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.

@radaar Assuming that the file contains commands that would ordinarily display output (e.g., they end with semicolon), it should appear immediately on your screen.