@Venkat

Hi, I am glad that this conundrum about the speed at which dsolve/numeric computes solutions has arrived at a conclusion and you agree that it runs at the speed of C code (it actually *is* C compiled code for the ODE system you bring).

Your observations about other things like the implementation of the algorithm, choice of solver and the method for solving nonlinear DAEs, perhaps in the presence of mathematical functions (your system has exp(u[i]) all around where u[i] are the unknowns), are rather valuable and I am sure Allan is probably already thinking about them. I think this debate is relevant and not annoying at all.

Your comments about Matlab and Mathematica are relevant too, although I think that providing files to be sent to non Maple users (as you point that Matlab does) may not be so relevant as being able to tackle large systems within the Maple workshet (when the system is tackable - see next paragraph).

I note however only one thing, Venkat: not being the expert in DE numerical methods but being sufficiently familiar with them, the system you posted here, where the algebraic equations contain many exp(u[i]) and u[i] are the unknowns, is *very hard* to decouple symbolically, I don't think that a program for handling that kind of coupling with N >= 8 (implies on ~20 equations) exists, and tracing dsolve/numeric I see that its symbolic preprocessing gets stuck precisely there, before writing or compiling any C code. So my question to you: do you say that you can tackle exactly this same system using Matlab or writing C code yourself for it? Or is this one example that you don't know who to solve using Maple and anything else?

Independent of that, using Maple, in my laptop, dsolve/numeric takes 46 seconds to solve the problem for N = 7 (i.e., 18 equations), with the symbolic preprocessing of these algebraic equations having exp(u[i]) taking most of the time. Then it hangs for N = 8 as said. There may be a problem in the symbolic preprocessing.

Edgardo S. Cheb-Terrab
Physics, Maplesoft

@Alejandro

We have a different view on this one. I don't think Markiyan showed "an algorithmic way of finding a simpler solution for a class of problems". No class of problems is identified in this post. His approach also has human educated guess, a transformation cos and sin -> tan, actually based on dsolve's userinfos, that works fine in the example he presented but that can other times produce the opposite, i.e. a more complicated solution. In the same line, Preben nicely refined Markiyan's presentation, but also didn't identify a class of problems for which the linear combination sys[1] + sys[2] followed by u = y+z would systematically result in a simpler form of solution.

Not found in that post, you elaborate here, in this thread, on top of Preben's approach to that example, saying that the system could be written in matrix form and decoupled. There are other ways of decoupling, Alejandro. Concretely, dsolve also decouples the system before solving it: check PDEtools:-casesplit(sys). Again, depending on the example, one way of decoupling may work better in one case, the other in other cases, and until you identify the class of problems for which one approach works better than another one you have nada.

Having said all that, I do not discard that someone could come forward and show concretely, even easily, that "for cases where "these" conditions are satisfied: a) "this" transformation is most of the times better than the one used by dsolve and b) "a matrixDE decoupling" will result in simpler solutions than "the differential algebraic lexicographical decoupling used by dsolve". I'd be happy to read such a post and consider implementing the underlying ideas. However I do not think such an approach could be *derived* [i.e. appear as a consequence] of using DE packages as Markiyan asked and so my answer to him: perhaps his question is based in too high expectations.

Anyway, at this point, being that the ODE posted by Markiyan is actually solved correctly by dsolve - all this debate is only about getting a simpler solution from a command that more frequently than otherwise already returns simpler solutions - I personally prefer to focus other things that look to me really more urgent or relevant.

Edgardo S. Cheb-Terrab
Physics, Maplesoft

This presentation, about ODEs, PDEs and Special functions, claims among other things that dsolve numerically solves ODEs - not PDEs - at the speed of C code. This is rather relevant information. Most people are not aware of it. In the presentation, an example of a system with 50 coupled ODEs, 50 unknowns, and 50 Lagrange multipliers, is solved as an illustration.

@Mac Dude and @Venkat

On the debate about the claim above: it is indisputable that dsolve writes and compiles C code on the fly for solving ODE systems when you pass compile = true. In this context, whether you think you can write better C code for a particular ODE system is a different issue. And I do not dispute it, although note that I've not seen an example of that in the 10 comments above. One example, concrete, complete, is always helpful to make a point. It also helps improving things. Likely, to have more options than currently available to indicate to dsolve to write the C code in this or that manner - a valuable suggestion, I really like it - is however also a different issue.

