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Aaron Meurer On Mon, Apr 29, 2013 at 11:53 AM, Aniket Anvit <[email protected]> wrote: > > > > On Mon, Apr 29, 2013 at 5:02 PM, <[email protected]> wrote: >> >> Today's Topic Summary >> >> Group: http://groups.google.com/group/sympy/topics >> >> Diophantine Equations Module [4 Updates] >> GSoC 2013: Equation editor [4 Updates] >> GSoC 2013: Univariate polynomials over algebraic domains [5 Updates] >> Lie algebras GSOC application [4 Updates] >> Introduction and a possible project [2 Updates] >> GSoC 2013: Error-correcting codes [1 Update] >> Diophantine Equations Module [1 Update] >> sympy.nsolve seems to solve system of non linear equations incorrectly [1 >> Update] >> >> Diophantine Equations Module >> >> Thilina Rathnayake <[email protected]> Apr 28 01:07PM -0700 >> >> Hi, I would like to implement a Diophantine equations module for Sympy. >> You can find my pull request here >> <https://github.com/sympy/sympy/pull/2024>. >> It's still not merged. >> >> I hope to solve following classical Diophantine equations. All the >> variables and constants >> used here are integers. >> >> 1)* **a1x1 + a2x2 + a3x3 + ...+ anxn = b* (Linear diophantine equation) >> Here *a1, **a2, ... **an* and b are constants.If solvable (there is a >> condition to determine this), >> solving this equation means expressing any two variables using other >> variables and an >> arbitrary integer* *n. i.e. solution is given by >> x1 = x1, x2 = x2, ... xn-2 = xn-2 , xn-1 = f( x1, x2, ... xn-2, n), xn-1 = >> g( x1, x2, ... xn-2, n) >> f and g are functions to be determined. >> >> 2) x12 + x22 + x32 + ... xn2 = k >> Here k is a non-negative constant. There will be a number of solutions >> depending on >> n and k. Solving this means assigning constants *a1, **a2, ... **an* to >> x*i >> *'s respectively. >> >> 3) x12 + x22 + x32 + ... xn2 = xn+12 (extension of Pythogorean equation) >> Solving this is pretty standard. There is a general primitive solution set >> using n relatively >> prime integers. All other solutions can be obtained by multiplying those >> equations by >> an arbitrary integer. >> >> 4) x2 + axy + y2 = z2 >> Here a is a constant. If z is a variable, a general solution can be given >> to this equation >> using a and three arbitrary integers. If z is a constant actual solutions >> can be given. >> >> 5) x2 - Dy2 = m2 (Pell's equation) >> Here D and m are constants. This has either no solution or infinitely many >> solutions. >> ax2 - by2 = 1 and ax2 + bxy + cy2 + dx + ey + f = 0 can also be solved >> with the >> light >> of Pell's equation. >> >> Lot of Diophantine equations can be converted to one of these forms. >> Addition >> of this kind >> a module will be a huge enhancement for Sympy. I would like to know how I >> can improve >> this. Thanks in advance. >> >> >> >> References >> [1] An Introduction to Diophantine Equations*, Andreescu*, Titu, >> *Andrica*, >> Dorin, *Cucurezeanu*, Ion >> [2] http://mathworld.wolfram.com/DiophantineEquation.html >> [3] http://en.wikipedia.org/wiki/Diophantine_equation >> >> Regards, >> Thilina Rathnayake. >> >> >> >> Aaron Meurer <[email protected]> Apr 28 02:22PM -0600 >> >> Just to be clear, is this for GSoC, or is this just something that you >> want to implement on your own? >> >> On Sun, Apr 28, 2013 at 2:07 PM, Thilina Rathnayake >> > x1 = x1, x2 = x2, ... xn-2 = xn-2 , xn-1 = f( x1, x2, ... xn-2, n), xn-1 >> > = >> > g( x1, x2, ... xn-2, n) >> > f and g are functions to be determined. >> >> The n in the indices is different from the other n, right? >> >> Is the rank of the system always 2 in this case? >> >> > a module will be a huge enhancement for Sympy. I would like to know how >> > I >> > can improve >> > this. Thanks in advance. >> >> It sounds like a good start. If this is for GSoC, I'd like to see more >> details. >> >> One thing to be aware of is that you will probably spend