@berkeliox It looks like you may be trying to numerically solve an initial-value problem (IVP), particularly a system of three differential equations, and plot the three solution curves. Rather than trying to rehabilitate your code, it may be quicker for us to show you how to solve the system using Maple's stock commands for that purpose. To do that, we'd need to see the differential equations.

@acer I just noticed that the original Question contains the plaintext of the code. It isn't formatted, but at least it's there.

@berkeliox There are many problems with your code. Acer was only correcting the error that generated the error message that you originally presented.

The most-obvious problem to me is that i has no meaning outside of the for loops for which it is the index. In particular, it has no meaning in the plot command.

@Elisha Your code can't possibly be correct because you re-use the variable f, and you even use it recursively (occuring in the same assignment in which it is defined).

Unless your purpose is to learn about RK4, there's absolutely no reason for you to write code to do it. You stated earlier that your purpose was something other than learning about RK4. Even if you do have good reason to use RK4, there's still no reason to write code to do it because it's already part of dsolve.

I asked you to post your differential equations, initial conditions, and parameter values in plain mathematical form. I can't help you until to do that.

I saw the Question. Perhaps it was deleted by the same person who posted it. It was obviously a pair of homework problems posted in exactly the same form as the professor had assigned it. Perhaps the student decided that such plagiarism was wrong, or dangerous, and decided to delete it.

@shatha Did you receive the email that I sent to you? If so, then simply reply to that email.

@brian bovril A formula for counting partitions of fixed block sizes is

PartCount:= (B::list(posint))-> mul(binomial~(ListTools:-PartialSums(B),B))/mul(rhs~(Statistics:-Tally(B))!~):     
The number of partitions of a set of size 14 into 7 blocks of size 2:
PartCount([2$7]);
                             135135
The number of partitions of it into 5 blocks with the sizes as even as possible:
PartCount([3$4, 2]);
                            1401400

I just derived the formula from fundamental counting principles. I don't know if there's a more-efficient form for it. A recursive form may be more efficient computationally.

@brian bovril You haven't set up the distance matrix correctly. In the paper, read very carefully the first two sentences after the graph on page 43. Note how the main diagonal entries are defined differently. This means that in your Maple Matrix, the first superdiagonal is completely wrong. The main diagonal from the paper should become the first row of your Maple Matrix. This being a real-world problem, there can be no violations of the triangle inequality.

I'm convinced that both problems can be easily solved in Maple by exhaustive search. Use Iterator:-SetPartitionFixedSize to step through the partitions. For the first problem, use 7 blocks of size 2. There are only 135,135 such partitions. For the second problem, use 4 blocks of size 3 and 1 block of size 2. There are only 1,401,400 such partitions. For each block of size 3, there are 3 permutations to check (because reversing a permutation must give the same distance). But there are many repeats of the size-3 blocks! There are only 364 possible 3-blocks. For each of these, find the min of its 3 permutations and store this in a remember table.

So I expect the first problem to take about 10 lines of Maple code, and the second problem about 15. I'll be able to work on this after I sleep some.

@vv If the partitions are generated using Iterator:-SetPartitions and processed one at a time, then the memory usage will be trivial. I'd guess that once that memory issue is addressed, processing 28 million partitions wouldn't take an unreasonable amount of time. However, I haven't had a chance to try this yet.

Since the distance matrix violates the triangle inequality, yet the distances are so precisely specified (all to 3 significant digits), I wonder where this problem comes from.

Also, would you please state the problem in (mathematical) English rather than just showing Maple commands?

@shatha Yes, let's ignore the case of resonance for now. It's just as unlikely to happen as the coefficient matrix A being singular.

As you can see above, your pictures didn't come through. I'm going to send you a private email so that you can reply and send me the pictures directly.

@shatha Okay, now we're getting somewhere. You're trying to solve a system of differential equations by using the Poincare-Dulac Theorem (see this Encyclopedia of Mathematics article) to derive an iterative algorithm. I will change the title of this Question to reflect this, unless you object. Now please write your system of differential equations in standard mathematical notation and post them here.  DO NOT attempt to write them as Maple code. Then write an overview of the algorithm, in standard mathematical notation. Then write out the first several steps of the application of the algorithm to the particular system of differential equations at hand. Do not attempt to write any of this in Maple code!

You misspelled a word that I couldn't figure out. In the phrase "it is beuase deuation of dgree three", I can't figure out the word "deuation". What is it?

In your first line of code, the Matrix contains the symbol i. I can't figure out what you mean by that. Is it an integer iteration index?

Is your goal to solve a particular system of differential equations? Or is it to write a procedure to apply this Poincare-Dulac method in general?

@tomleslie You're absolutely right,Tom, about sequencing presenting a completely different situation with respect to the timings. Sorry that I forgot to include that case. Indeed, my first thought upon awakening this morning was that I had forgotten it.

This may seem counter-intuitive at first, but if you assign values to D__0 and R with assignment statements (using :=), then you guarantee that D__0 and R cannot appear in any subsequent results returned by Maple.

You are confusing moderators with administrators. You and I are both moderators. Moderators don't have the power to remove the material that you're referring to, but administrators do.

But, yes, I wholeheartedly agree that those accounts should be removed.