I also read carefully the comments by Venkat and Allan's response. I do not think the technical details presented - valuable perhaps regarding improvements in dsolve/numeric - change the facts about the option compile, neither change my opinion that tackling ODE systems within a computer algebra worksheet is today possible at the speed of C code and tremendously advantageous.

By the way, Venkat, everywhere in computer algebra you have a similar situation where more specific problems may require non-trivial use of optional arguments to get an optimal result, optional arguments that may even not exist today. And yes, having expertise we sometimes cross with or can formulate problems that beat a computer algebra system. We grow studying these concrete examples when posted. That makes this debate potentially so valuable. But, hey, that also doesn't change the status of things about the speed at which dsolve/numeric runs. And by the way: no, I do not think at all that Maple's dsolve/numeric writes bad or mostly inconvenient C code when the system includes many ODEs, regardless of how much this automatic generation of C code can be improved.

But I don't want to close my summary with just reafirming that dsolve/numeric does write, compiles and runs C code to solve ODE systems. I also see a very detalied debate that perhaps triggers valuable further developments/improvements, and I am frankly glad with that and all your participation.

@Markiyan
In "Play a simple melody" you show an ODE system and the solution. You do not show how to obtain that solution. Perhaps the advanced status of things in this or that matter creates an expectation difficult to fulfill .. if the logic underlying a result is not understood (as in your post), then it cannot be coded. Your other post is more about computing a limit, and using Green functions for solving PDEs & Boundary conditions - some things are already in place, others not. I presented this post about ODEs, PDEs and Special functions aiming at illustrating the current status of things, that is really extraordinary, I think, but there is still a long way to go improving things; again, thanks for your comments Markiyan.

Edgardo S. Cheb-Terrab
Physics, Maplesoft

Hi
As mentioned in a previous comment in this thread we now provide access to the version of Physics under development, including adjustments/fixes/novelties as they are ready. A new update (today Aug/1) is available for download at http://www.maplesoft.com/products/maple/features/physicsresearch.aspx.

Edgardo S. Cheb-Terrab
Physics, Maplesoft

@Mac Dude

Taking real/good advantage of computer algebra in physics courses is to some point pioneering activity. As mentioned in the presentation it is also one of the main targets of the Physics project. Please feel free to bring your related questions/suggestions, here or writing us at physics@maplesoft.com, and we may be able to help you / join efforts.

Edgardo S. Cheb-Terrab
Physics, Maplesoft

@Mac Dude

When using computer algebra in physics, "casting expressions into certain forms" is the very relevant type of thing we (students or not) need to do all the time. One needs to learn these basic operations in order to take advantage of the computer in physics. And for these basic operations I think that a basic and really short, to-the-point, textbook/tutorial is best. Your first comment made me consider revamping/updating that tutorial I wrote for physics students years ago. In your second comment, however, you mention something else, like "pushing the limits of Maple", e.g. simplifying an expression for which simplify fails, etc. I think these are more advanced operations, or more technical things like the evaluation levels detail you linked, more relevant for people who go deeper into using computer algebra in general - as such I feel these topics are outside the sphere of the Physics package.

Edgardo S. Cheb-Terrab
Physics, Maplesoft

Hi
Just revising the documentation about dsolve's speed to see if the information I have is also available to you, Mac Dude. Check please ?dsolve/rkf45 and ?dsolve/numeric/efficiency: current dsolve runs as fast as C code when specifying the compile = true option in the call to dsolve/numeric. Using this option actually makes dsolve write the C code for you, compile it, and use it for solving the DE system, without you perceiving. It is all automatic, as said in the presentation. The limitations of "dsolve at the speed of C code" are also just those for regular C code you may nevertheless prefer to write yourself to use with a connection Toolbox: it only works with real valued DE systems that contain at most elementary functions (no special functions allowed) - see the list of functions in ?Compile,compile under Runtime Support.

Edgardo S. Cheb-Terrab
Physics, Maplesoft

Hi
These polemic questions and the answers I present in this talk actually passed through my mind at different times and reflect, honestly, what I think.

In 1996 I was interested in computing Poincare sections in the context of a general relativity problem. There was Fortran code for that (see Calzetta and El Hasi, Phys.Rev. D 51 - 1995). After struggling badly because the number of numerical experiments to be done was really large, it resulted to me much simpler to write a complete program to compute Poincare sections within a computer algebra system than trying to use the existing one for Fortran. The program is today found in Maple, DEtools[poincare].

