailed explanations and propose an alternative visualization (that you might consider this a "positive response" or not is irrelevant to me and I'm done with this thread).

Help-2_sand15.mw

Another possibility is given here  I_am_done_after_that.mw (I don't have time to explain what I've done, particularly as I have the feeling that "positive response" simply means 'result' and doesn't include the explanations that come with it).

@NIMA112

With @acer's suggestion L writes cos(v*y)*C where C does not depend upon y: so you don't even need Maple to integrate this.

The smoothing dependes upon two parameters (s__M and s__T) which controls thesmoothing in directions HAMM and HAMT respectively.
The higher their values the lower the smoothing.
For s__M = s__T = 2 the suface presents some spurious oscillations wich are removed when s__M = s__T = 1.

If you are interested only in qualitative results, you can search for the couple (s__M , s__T) which suits you the best.
Help_smoothing.mw

If you are interested in quantitative results, (for instance to do prediction) you must an approximative reconstruction with very specific models.
As @acer mentionned using cubic interpolation will be a poor idea.

EDITED 2024/03/22 GMT 19:00
Here is a variant of my file Help_smoothing.mw where the approximating surface is build using Kernel Regression.
If you set the parameter shrink to a small value, let's say 0.1 (I infer that much smaller values could lead to numerical problems) you will get a surface close to what ArrayInterpolation(..., method=nearest) gives.
Kernel_regression.mw

By the way, given the values of MAE[i, j, k] I adice you to do this

surfdata(
   [ seq([seq([op(i1, HAMM), op(j1, HAMT), log[10](MAE[1, i1, j1]) ]
   , j1 = 1 .. nops(HAMT))]
   , i1 = 1 .. nops(HAMM))]
   , axes = boxed, color = cyan
   , labels = ["M", "N", "MAE"]
   , labeldirections = [horizontal, horizontal, vertical]
   , labelfont = [Arial, 16, bold]
   , axesfont = [Arial, 14]
   , transparency = .1
   , style = pointline
)

Given the logarithmic range, this plot is almost like the one you could obtain by plotting only data above 1.
If you want something more correct you can do this

surfdata(
   [ seq([seq([op(i1, HAMM), op(j1, HAMT), log[10](MAE[1, i1, j1]) ]
   , j1 = 1 .. nops(HAMT))]
   , i1 = 1 .. nops(HAMM))]
   , axes = boxed, color = cyan
   , labels = ["M", "N", "MAE"]
   , labeldirections = [horizontal, horizontal, vertical]
   , labelfont = [Arial, 16, bold]
   , axesfont = [Arial, 14]
   , transparency = .1
   , style = pointline
   , view=[default, default, 1..143]
)

I don't understand what yiu are trying to achieve with

nminz, nmaxz := (min, max)(newz); C := .666*(1-ImageTools:-FitIntensity(newz)); PC := PLOT(GRID(0 .. 1, 0 .. 1, newz, COLOR(HUE, C)), STYLE(PATCHNOGRID))

and with what comes before?

What does "I want to plot the density plot for all points with the range ..." mean?

I was quite surprised that Maple 2024 offered such a feature. All the more so if it's not removable (but I only have Maple 2021 at my office  and can no longer judge).
Nevertheless scrolling matrices can be of great help and this could be done easily in previous version by using package Spread Spread.mw

@emendes
@dharr : sorry for interfering

This a rough version of a code which seems to do the job (if I understood correctly your problem).
It is likey that a lot of improvements can be done to make it faster.

A core problem remains: it may happen that not all the twins are assigned or that not all the elements of L are assigned a twin.
What do you wish to do when these situations occur?

Here is the raw code:

Download Twins_mmcdara.mw

@dharr : maybe you'll find it interesting to go further.

@Joe Riel 

ifactor must indeed contain an implicit loop.

By the way, here is a simplified version of my previous code. Its performances are still poor compared to a with-while-loop-code (as @vv  mentioned).
 

NT := proc(p, q)
  local fp, fq:
  if divide(p, q) then
    fp := ifactor(p); 
    fq := ifactor(q);
    algsubs(fq=X, fp);
    degree(%, X);
    printf("%d divides %d  %d times \n", q, p, %)
  else 
    printf("%d does not divide %d \n", q, p)
  end if;
end proc:

@vv 

I know, but this was not my purpose. I only wanted to provide a solution which requires no explicit loop (even if on might comsider  map and "~" are disguised loops).
In my mind it was just a kind of stylistic exercise.

@Scot Gould 

What I like about your videos is that you do the course live, filling in the worksheet step by step.

I myself have developed a series of courses for corporate training courses on statistics which are based on the use of Maple (which implies that people already have some knowledge of Maple).
And the main question is "Is it better to build the worksheet (or document) live, or to run a step-by-step worksheet that's already been written (as Robert Lopez does)?"
I feel that your choice is well suited to short videos and I wonder whether it would be appropriate for one-hour training sessions, for example.