more time >> worrying about how to pattern match these things than writing the >> algorithms to solve them. SymPy has .match, but it can be limited. >> For example, you can easily write a pattern to match >> >> x1**2 + x2**2 = k >> >> But if you can solve that, then you can also solve >> >> (x1 - 1)**2 + (x2 - 1)**2 = k >> >> or >> >> x1**2 - 2*x1 + x2**2 - 2*x2 = k >> >> by a simple shift. But the same simple pattern won't match either of >> these. I ran into this kind of issue all the time when I wrote the ODE >> module, which works very similarly (by the way, you should take a look >> at the design there, as I think the design of this module can be very >> similar). We should extend the pattern matcher's abilities to be able >> to still succulently write patterns, but to allow them to match more >> advanced things like shifts automatically. >> >> Of course, the worst case scenario is that it won't recognize a given >> equation as being in a certain form, so it just won't be able to solve >> as much as it could. So if you want, you could focus on this, or you >> could leave it to someone else. >> >> Aaron Meurer >> >> >> >> >> Thilina Rathnayake <[email protected]> Apr 29 11:21AM +0530 >> >> Thanks Aaron for your reply. I am proposing this as a GSoC project. >> Sorry, I didn't mention it earlier. However, even if this is not accepted >> as a >> GSoC project, I would like to work on this. >> >> The two n's are different. I should have used distinct letters instead of >> the n's. >> Sorry for the confusion. >> >> I have assumed that all ai's are nonzero and the above linear Diophantine >> equation (DE) satisfies the condition for the existence of solutions. If >> n>=3, >> then solutions to the above linear equation can always be given using n-1 >> parameters. If we define first n-2 xi's as parameters, then with another >> parameter m, we can express the solutions for the other two. For n=2, >> solutions >> can be determined using only one parameter, m. For n=1, If a solution >> exists, >> it can be computed by dividing b by a1. I guess I answered your question. >> If not please let me know. >> >> I checked the ODE module and indeed it can be used as a reference model >> for this project. I hope to implement a method DiophantineSolve(), which >> would >> take DE to be solved in the form f(x1*,* x2*,* x3*,* ...xn) = 0 or f(x, y, >> z) = 0 as an >> argument. All of the proposed equations can be expressed in such a way by >> a >> simple >> rearrangement. It will return the solutions/solution if they/one exist(s). >> >> For inner computations a classification function would be needed as in ODE >> module. >> That can be used to determine whether the DE is linear, quadratic, >> Pythogrean, .. etc. >> Algorithm used for the solution will depend on the result of this >> classification. >> >> I am still studying the ODE module. I am not familiar with pattern matcher >> either. >> I would take a look at that also. Comments and suggestions are >> appreciated. >> >> Thanks in advance. >> >> Regards, >> Thilina Rathnayake. >> >> >> >> >> >> Aaron Meurer <[email protected]> Apr 29 12:02AM -0600 >> >> On Sun, Apr 28, 2013 at 11:51 PM, Thilina Rathnayake >> > Sorry, I didn't mention it earlier. However, even if this is not >> > accepted >> > as a >> > GSoC project, I would like to work on this. >> >> OK. In that case, you should get writing on your application, as the >> deadline is Friday. What you wrote here is a good start. You will also >> need to delineate it into a timeline, think (a lot) about the >> interface (my personal opinion is that you should mimic the ODE module >> exactly, but your application should demonstrate that you understand >> that structure), and give more details to demonstrate that you really >> do understand the theory (maybe show how to solve an example problem). >> >> > Pythogrean, .. etc. >> > Algorithm used for the solution will depend on the