[Date Index]
[Thread Index]
[Author Index]
Re: question on diophantine equations in Mathematica
*To*: mathgroup at smc.vnet.net
*Subject*: [mg115545] Re: question on diophantine equations in Mathematica
*From*: Ivan Smirnov <ivan.e.smirnov at gmail.com>
*Date*: Fri, 14 Jan 2011 06:17:49 -0500 (EST)
If so, why then your PC computed it from 1 to 10^10 (in such big interval)
with only 269.967 and my is still computing yet, more than 20 minutes?
I asked it only from this point. It can be only from hardware difference.
What is your CPU&RAM&bitness, Adam?
2011/1/14 Adam Strzebonski <adams at wolfram.com>
> For Diophantine equations Solve and Reduce use the same code, so there
> should be no speed difference.
>
>
> Best Regards,
>
> Adam Strzebonski
> Wolfram Research
>
>
> Ivan Smirnov wrote:
>
>> Am I right that with such constraints Reduce now is many faster than
>> Solve?
>>
>> 2011/1/14 Adam Strzebonski <adams at wolfram.com <mailto:adams at wolfram.com>>
>>
>>
>> Reduce does a case-by-case search here, but it chooses the variables
>> with smallest ranges of values for the search.
>>
>> In[1]:= form = x^10 + y^10 + z^10 == t^2 && x >= 0 && y >= 1 &&
>> x <= y && y <= z && 1 <= t <= 250000;
>>
>> In[2]:= vars = {x, y, z, t};
>>
>> In[3]:= {Ceiling[MinValue[#, form, vars]],
>> Floor[MaxValue[#, form, vars]]}&/@vars
>>
>> Out[3]= {{0, 10}, {1, 11}, {1, 12}, {2, 250000}}
>>
>> Reduce does case-by-case search for the first 3 variables,
>> so the problem reduces to solving
>>
>> In[4]:= 11 11 12
>> Out[4]= 1452
>>
>> univariate quadratic equations. The most time-consuming part
>> here is computing the MinValue/MaxValue (8 CAD problems with
>> 4 variables).
>>
>> This method will do exhaustive search on {x, y, z} as long as
>> the number of possible {x, y, z} values does not exceed 10000.
>>
>> If you want it to do larger searches you need to change
>> the second value of the system option
>>
>> In[5]:= "ExhaustiveSearchMaxPoints"/.("ReduceOptions"/.SystemOptions[])
>> Out[5]= {1000, 10000}
>>
>> This increases the maximum number of search points to 10^6.
>>
>> In[6]:=
>> SetSystemOptions["ReduceOptions"->{"ExhaustiveSearchMaxPoints"->
>> {1000, 10^6}}];
>>
>> This proves that the problem has no solutions with t <= 10^10
>> (a search over 828630 cases).
>>
>> In[7]:= Reduce[x^10 + y^10 + z^10 == t^2 && 0 <= x && 0 < y &&
>> x <= y && y <= z && 1 <= t <= 10^10, {x, y, z, t}, Integers] // Timing
>>
>> Out[7]= {269.967, False}
>>
>>
>> Best Regards,
>>
>> Adam Strzebonski
>> Wolfram Research
>>
>>
>>
>> Andrzej Kozlowski wrote:
>>
>> Actually Solve (and Reduce) are remarkably efficient at solving
>> this problem. It is better to reformulate it by requiring that
>> the two smallest values be larger than 0. This eliminates all
>> trivial solutions. Solve (and Reduce) then work remarkably fast
>> for very large numbers t, e.g.
>>
>> Solve[
>> x^10 + y^10 + z^10 == t^2 && 0 <= x && 0 < y && x <= y && y <=
>> z && 1 <= t <= 250000, {x, y, z, t}, Integers] // Timing
>>
>> {1.80372,{}}
>>
>> Now look at this:
>>
>>
>> In[1]:= Solve[
>> x^10 + y^10 + z^10 == t^2 && 0 <= x && 0 < y && x <= y && y <=
>> z && 1 <= t <= 350000, {x, y, z, t}, Integers] // Timing
>>
>> Out[1]= {1.90608,{}}
>>
>> This is so fast that it almost excludes the possibility of "case
>> by case" verification. Moreover, solving the problem for t=
>> 350,000 took only slightly longer than for t= 250,000.
