Maple 11 solve command does not seem to be able to solve systems Maple 10 can
The following system of equations can be solved by Maple 9.5 and Maple 10, but either takes a long time in Maple 11 or else simply won't solve. Is anyone else experiencing problems with the solve command? Earlier I noticed that Maple 11 was returning duplicates in solutions, but this is more serious. We tried this system on Maple 9.5 and Maple 10, and it solves easily, but on two computers with Maple 11, in classic worksheet mode, minutes went by with no solution.
{2*a[2,6]*a[2,9]-2*a[2,8]*a[2,9], 2*a[1,5]*a[3,1]+2*a[1,7]*a[3,1], 2*a[2,6]*a[3,4]-2*a[2,8]*a[3,4], 2*a[1,5]*a[2,1]+2*a[1,7]*a[2,1], 2*a[1,2]*a[3,1]+2*a[1,3]*a[3,1], 2*a[1,2]*a[2,1]+2*a[1,3]*a[2,1], -2*a[3,4]*a[3,6]+2*a[3,4]*a[3,8], 2*a[2,8]*a[3,8]-2*a[2,9]*a[3,4], -2*a[2,8]*a[3,8]+2*a[2,9]*a[3,4], 2*a[1,2]*a[1,5]+2*a[1,2]*a[2,6]+2*a[1,5]*a[3,6], 2*a[2,6]*a[3,6]-2*a[2,9]*a[3,4], -2*a[2,6]*a[3,6]+2*a[2,9]*a[3,4], -2*a[2,6]*a[3,4]+2*a[2,8]*a[3,4], 2*a[2,9]*a[3,6]-2*a[2,9]*a[3,8], -2*a[2,9]*a[3,6]+2*a[2,9]*a[3,8], 2*a[1,2]*a[2,1]+2*a[2,1]*a[2,4]+2*a[2,8]*a[3,1], 2*a[1,2]*a[2,9]+2*a[1,5]^2+2*a[1,5]*a[3,9], 2*a[1,2]^2+2*a[1,2]*a[2,4]+2*a[1,5]*a[3,4], -2*a[2,4]*a[2,9]+2*a[2,6]^2-2*a[2,6]*a[3,9]+2*a[2,9]*a[3,6], 2*a[1,2]*a[1,5]+2*a[1,2]*a[2,8]+2*a[1,5]*a[3,8], -2*a[2,4]*a[3,8]+2*a[2,8]*a[3,4]-2*a[3,4]*a[3,9]+2*a[3,8]^2, -2*a[1,3]*a[1,7]+2*a[1,3]*a[2,6]+2*a[1,7]*a[3,6], -2*a[1,3]*a[1,7]+2*a[1,3]*a[2,8]+2*a[1,7]*a[3,8], 2*a[1,3]*a[2,1]-2*a[2,1]*a[2,4]-2*a[2,6]*a[3,1], 2*a[1,3]*a[2,9]-2*a[1,7]^2+2*a[1,7]*a[3,9], 2*a[1,5]*a[2,1]+2*a[2,1]*a[2,6]+2*a[2,9]*a[3,1], 2*a[1,7]*a[2,1]-2*a[2,1]*a[2,8]-2*a[2,9]*a[3,1], 2*a[1,7]*a[3,1]-2*a[2,1]*a[3,8]-2*a[3,1]*a[3,9], 2*a[1,2]*a[3,1]+2*a[2,1]*a[3,4]+2*a[3,1]*a[3,8], 2*a[1,5]*a[3,1]+2*a[2,1]*a[3,6]+2*a[3,1]*a[3,9], 2*a[1,2]*a[2,1]+2*a[1,3]*a[2,1]+2*a[1,5]*a[3,1]+2*a[1,7]*a[3,1], -2*a[1,3]^2+2*a[1,3]*a[2,4]+2*a[1,7]*a[3,4], 2*a[2,4]*a[2,9]-2*a[2,8]^2+2*a[2,8]*a[3,9]-2*a[2,9]*a[3,8], -2*a[2,6]*a[3,8]+2*a[2,8]*a[3,6]-2*a[3,6]*a[3,9]+2*a[3,8]*a[3,9], 2*a[2,4]*a[2,6]-2*a[2,4]*a[2,8]-2*a[2,6]*a[3,8]+2*a[2,8]*a[3,6], 2*a[1,3]*a[3,1]-2*a[2,1]*a[3,4]-2*a[3,1]*a[3,6], 2*a[2,4]*a[3,6]-2*a[2,6]*a[3,4]+2*a[3,4]*a[3,9]-2*a[3,6]^2}
- A procedure
2 hours 31 min ago - ellipse
3 hours 54 min ago - Also try the options scaling, gridrefine and crossingrefine
5 hours 11 min ago - Try implicitplot
5 hours 54 min ago - The patch
7 hours 30 min ago - thx!