result of this >> > classification. >> >> Yes, I believe the structure can match the ODE module almost exactly, >> at least as far as matching having various hints goes. You might also >> want to look at the constantsimp stuff if your solutions will result >> in a dependence in new, arbitrary parameters. I wouldn't worry about >> that for the outset, though, even if that is the case. >> >> >> > I am still studying the ODE module. I am not familiar with pattern >> > matcher >> > either. >> >> The pattern matcher is very simple. Just set up the variables you want >> to match as Wild, create the pattern expresion, and use matches. The >> issue is that it is very stupid in that it matches almost nothing >> beyond what you explicitly tell it to. You also need to be careful to >> always set the exclude parameter on the Wild objects to exclude your >> variables. Otherwise, if you do something like try to match a*x + b*y >> against 3*x + 2*y (here a and b are the Wilds and x and y are the >> Symbols), you will expect to get {a: 3, b:2}, but you might instead >> get {a:0, b:3*x/y + 2}. >> >> Aaron Meurer >> >> >> >> >> GSoC 2013: Equation editor >> >> Alexander Gudulin <[email protected]> Apr 28 01:40PM -0700 >> >> Hi everybody, >> >> My name is Alexander Gudulin, I am a second year student at Saint >> Petersburg State University, Russia. >> I'm interested in solving "Equation editor" problem. Wiki says it would be >> great to discuss this idea with you guys. >> >> Have you got any additional thoughts about the problem or should I suggest >> any ideas? >> >> >> >> Aaron Meurer <[email protected]> Apr 28 09:38PM -0600 >> >> It's fairly open ended, in that you'll need to come up with most of it >> yourself (we haven't thought too hard about it). You should probably >> focus on the IPython notebook. >> >> Aaron Meurer >> >> On Sun, Apr 28, 2013 at 2:40 PM, Alexander Gudulin >> >> >> >> Chris Smith <[email protected]> Apr 29 10:14AM +0545 >> >> I wonder if the desmos editor could be used to do this. Perhaps code that >> communicates between sympy and a web page? >> >> https://www.desmos.com/frontpage?utm_expid=43664205-2 >> >> >> >> Aaron Meurer <[email protected]> Apr 28 11:53PM -0600 >> >> Maybe I missed it, but can that even be used as a library? It looks >> like it is all closed source, and stuck within its own ecosystem. >> >> But I agree that if you can find an already existing equation editor >> that can be hooked up to SymPy that that would be a good alternative >> to writing one from scratch. >> >> Also, I don't think it would be too much work to extend the Unicode >> pretty printer to get a curses based equation editor. If you search >> the list, I've explained the idea in a little more depth in the past. >> >> Aaron Meurer >> >> >> >> >> >> GSoC 2013: Univariate polynomials over algebraic domains >> >> Katja Sophie Hotz <[email protected]> Apr 28 01:27PM >> -0700 >> >> I just finished a first version of my GSoC application. As it turned out, >> some of the stuff I wanted to do is already implemented, so I changed the >> direction of my proposal a bit. >> The new title is Faster Algorithms for Polynomials over Algebraic Number >> >> Fields<https://github.com/sympy/sympy/wiki/GSoC-2013-Application-Katja-Sophie-Hotz:-Faster-Algorithms-for-Polynomials-over-Algebraic-Number-Fields> >> . As far as I can see these algorithms would be new to SymPy. >> >> I would be very grateful for any feedback. >> >> Thank you in advance, >> Katja Sophie >> >> >> >> Aaron Meurer <[email protected]> Apr 28 03:05PM -0600 >> >> Don't forget to submit this in Melange. >> >> Aaron Meurer >> >> On Sun, Apr 28, 2013 at 2:27 PM, Katja Sophie Hotz >> >> >> >> David Joyner <[email protected]> Apr 28 05:30PM -0400 >> >> On Sun, Apr 28, 2013 at 4:27 PM, Katja Sophie Hotz < >> > >> > Fields<https://github.com/sympy/sympy/wiki/GSoC-2013-Application-Katja-Sophie-Hotz:-Faster-Algorithms-for-Polynomials-over-Algebraic-Number-Fields> >> > . As