@a_simsim Off the top of my head (someone please correct if I've forgotten some), Maple's only mutable structures are tables, rtables (which includes Vectors, Matrixes, and Arrays), modules (which includes objects and Records), and procedures (which includes arrow expressions such as x->x^2); all other structures, including containers such as sequences, lists, and sets, are immutable.

A benefit of an immutable structure is that only one copy can exist (per kernel) in memory, with all references being pointers to that single copy. Part of the process called automatic simplification is to match nascent structures with those already stored (in a master table called the simplification table) and eliminate duplicates. Thus, checking equality of immutable structures becomes as simple as comparing their addresses, and is thus very quick. All this happens in the background, but being aware of it helps one to write efficient code.

A drawback of an immutable structure is that any change made to it necessitates that the entire structure be recreated, no matter how large it is. So, one of the most-common causes of slow performance is to add elements one at a time to a sequence, list, or set. In almost all such cases, a table or Vector should be used instead.

The benefits and drawbacks of mutable structures are just the reverse of those of immutable structures: Modifying (or adding to) the structure is trivial, but equality comparisons are not, and duplicate copies consume memory.

I get this plot:

It's a circle of radius 5 centered at the orgin bisected by a diameter pitched at angle Pi/9. That seems to be exactly what you specified. So, what's wrong?

@Racine65 Don't be misled that that's giving you the exact value of (1/2)! just because you used the keyword exact. Rather, it's giving you the exact rational value of 0.8862269255, which is the 10-digit decimal approximation of (1/2)!.

The simplest form of the exact value is sqrt(Pi)/2. It's an irrational number, and Maple has many ways to get there starting from (1/2)!. Is that not acceptable to you?

@izhammulya Sure: You can color particular edges or all edges the same color or each edge a different color all with GraphTheory:-HighlightEdges. In the example below, I continue from the code above and color each edge based on its relative length:

Dist:= (v1,v2)-> evalf(norm(<P[v1]-~P[v2]>, 2)): #P is your list of vertex coordinates.
E:= [GT:-Edges(G)[]]: # [...[]] converts set to list
dists:= (Dist@op)~(E): #List of edge lengths.
(maxD,minD):= (max,min)(dists):
GT:-HighlightEdges(G, E, [seq(COLOR(HSV, .85*(d-minD)/(maxD-minD), .85, .85), d= dists)]);
GT:-DrawGraph(G);

A similar treatment can be given to the vertices using GraphTheory:-HighlightVertex.

@muchlove What data should I use for the x-errors? Do you have a separate list of those?

@garth 
 

Download Euler_fieldplot.mw

This site renders the plots (rather than just using a GIF) and makes a few errors in doing so. The plot actually looks like this:

@radaar Please post your code. I may be able to re-arrange your evalhf so that it works. However, I won't be able to do it if the computation is inherently symbolic. You just need to post the plaintext file that you pass to read; I don't need to see any output.

@radaar You cannot use a restart statement in "a procedure or from a file being read by the read statement." Recalling your other recent Question, I believe that you are using a read statement, on my recommendation.

Regarding evalhf: It often requires some arcane and lengthy syntax to workaround evalhf's severe restrictions, but it is very often worth it. I often get a factor-of-25 speedup from it. 

@colin12345678 The scope of the derivative in the reference that you linked does not include the integral. The derivative is specified as dK/dbeta, which is just a simple derivative rather than a differential operator.

Your 25%-75% x 100x algorithm sounds very strange to me. Do you have a reference for it, or did you just make it up? Is there something unusual about this dataset?

@Hassan Alkomy Your work is okay. It wasn't clear in your original Question that FyjR is an expression that depends on j. From what you've shown, it appears to be a simple variable name. If it depends on j, everything is fine with the sums.

Please re-enter both your original expression for Gc and your desired simplified form with all multiplications explicitly specified.

Also, you say that a, b, and c are to be determined by Maple, yet b (definitely) and c (possibly) appear in the original Gc, so it doesn't make sense to me to "determine" them. So please clarify that. 

I think that the fastest code to calculate an arbitrary position^[*1] in the Fibonacci sequence is also the briefest:

F:= n-> (<1,1;1,0>^(n-1))[1,1];

(For the OP: <1,1;1,0> is the 2x2 matrix

1  1
0  1,

and the [1,1] selects the row 1, column 1 entry.)

I believe that the TI-84 handles small-matrix arithmetic. You may need to make a free download from TI to get a matrix upgrade.

A similar matrix method can be constructed for any linear recurrence with constant coefficients (regardless of order).

^[*1]Direct application of the recurrence is probably fastest for generating an entire initial segment of the sequence.

@radaar So, nothing at all happens when you enter the read command at the prompt, then press Enter? You don't even get a new prompt? What happens if you enter 2+2;? What about sin(Pi);?

@Kitonum The OP means that given a positive rational number u, find rational numbers a, b, c, such that u = a*b/2 and a^2 + b^2 = c^2.