I learned two things: 1) the advantages of running numerical computations within a computer algebra environment were just enormous. 2) the drawback at that time: speed. I didn't like the speed. No, it was not satisfactory. C was abut 30 to 100 times faster.

In 2000 this speed issue got on my nerves. I talked to Allan Witkoppf, my friend, at that time a Ph.D student and an expert in C. We came up with some interesting ideas that ended up in the Maple DNA project for fast numerical ODE solutions (google "DNA Maple 6"). Surprisingly, with simple ideas we got 15x to 30x speed up - hey! I understood one more thing: the numerical slowness of computer algebra systems was not a real or serious obstacle.

Along the next ten years Allan became the Maple numerical DE guy, great - and voila the speed, Dude. It is there. For real. Even DNA is now in the dust. Try your examples. Compare. I - honestly - believe and fully stand by this answer for polemic question 1. Yes, you can also use a connection Toolbox instead, as you say, and some people will prefer it, perhaps because already have C or Fortran code that they can reuse. Still I think the CAS worksheet is really better in the small and large scale, for the opportunity of reusing, symbolic preprocessing not available in Fortran or C, and including the speed.

About polemic question #2: I agree with you in that the computer is not exclusive of books. I see from your comment that the presentation could be misunderstood as suggesting "this or that". The intention is only to reflect a practice, though: people tend more and more to use one or the other, as in "only go to the books if the computer (including web) is insufficient". And not few people ask me about this regarding DEs. The last time I consulted DE books I believe it was ~10 years ago. For ODEs the computers went just far beyond textbooks. Not beyond humans. I currently do consult the arxiv.

You mentioned Abramowitz. It is also true, I have my copy. And always looked at it as a monumental piece of work. But then in 1999 I moved away from the paper version of it (google "abramowitz pdf", second link). Then I saw the DLMF (Digital Library of Mathematical Functions) and Marichev's and others emerging as static repositories of special function information. The FunctionAdvisor in Maple came after all that, with a different idea in mind: do not present 'static' information. Instead: process information, interrelating it on the fly, according to user's input, using an increasing number of new algorithms popping up all around.

There is still a long way to go. I think however that the paradigm has shifted already. Textbooks and static presentations are entering the rear mirror. Core pieces of mathematical information processed on the fly with each-day-more-varied-algorithms is in front of us. This is what is illustrated in the presentation attached.

The core of the special functions part was also presented in "Special Functions in the Digital Age" (IMA 2002), while a previous version of most of this DE material, including the questions, was also presented in the session for "Teaching and Learning Differential Equations" of the meeting of the CMS 2000.

Edgardo S. Cheb-Terrab 
Physics, Maplesoft

Hi
Thanks for your comments. The learning curve: basic or advanced quality textbooks? Struggling to do seemingly trivial things like casting expressions into certain forms … I'd say compact-and-basic is the remedy, with compact & quality in bold. For teaching in Brazil, State University of Rio de Janeiro, I tried to write such a mini-and-basic for physics students; I may revamp it. For advanced: what is what you'd like to see in such a text that would smooth out the learning curve?

Examples and textbook notation: I'm glad to hear that you find these examples useful. The are mostly those shown in ?Physics,Examples. Physics and mathematical methods are not the same thing, and Feynman's and Landau's books (not only them, of course) are great in that in their problems illustrate both for real. BTW three of the four examples under "Mechanics" are from Landau's vol.1.

The 'notation' issue actually refers to keyboard input, and the display of results on the screen, so not palettes. To enter things, say, as you do when computing with paper and pencil, and see the output as you see it in textbooks. It is impressive for me how this speeds up matters in my brain. Even after working with computer algebra for so many years, I'm still not used to (sounds alien to me) computerish style representing mathematical objects so differently than the way we do it by hand, ditto for the output. All the efforts are put in Physics to not do that.

New: we now provide access around the clock to the version of Physics under development (http://www.maplesoft.com/products/maple/features/physicsresearch.aspx). This includes ways for you to present your suggestions/report bugs/feedback. The downloadable package, post 17.01, is at "zero known bugs" in this moment and we intend to keep it this way. Not having to wait for adjustments until the next release anymore.

Edgardo S. Cheb-Terrab
Physics, Maplesoft

Download you_can_set_g_to_wha.mw

@ecterrab 

Download you_can_set_g_to_wha.mw

@ecterrab 

@peter137 Yes I have in mind additional documentation; that is key in this project and chunks are added at each release, with some people now complaining because of the current large size of ?Physics,examples ... These examples are mainly from the Landau books or from the other books shown in the references (footnote of ?Physics).