far as I can see these algorithms would be new to SymPy. >> >> > I would be very grateful for any feedback. >> >> >> >> I realize that this is not your fault, but the fact that the Sympy >> documentation [1] lacks >> examples of algebraic numbers will make it more difficult to evaluate the >> feasibility of >> your proposal. >> >> >> [1] >> >> http://docs.sympy.org/dev/modules/polys/reference.html#algebraic-number-fields >> >> >> >> >> >> Aaron Meurer <[email protected]> Apr 28 04:04PM -0600 >> >> Algorithms like factor can handle algebraic numbers using the extension >> flag >> >> In [182]: factor(x**2 + 1, extension=[I]) >> Out[182]: (x - ⅈ)⋅(x + ⅈ) >> >> In general, algebraic numbers can be slow, because minpoly is slow >> (this is being fixed at https://github.com/sympy/sympy/pull/2038). I >> think multiple extensions are also slow for other reasons. >> >> Currently, only algebraic numbers are supported, but it would be great >> to support algebraic functions (like sqrt(x) instead of sqrt(2)). >> >> Aaron Meurer >> >> >> >> >> Katja Sophie Hotz <[email protected]> Apr 28 03:55PM >> -0700 >> >> As far as I know, the modular gcd algorithm and the factorization >> algorithm >> from my proposal >> can be extended to algebraic function fields, but I don't think there will >> be enough time to go that far in one summer. >> >> >> >> Lie algebras GSOC application >> >> Mary Clark <[email protected]> Apr 28 01:38PM -0700 >> >> Hi all, I've written up a preliminary version of my application: >> >> https://github.com/sympy/sympy/wiki/GSOC-2013-Application-Mary-Clark:-Lie-Algebras >> Any comments/suggestions/etc would be welcome. >> >> Mary >> >> >> >> Aaron Meurer <[email protected]> Apr 28 03:05PM -0600 >> >> Don't forget to submit it in Melange. >> >> Aaron Meurer >> >> >> >> >> Alan Bromborsky <[email protected]> Apr 28 06:04PM -0400 >> >> On 04/28/2013 04:38 PM, Mary Clark wrote: >> > To post to this group, send email to [email protected]. >> > Visit this group at http://groups.google.com/group/sympy?hl=en-US. >> > For more options, visit https://groups.google.com/groups/opt_out. >> >> You might find the attachment of interest - >> >> >> >> Prasoon Shukla <[email protected]> Apr 28 03:33PM -0700 >> >> You should also add a link to your application here: >> https://github.com/sympy/sympy/wiki/GSoC-2013-Current-Applications >> >> >> >> Introduction and a possible project >> >> Stefan Krastanov <[email protected]> Apr 28 10:30PM +0200 >> >> Hi, >> >> Check our github wiki pages for GSoC 2013 information. Be aware that >> we require all applicants to submit a patch on github (to gauge their >> abilities with python and git). >> >> >> >> >> Saurabh Jha <[email protected]> Apr 28 01:34PM -0700 >> >> Hi Saurav, >> >> You may find this discussion helpful >> >> >> http://groups.google.com/group/sympy/browse_thread/thread/fbbb7effcaf92fe4/c6f6d9c1a4543507?lnk=gst&q=Karr+algorithm+Saurabh+Jha#c6f6d9c1a4543507 >> >> Also, I would advice to get started with fulfilling patch requirement, >> check application template on wiki pages >> >> -Saurabh Jha >> >> >> >> >> GSoC 2013: Error-correcting codes >> >> Aaron Meurer <[email protected]> Apr 28 02:06PM -0600 >> >> I don't know (it depends on how much work Matuesz does). A contingency >> would be to find suitable work-arounds. >> >> Note that even the cyclic finite fields also need a lot of work. For >> example, they are currently not instances of Basic, meaning that just >> doing Matrix([[FF(2)(1)]]) does not work because of >> https://code.google.com/p/sympy/issues/detail?id=3784. So I would >> plan on doing a good chunk of work on just fixing those. >> >> If those objects are Basic, then I think that they should just work if >> you dump them into the matrices. It won't be as fast as it will be >> when the matrices get proper domains, but at least it will work, and >> as soon as the matrices are rewritten, it will automatically become >> faster. >> >> Aaron Meurer >> >> >> >> >> Diophantine Equations Module >> >> Thilina Rathnayake <[email protected]> Apr 28 01:04PM -0700 >> >> Hi, I would like to implement a Diophantine equations module for Sympy. >> You can find my pull request here >> <https://github.com/sympy/sympy/pull/2024>. >> It's still not merged. >> >> I hope to solve following classical Diophantine equations. All the >> variables used >> here are integers unless otherwise stated. >> >> 1)* **a1x1 + a2x2 + a3x3 + ...