What you are looking for is a kind of memory dump and restore. I am not aware of such an option but did you know that output with an equation label is available without executing the worksheet?

If not try the following:

Download GUI_Output.mw

I have tried

sumpro := proc(i) local a, b; a := rand(1 .. 10)(); b := rand(1 .. 10)(); print("Sum of ", a, " and ", b, " is ", a + b); end proc;
CodeGeneration:-CSharp(sumpro);
Error, (in Printer:-Print) invalid input: Printer:-Function:-Get expects its 1st argument, fname, to be of type string, but received CodeGeneration:-Names:-Name("rand")

Looks like that CodeGeneration:-CSharp cannot translate Maple library function rand to C#

You could try LLMs. Note that i is not used in your code. You can remove it.

Chat CPT:

using System;

class Program
{
    static void SumPro(int i)
    {
        // Generate random numbers between 1 and 10
        Random random = new Random();
        int a = random.Next(1, 11); // Upper bound is exclusive, so use 11 for 1 to 10
        int b = random.Next(1, 11);

        // Print the sum
        Console.WriteLine($"Sum of {a} and {b} is {a + b}");
    }

    static void Main(string[] args)
    {
        SumPro(0); // Example usage with a dummy parameter
    }
}

Gemini:

using System;

public class MapleToCSharp
{
    public static void SumPro(int i)
    {
        Random rand = new Random(); // Create a Random object outside the loop for better randomness
        int a = rand.Next(1, 11); // Generates random integers between 1 (inclusive) and 11 (exclusive), i.e., 1 to 10
        int b = rand.Next(1, 11);

        Console.WriteLine($"Sum of {a} and {b} is {a + b}"); // Use string interpolation for cleaner output
    }

    public static void Main(string[] args)
    {
        // Example usage (equivalent to calling sumpro(i) in Maple):
        SumPro(0); // The 'i' parameter isn't actually used in the Maple code, so we can pass any value.
        SumPro(1); // Call it again to show it generates different random numbers each time
    }
}

I have added a few (t) to some variables in the equations of your worksheet.
The equations were in non executable mode. I changed that to executable and applied dsolve to the equations.

Below is the solution. Since you have not given initial conditions for the variables, the solution contains integration constants c__i. Maple cannot solve the second differential equation symbolically and returns the equation unchanged. This does not mean that there is no such solution. It simply means that Maple does not provide such a solution. You have to give initial conditions for the variables, numerical values for the parameters of the equation and solve numerically with dsolve.

Download System_Of_Differential_Equations_reply.mw

An example would be helpfull. Other users could compare the perfomance of your system to their system.

To start you can compare your system to this one (i7-10710U CPU @ 1.10GHz):

f := sin(x + y) - exp(x)*y = 0:
g := x^2 - y = 2:
CodeTools:-Usage(fsolve( {f, g}, {x = -1..1, y = -2..0} ));

memory used=7.79MiB, alloc change=33.00MiB, cpu time=94.00ms, real time=87.00ms, gc time=0ns

A recent processor model i9-13900H 2.6 GHz ("high" performance mobile) is about 3 times faster in cpu time. You will not get much more speed on a laptop.

The graphics card is probably not used for your problem. You can check in task manager for GPU usage.

Q1: Your setup looks good, however for a better overview on the structure of the problem it is helpfull to replace all subexpression that do not depend on the unkowns lambda[i] by new parameters (new names). Do denom(eq1) for example to see what we are dealing with. These are big expressions.

Q2: Ok

Q3:Yes, see Q1. You can for example use innert operators when defining the equations

pTRw:=((p%^T) %. R) %. w

In this case I would not use the assignemt operator (":=") to define p, R, w. Use equations for later substitution/evaluation like p=...

Concerning the problem:

You want to solve a system of polynominal equations. The equations are of 4th degree. Try for all equations and unknowns the following to investigate:

degree(simplify((eq1-lambda[1])*denom(eq1)),lambda[1])
                               4

As you posed the problem Maple assumes all parameters complex. This adds, besides the size of the equations, an addtional burden to Maple. It could be that there are symbolic solutions in the complex domain but Maple has to investigate many cases. This takes time.

Therefore, reducing the complexity of the problem by using assumptions and upfront simplification of the equations helps allot.

Maybe a soluton by hand is possible but for that we have to see the structure of the equations in the first place

Here are some more but not all solutions.
To find at least some non-trival solutions, I have attached a way to reduce the number of equations from 8 to 4. This was possible because your system of equations has a sparse structure (i.e. not all equations depend on all unknows -> see the white gaps).

It is now much easiere to find some non-trivial solutions for the remaining unknowns

It also looks like that for x5=0 and x7=0 any real value for x1 and x4 is allowed.

For an extended search of solutions I have to hand over to someone more gifted in using fsolve and other rootfinding tools of Maple. If you do not know ranges with roots you can deduce them from the attachement.
I hope this is a good start.

question1118_reply.mw

Compact display seems to prevent rendering of partial differentials. Attached is a version that does not use compact display but alias instead.

Update:
I have attached a second version that resolves the issue with varphi not beeing rendered compact

Download error_display-2.mw

error_display-2-1.mw

Partial derivative of a summation: why it is not just 2*`X__i`?

Answer: The output of Maple is correct because you differentiated for all i from 1 to n. How can Maple know which i you are interested in? If you expect only 2*X__i as output the information about the range for i from 1 to n is lost.