What would be your suggestion for showing the use of the package that at the same time is of your interest and you think would be of wide range interest?

Edgardo S. Cheb-Terrab
Physics, Maplesoft

@peter137 Yes I have in mind additional documentation; that is key in this project and chunks are added at each release, with some people now complaining because of the current large size of ?Physics,examples ... These examples are mainly from the Landau books or from the other books shown in the references (footnote of ?Physics).

What would be your suggestion for showing the use of the package that at the same time is of your interest and you think would be of wide range interest?

Edgardo S. Cheb-Terrab
Physics, Maplesoft

@Alejandro Jakubi as said in this and another thread this problem is fixed. But you are suggesting something else. Note that in the output you show by simplify/size you have w * conjugate(w) -- that is a non-simplified product, and as such it is not good as output of a simplify command. The output after the fix in fact is as the one you show by simplify/size but with that w * conjugate(w) also simplified.

Besides that there is simplify/size itself being discovered. To have in mind: when writing it tried thoroughly to avoid performing any mathematical operation - nothing - only collect coefficients (including of frozen subexpressions) and at most decomposition of fractional powers, that can be done fast. Anything else would ruin the performance of this wonderful routine, that was originally developed for, and is systematically in use in all the Maple DE, Physics and MathematicalFunctions code, and it is not used by simplify by default only because of backwards compatibility debates.  

And one idea to keep in mind: wouldn't it be beautiful if *every* output, not just the one of simplify, would always be preprocessed with simplify/size? Exception made only (from the top of my head) of the output of collect, factor and expand. I am talking of a kind of a setting. One that you could turn ON/OFF optionally, say as simply as typing > on;, or >off; and voila the thing activated/deactivated :) No? :) I think so, and work everyday with this feature implemented - it becomes addictive. At every release I consider adding it to the system. Perhaps ... It would need some tunning but mostly facing a significant break with so many years of (for me boring or frequently wallpaper) computer algebra output.

Edgardo S. Cheb-Terrab
Physics, Maplesoft

Hi
The comparison you mention consists of mainly 3 pages, of which less than 1/2 is about mathematical features (starting at 'Graph theory' and finishing at 'Cayley tables'), so to begin with there are obvious and significant omissions.

To mention but a few in areas I am more familiar with, the new Maple 17 functionality in Physics, Differential Equations and Differential Geometry - not mentioned at all in the comparison - is not only not available in Mathematica 9 or previous ones, but it is also probably 5 to 10 years of development ahead of what you see in Mathematica. This is the opposite of what is claimed in that comparison advertisement. Moreover, this distance is favorable to Maple, consistently, since year 2000, and consistently increasing every year. For Differential Equations the difference started shockingly (that is the word) in 1997 at least and Mathematica just never recovered.

Independent of these stark omissions, regarding the half page of mathematics that is mentioned in that comparison, either the person who wrote it does not understand the topics, or distorted the facts.

For example, quoting literally from this comparison you mention, it says that Maple 17: "a) introduced Visualization: visualize branch cuts, b) this Maple's functionality is not equivalent to Mathematica's automatic branch cut detection—it allows you to request a visualization of the branch cuts of individual functions but does not detect branch cuts in functions that you try to plot, c) that Mathematica introduced this in 2007". Statement c) is supposed to indicate that Mathematica introduced this functionality before Maple.

Both statements a) and b) about Maple are incorrect and the implication of statement c) is false.

Maple introduced visualization of cuts with its plots:-plotcompare not in Maple 17 but already in 2002, so 5 years before Mathematica and in a way still not seen in Mathematica 9; the Maple visualization is not restricted to cuts of individual functions, and of course the Maple plotting routines detect the branch cuts of the expressions you try to plot. See for instance the examples in the help page for ?plotcompare. But more important, Maple 17: a) introduced *algebraic representations* for the branch cuts of *algebraic expressions* not just individual functions, and this functionality does not exist at all in Mathematica - in fact it is new research work, see references in ?FunctionAdvisor,branch_cuts - and b) Maple introduced *algebraic representations* for branch cuts of individual functions already in 2003, with the FunctionAdvisor project, and neither those algebraic representations for the cuts even of individual functions, nor anything like the FunctionAdvisor project, ever existed in Mathematica, also not in the new Mathematica 9.

You see ...

Edgardo S. Cheb-Terrab
Physics, Maplesoft