7 hours 56 min ago - The source of the bug
8 hours 14 min ago - my concern is ...
9 hours 10 min ago - plotting with the "Standard interface"
9 hours 29 min ago - variant
9 hours 47 min ago
solving polynomial systems
At first I assumed this was due to the Groebner changes in Maple 11, but that seems to not be the case. Thank you for the interesting benchmark. Try the following:
sys := {...your system...}: sol := Groebner[Solve](sys): sol := map(a->[solve(a[1],[op(a[2])]), map(`<>`, a[3], 0)], sol): sol := {seq(seq([i,j[2]], i=j[1]), j=sol)}:This should write the solutions in a nice form. There may be duplicate or equivalent solutions, but you shouldn't lose any.
Another Workaround
This looks like a bug in one of the internal solve routines in Maple 11. Here is another way to solve this:
sys:={...your system...}; SolveTools[PolynomialSystem](sys, indets(sys));From ?SolveTools[PolynomialSystem]:
Issue has been tracked
The issue mentioned in this post is now tracked in Maple's internal bug-tracking system. While we are fixing this bug, please try the workarounds mentioned above.
Solve still defective
Maple 9.5 and Maple 10 can solve the following system of equations found at this link Download 2448_system.mws
in about 5 or 10 minutes. They both properly come up with 128 solutions.
Maple 11, and now Maple 12, cannot solve the system.
The workaround given by jpmay does not work since it only returns 1 of the 128 solutions.
It is disappointing that this tracked bug was not fixed before Maple12 was released
Clarifications
The solve problem reported in penkavmr's original post in has been fixed. In Maple 12, solve finds solutions for that system in less than one second. Try it:
As far as the new system given by Alex, it looks like Maple versions 10-12 cannot solve it (I stopped 10 and 12 each at 3000 seconds) while Maple 9.5 solves it in 200 seconds or so. Even though it is a system similar to penkavmr's, the underlying problem appears to be completely different.
Also, in Maple 12, SolveTools[PolynomialSystems] does solve Alex's system (in about 250 seconds), and since solve pretty much immediately calls SolveTools[PolynomialSystems] in this case, that means the problem here is much weirder than the problem penkavmr's system was encountering (I suspect it is subtle interaction due to the renaming of the indexed variables).
As for returning only one solution, that is the default, but SolveTools[PolynomialSystems] will return more if asked. Take a look at its help page.
John May
Math Developer
Maplesoft
Subtle Problem + Work around
In fact it is a very subtle problem. Indexed variables a[i,j] can cause problems in some solve subroutines, so one of the first things solve does is rename these variables to new symbols like x_nn. In the case of this system, the order in which the polynomial system solver handles the variables matters a lot, and when the variables get renamed, they get ordered differently. It seems more or less chance that the original variable names get ordered in a way so the polynomial system solver returns quickly.
Here is another work around (use symbols instead of indexed names):
John
evalindets
That could be done more simple using evalindets.
Alec
as one line
*edit: or that ---^
True, a one-liner would do it:
subsindets(sys, 'indexed', x->cat(op(0,x),'_',op(1,x),'_',op(2,x)) );
John
with underscores
Or, I missed the underscores! With underscores, that could be also done as
Alec
Very nice.
Very nice.
John
Thank you!
Thank you,
Originally, I hadn't realized that both a{1,11] and a[11,1] would get converted to the same variable a111 if underscores were not used.
Also, in the second version, I forgot to quote the underscore. It should be '_' instead of _, I just noticed that you quoted it in your version of the code above.
Alec
It is better
The workaround that John May and Alec propose is productive. When I ran it, Maple 12 returned 92 solutions, whereas Maple 9.5 returned 128 solutions, but we will check theory, etc. to see if the discrepancy is real.
I suppose that my colleagues and I could figure this next step out on our own ("exercise for the reader"!), but John May and Alec seemed primed to do this four orders of magnitude faster:
How can we take the returned solutions that contain variables such as a_1_1 and efficiently convert the solutions to variables such as a[1,1]?
Thanks!
No problem
Suppose that sol is the list of solutions with a_1_1, then the following should convert them to the list of solutions with a[1,1],
f:=proc(x) local a; a:=StringTools:-Split(x,'_'); if nops(a)>1 then parse(cat(a[1],sprintf("%a",map(parse,a[2..-1])))) else x fi end: sol:=evalindets(sol,'symbol',f);Not tested, because I couldn't login to the server with Maple 12 that I used last few days and I don't have Maple installed at home.
Alec