` `*`Partial derivative of a double summation: how to define the nested structure of a double summation where j<>i?`

Answer: B__wrong is correct. Maple only automatically simplified it. You can use Sum instead of sum to prevent this

` `*`System of n equations: how to define and solve for it?`

You must give n a number. Example for n=m=5

m:=5;
A := sum(X[i]^2, i = 1 .. n);
eqs:=seq(diff(eval(A = 0,n=j),X[j]),j=1..m);
vars := seq(X[i], i = 1 .. m);
solve({eqs}, {vars})
                             m := 5

                               n        
                             -----      
                              \         
                               )       2
                        A :=  /    X[i] 
                             -----      
                             i = 1      

eqs := 2 X[1] = 0, 2 X[2] = 0, 2 X[3] = 0, 2 X[4] = 0, 2 X[5] = 0

              vars := X[1], X[2], X[3], X[4], X[5]

       {X[1] = 0, X[2] = 0, X[3] = 0, X[4] = 0, X[5] = 0}

What you experience are numerical integrations errors.

method=rosenbrock gives a slightly deceasing energy.

Forward Euler is in this respect particulary bad (and maybe instructive for your students). I came accross a similar behaviour when exloring solver settings in MapleSim

In this context I ask myself: What has to be invested in numerical fidelity when it comes to predictions of the future of the solar system?

Below is way using names p1 and m1 for 1 and -1 to prevent automatic simplification. These names are assigned to other names that typeset nicely. This assignement is not mandatory but facilitates input (macro could have been used alternatively).

In addition auxilliary equation expressions are used to counteract automatic simplification (ax1) and to simplify names with brackets (ax2). The equation expressions are generally valid and do not create overhead. All other manipulations are performend on equations (i.e. respecting equality).

(Initially I tried to work with inert operators but this did not work well because I could not expand such expressions.)

Download m1_times_m1.mw

What about:

e:=_C1^2+_C1+_C2^3+a+b;
indets(e, And(symbol, suffixed(_C, nonnegint))^anything);#extract powers of
indets(e-add(%), And(symbol, suffixed(_C, nonnegint))); #remove powers of and scan for remainders
%% union %

{_C1, _C1^2, _C2^3}

Edit:
This one works also for products

e:=_C1^2+_C1+_C2^3+a+b:
pw_ind:=indets(e, And(symbol, suffixed(_C, nonnegint))^anything):
lin_ind:=indets(subs(seq(pw_ind[i]=0,i=1..nops(pw_ind)),e), And(symbol, suffixed(_C, nonnegint))):
pw_ind union lin_ind
                       /        2     3\ 
                      { _C1, _C1 , _C2  }
                       \               / 

Maple assumes all names complex. Just look at the two first terms of the sum when we force Re on it

[op(1..2,lhs(P))];
map(Re,%);
add(%);

Without assumptions Maple cannot tell which term is real and which is imaginary.

PS.: In one of the terms there might be a space missings

In case you are interested in a way to get all solutions in one go (Roubens answer contains them in the text passage: i.e.: m=0 is allways a solution and the two other solutions are only opposite in sign)

The variant below creates a list of time stamps, solves your equation eqm for m, substitutes both into eqG and sorts the results.

eqm := m = tanh(6*m/T);
eqG:=G = -T*ln(2.0*cosh(6*m/T)) + 3*m^2;# 
T_list:=[seq(t,t=0.1..4,0.1)]:
m_list:=[seq((op~)([fsolve(subs(T=t,eqm),{m=-2..2},maxsols =3)]),t in T_list)]:
G_list:=[seq([seq(evalf(subs(m_list[j][i],T=T_list[j],eqG)),i=1..nops(m_list[j]))],j=1..nops(T_list))]:
for i to nops(T_list) do T=T_list[i],zip((x,y)->[x,y],m_list[i],G_list[i])[]; end do;

I have derived the answer from an attempt of using solve and getting all the roots from a RootOf expression of the symbolic solution. Neither allvalues nor evalf(RootOf) worked as I had hoped-for. In the attached document I have solved the argument of the RootOf expression with fsolve. This turned out the be numerically more demanding (Digits had to be increasesd to find all roots) than directly solving the problem numerically (as above). Such a hybrid approach might be usefull in other situations but not here.

tanh_cx_-_x.mw

Using op, you can disasemble expressions into their operands. I have highlighted this in yellow below.

The result represents a sum over _alpha where the summation index _alpha equals a RootOf expression. The RootOf expression is a placeholder for all roots of an expression. For some expressions RootOf can be evaluated.See ?RootOf for more. In your case you get 4 roots to choose from. 

allvalues(RootOf(1 + C1*C2*L1*L2*_Z^4 + (C1*C2*L1*R2 + C1*C2*L2*R1)*_Z^3 + (C1*C2*M^2*omega^2 + C1*C2*R1*R2 - C1*L1 - C2*L2)*_Z^2 + (-C1*R1 - C2*R2)*_Z));

This answer comes delayed (MaplePrimes service was not available for me).

If op is an option for you:

op([1,1],Zin)*op(2,Zin)+op([1,2],Zin)*op(2,Zin)

or a bit more generic for longer sums (if map is acceptable as well)

map(op(0,Zin), op(1,Zin), op(2,Zin))

For code readability and avoidance of errors, I am not a big fan of using op. In this case it might be an acceptable compromise. Whenever I have to use op, I check the simplified result for equivalence.

acers variants shine a light on how a custom-made "simplification" routine could be implemented not using "low level" command as op.

I agree that a command would be desirable that expands a quotient (having a sum in its numerator) without applying any further algebraic transformation to the expression (as denom and numer do to Zin).

I would also welcome a command that can collect more complex expressions (like the denominator in your example) from an expression than it is possible today.