+ anxn = b* (Linear diophantine equation) >> Here *a1, **a2, ... **an* and b are constants.If solvable (there is a >> condition to determine this), >> solving this equation means expressing any two variables using other >> variables and an >> arbitrary integer* *n. i.e. solution is given by >> x1 = x1, x2 = x2, ... xn-2 = xn-2 , xn-1 = f( x1, x2, ... xn-2, n), xn-1 = >> g( x1, x2, ... xn-2, n) >> f and g are functions to be determined. >> >> 2) x12 + x22 + x32 + ... xn2 = k >> Here k is a non-negative constant. There will be a number of solutions >> depending on >> n and k. Solving this means assigning constants *a1, **a2, ... **an* to >> x*i >> *'s respectively. >> >> 3) x12 + x22 + x32 + ... xn2 = xn+12 (extension of Pythogorean equation) >> Solving this is pretty standard. There is a general primitive solution set >> using n relatively >> prime integers. All other solutions can be obtained by multiplying those >> equations by >> an arbitrary integer. >> >> 4) x2 + axy + y2 = z2 >> Here a is a constant. If z is a variable, a general solution can be given >> to this equation >> using a and three arbitrary integers. If z is a constant actual solutions >> can be given. >> >> 5) x2 - Dy2 = m2 (Pell's equation) >> Here D and m are constants. This has either no solution or infinitely many >> solutions. >> ax2 - by2 = 1 and ax2 + bxy + cy2 + dx + ey + f = 0 can also be solved >> with the >> light >> of Pell's equation. >> >> Lot of Diophantine equations can be converted to one of these forms. >> Addition >> of this kind >> a module will be a huge enhancement for Sympy. I would like to know how I >> can improve >> this. Thanks in advance. >> >> >> References >> [1] An Introduction to Diophantine Equations*, Andreescu*, Titu, >> *Andrica*, >> Dorin, *Cucurezeanu*, Ion >> [2] http://mathworld.wolfram.com/DiophantineEquation.html >> [3] http://en.wikipedia.org/wiki/Diophantine_equation >> >> Regards, >> Thilina Rathnayake. >> >> >> >> sympy.nsolve seems to solve system of non linear equations incorrectly >> >> Pushpak Dagade <[email protected]> Apr 28 09:08AM -0700 >> >> Thanks a lot chris for your timely help! >> I hope this helps me out in my work ahead. >> >> On Friday, 26 April 2013 01:44:21 UTC+5:30, smichr wrote: >> >> >> >> You received this message because you are subscribed to the Google Group >> sympy. >> You can post via email. >> To unsubscribe from this group, send an empty message. >> For more options, visit this group. >> >> -- >> You received this message because you are subscribed to the Google Groups >> "sympy" group. >> To unsubscribe from this group and stop receiving emails from it, send an >> email to [email protected]. >> To post to this group, send email to [email protected]. >> Visit this group at http://groups.google.com/group/sympy?hl=en-US. >> For more options, visit https://groups.google.com/groups/opt_out. >> >> > > > -- > You received this message because you are subscribed to the Google Groups > "sympy" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected]. > To post to this group, send email to [email protected]. > Visit this group at http://groups.google.com/group/sympy?hl=en-US. > For more options, visit https://groups.google.com/groups/opt_out. > > -- You received this message because you are subscribed to the Google Groups "sympy" group. 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