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Editor-in-Chief of Maple Transactions (www.mapletransactions.org), longtime Maple user (1st use 1981, before Maple was even released). Most obscure piece of the library that I wrote? Probably `convert/MatrixPolynomialObject` which is called by LinearAlgebra[CompanionMatrix] to compute linearizations of matrix polynomials in several different bases. Do not look at the code. Seriously. Do not look. You have been warned.

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These are replies submitted by rcorless

They are






I added .html versions of those, as well.

Note that this is the first real use I have had of the `diff/f` feature of Maple in a very long time, but boy is it useful here.

We have a refereed section, so not only would your work be proofread, it would be read critically (with the concomitant possiblity of rejection, although in my opinion that would be unlikely; your work looks very good to me).

We have an unrefereed section, so for that one it would be proofread by our copy editor (and such editors who are interested, such as me) with the purpose of improving its readability if necessary.  We also do a light code review, which would take care of that "student" package issue, although I am not aware of anything that would cause problems there. 

Dear C_R,

Please submit this as a paper for Maple Transactions!  We accept Maple documents as "publications" (they wind up in the Cloud, and readable that way) but they count as refereed publications---if you want.

I'd like to see this one in the journal.  I published a paper some years ago myself on a related topic (Jeffery-Hamel flow) https://doi.org/10.1145/1277548.1277564 and the issues you raise are related.

@C_R Maple 2023, build 1689885.  No updates available, so this must be the latest...

@pallav I rather suspect that you didn't actually define A.  I noticed in your original that you just had A = ... and not A := ... but I didn't comment on it because that is an obvious typo (although to be fair it's a common error).

But if you say that something is not working, you have to show everything, in order to let us help you find where the error is!

At the very least, you should give us the error message!

In this case, the error arises because the third argument to "resultant" must be a variable name, not an expression.  But you have put the expression (the value of B) in to the place where the code is expecting a variable name.

@ogunmiloro This forum is run by and for volunteers, and normally one asks for help with something and people freely give it.  It's a good place.

Your request " Please correct type t versus T and do the remaining computation. Thanks" made me smile a bit.  To answer that kind of question would require a bill: I do some consulting, and I normally charge $1000USD per day.  To do your computation for you would require me to ask you several more questions, to make sure that I was solving the problem you really wanted solved.  I'm fairly sure that I could get an answer for you within a day or two, once all the questions were answered.  I'd actually charge only if the answer was satisfactory to you.

It might be worth it for you.  I don't know why you are working on this particular problem, but I do know that many people are interested in similar models, and many of them (perhaps you, also) are well-funded.  

But then it might not be worth it for you, or you might not be well-funded.  In that case, there would have to be some other benefit offered (I don't know, maybe a coauthorship on a paper, but that would need some discussion), for me to do your work for you.  Send me an email if you want to pursue that further.

Or, you could solve the problem yourself, given the answers you've received here.  That's the usual model of interactions here, I believe.  As I said, it's a good place.

If you do manage to solve the problem without me, and write it up in a nice way for publication, please consider Maple Transactions as a place to publish it.  This is an open access journal with no page charges.  See mapletransactions.org . I think that people may well be interested in that, and I would be happy to help you publish it (this time, free of charge---that's my one-day-a-week job :)


I think what you want is the RealTriangularize command in the RegularChains package.  I am trying it now; it's taking a while because it's using exact arithmetic.  In the end, if it succeeds, it will give exactly all the solutions (and a proof that there are no more).  If it doesn't succeed, I will ask the designer (Marc Moreno Maza) if he can make it work with some extra polishing of the input somehow.

@justauser We rewrote the modified equations code at least twice since 1997.  The most recent version is, however, ten years old (it's in my 2013 book with Nic Fillion). 

Here are two implementations from 2015




These are from the paper "Variations on a Theme of Euler" which is available for free at



You will see that the code is not at all general: it is applied to a single first order equation.  It is, however, extremely simple.

I should work up some general code, but most of my work nowadays is "ad hoc".



Yes, I have done work on modified equations :)

I have a few papers, but the most recent stuff is in Chapter 13 of my book with Nic Fillion.

Chapter 11 is on finite differences.


If your library has the Springer Bundle, then you will have free access to the PDF of the book (and can buy a paperback copy for $25, or 25 euros if you are in Europe, or 25 pounds if you are in the UK).

What I am hearing from you is that there is scope for a post here on Maple Primes on this, or maybe a paper in Maple Transactions.  

Ok, will put it on my list.  If you have a specific example that you would like to analyze, I'd like to try that.  Modified equations were first invented for PDE but they work better for ODE.



I've dug it up, but not the worksheet it came from (I have the Maple program for it).  It might be of interest; I'll try to update it for Maple 2023.  It was based on 1995 work with Jacek Rokicki of TU Warsaw, who sadly passed away not too long ago.  Updated in 2004 and 2005 by myself and Jichao Zhao we used it for compact finite difference schemes for finance and cardiac modelling.


The trouble with this kind of work is usually notation: what one person likes and understands, another person thinks is opaque and clumsy.  At least I have some historical quotes and references in the talk, anyway.


@sursumCorda Any nontrivial computer program has bugs.  It is *always* the user's responsibility to ensure correctness and fidelity.

(That maxim is well-known: the corollary is that if your program is bug-free, it is trivial :)

@Zeineb I see no variable Pe.  Later you say that one should take Z to infinity.  But you don't specify anything about the functions, so taking the limit of (say) f( x, y*Z^(1/3)) depends on the asymptotics of f as its second variable goes to infinity.  So I think that you have not specified enough information for anyone to get an answer.


Unless I am misunderstanding.

@abdulganiy I learned how to do this many years ago in an undergraduate numerical analysis course.  This was about the time that the wonderful book by Ascher, Mattheij, and Russell was being written on the subject.  Learning from that book would be the most thorough way to learn this material even today. https://epubs.siam.org/doi/book/10.1137/1.9781611971231


For a gentler introduction, you might do well to work from other sources.  My own chapter in my book (...Chapter 13 in a Graduate Introduction to Numerical Analysis) is very short and gives only the barest introduction, and depends heavily on other material.  There are the wonderful videos by Gil Strang and Cleve Moler (they start gently but get to interesting places very quickly) https://www.youtube.com/watch?v=ghjOS7Q82s0 

(full course at https://ocw.mit.edu/courses/res-18-009-learn-differential-equations-up-close-with-gilbert-strang-and-cleve-moler-fall-2015/ )

More references at http://www.scholarpedia.org/article/Boundary_value_problem


@Pepini Following the instructions in "parse" it works for me (to make the last one work one would have to insert a * in front of the microamp symbol, or otherwise deal with it).  Of course you can put the answers whatever you want; I just put them in a table called "piece" for convenience.  


This is an example of a problem with a "mass matrix", of the form M f' = K f  where M and K are both matrices.  In your case the matrix is a Vandermonde matrix (using nodes 1, 1/3, and 2/3).  Applying RK2 to the system *without* inverting M is an interesting thing to do; this approach (not with RK2 but with other RK methods) leads to good methods for stiff systems.

Is that what you were thinking about?  

If you use the approach for larger dimension, the condition number of the matrix M gets exponentially large, although there are very good and stable ways to solve it even so (I can point you to papers by Demmel and Koev if you are interested).  

A nice kind of problem to think about.  Might investigate later.

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