Hi Ondrej, I don't know much about solving recurrence relations on two variables, but the sources I am referencing provides the solutions for the recurrences involved with Pell equation. So, I think that would be enough.
Concerning the solutions of x**2 - 13*y**2 = 27, solutions returned by diop_pell are: >>> diop_pell(13, 27) [(220, 61), (40, 11), (768, 213), (12, 3)] Here is how wolfram alpha represents it: http://www.wolframalpha.com/input/?i=x**2+-+13*y^2+%3D+27 They provide general solutions for each class. (Look in the box named "Integer solutions") Regards, Thilina On Wed, Jul 3, 2013 at 2:55 AM, Ondřej Čertík <[email protected]>wrote: > > I am really excited to tell you that I implemented the algorithm for > > solving generalized Pell equation. For the past week or so I was working > > on the case B**2 - 4*A*C > 0 in quadratic DEs. Now, since the pell > > equation is solved, I can solve the above case by transforming it to a > > Pell equation. I looked a bit at the transformation and it's not that > hard. > > I will be able to code it and finish implementing quadratic DEs at the > > end of this week. > > Excellent, great job! > > > > > Currently, solutions returned for the Pell equations are the basic > solutions > > of the particular equation passed to the Pell equation solver. We can > > represent > > other solutions by a recurrence. Both you and Aaron had answered on how > > to represent the recurrence in the solution. If I am not mistaken > rsolve() > > currently > > solves the recurrences in one variable. But recurrences we are talking > here > > involves two variables. So returning the recurrence itself won't be a > good > > idea. > > What Wolfram alpha currently does is, it solves the recurrence and > returns > > the general solution without returning any other specific solutions. > Would > > that > > be a bad idea since I am implementing lower level API's? > > Do you know how to solve the recurrence of two variables? Do you have > some examples for Wolfram Alpha that you tried? Let me see some > examples and think about the best way. > > > I coded the algorithms mostly looking at these two papers. > > > > [1] Solving the generalized Pell equation x**2 - D*y**2 = N, John P. > > Robertson, > > July 31, 2004, Pages 16 - 17 and 4 - 8. http://www.jpr2718.org/pell.pdf > > > > [2] Solving the equation ax**2 + bxy + cx**2 + dx + ey + f = 0, by John > P. > > Robertson. http://www.jpr2718.org/ax2p.pdf > > > > I added a commit. I would love to have your feedback on that. Please take > > a look at that when you are free. > > I left some small comments. > > Ondrej > > > > > Regards, > > Thilina > > > > > > > > > > -- > > 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. > > 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. > 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. For more options, visit https://groups.google.com/groups/opt_out.