>> If this is indeed a valid proof (and I think it is - Reduce
>> gives the same answer) then it looks like Solve is really using
>> some knowledge of Diophantine equations to solve this. It would
>> be really interesting to know what is going on. I almost
>> inclined to believe that Solve is able to prove that there are
>> no solutions for all t, but running it without a bound on t
>> produces a useless (in this context) "conditional" answer:
>>
>>
>> Solve[x^10+y^10+z^10==t^2&&0<=x&&0<y&&x<=y&&y<=z,{x,y,z,t},Integers]
>> (Output deleted).
>>
>> I do hope we get some insight into what Solve is doing. It is
>> beginning to look fascinating, although I am probably missing
>> something simple...
>>
>> Andrzej Kozlowski
>>
>> On 12 Jan 2011, at 19:41, Andrzej Kozlowski wrote:
>>
>> Yes, but I think I unintentionally deceived you (and
>> myself). Mathematica caches its results and recall that I
>> tried solving this several times with different numbers
>> before I run Timing. mathematica obviously remembered all
>> the answers and when I tried measuring the time taken it was
>> amazingly fast. I should have found it suspicious but as I
>> was busy with other things I did not notice anything.
>>
>> Now that I tried again with a fresh kernel the results are
>> much less impressive: in fact much closer to yours.
>> But note that now it is clear that Solve is very much faster
>> than PowersRepresentatins (it looked the other way round
>> before). In fact Solve deals with this impressively fast:
>>
>> Timing[
>> Select[Table[
>> PowersRepresentations[t^2, 3, 10], {t, 1, 90000}], #1 != {}
>> & ]]
>>
>> {641.343049, {{{0, 0, 1}}, {{0, 0, 2}}, {{0, 0, 3}}, {{0, 0,
>> 4}}, {{0, 0, 5}}, {{0, 0, 6}}, {{0, 0, 7}}, {{0, 0,
>> 8}}, {{0, 0, 9}}}}
>>
>>
>> Timing[
>> Solve[x^10 + y^10 + z^10 == t^2 && 0 <= x && x <= y && y <=
>> z && 1 <= t <= 90000, {x, y, z, t}, Integers]]
>>
>> {1.161798, {{x -> 0, y -> 0, z -> 1, t -> 1}, {x -> 0, y
>> -> 0, z -> 2, t -> 32}, {x -> 0, y -> 0, z -> 3, t ->
>> 243}, {x -> 0, y -> 0, z -> 4, t -> 1024}, {x -> 0, y ->
>> 0, z -> 5, t -> 3125}, {x -> 0, y -> 0, z -> 6, t ->
>> 7776}, {x -> 0, y -> 0, z -> 7, t -> 16807}, {x -> 0, y
>> -> 0, z -> 8, t -> 32768}, {x -> 0, y -> 0, z -> 9, t
>> -> 59049}}}
>>
>> This suggests strongly that you should in fact use Solve.
>> However, you should not try to test for too large a group of
>> solutions at a time. For example, you can get the next
>> 10,000 quickly:
>>
>> Timing[
>> Solve[x^10 + y^10 + z^10 == t^2 && 0 <= x && x <= y && y <=
>> z && 90000 <= t <= 100000, {x, y, z, t}, Integers]]
>>
>> {1.48964,{{x->0,y->0,z->10,t->100000}}}
>>
>> But the time for this 10,000 is longer than for the previous
>> 90,000!
>>
>> With best regards
>>
>> Andrzej
>>
>>
>>
>> On 12 Jan 2011, at 16:03, Ivan Smirnov wrote:
>>
>> Oh, it's very cool computer. What model of CPU, Intel or
>> AMD?
>> I just thought that time was in seconds, but surprised
>> that it took so few time and asked.
>> I have only 1.6 Ghz Acer (Intel T5500) with 1 GB Ram, so
>> for 1..90000 it took 1034 seconds to compute...
>>
>> 2011/1/12 Andrzej Kozlowski <akoz at mimuw.edu.pl
>> <mailto:akoz at mimuw.edu.pl>>
>>
>> I am using Mathematica 8 on 2.66 Ghz Mac Book Pro with 8
>> gigabytes of Ram. The time is measured in seconds. With
>> Mathematica 8 you can also get the same answer with Solve:
>>
>> Timing[
>> Solve[x^10 + y^10 + z^10 == t^2 && 0 <= x && x <= y && y
>> <= z &&
>> 1 <= t <= 90000, {x, y, z, t}, Integers]]
>>
>>
>> {1.01969,{{x->0,y->0,z->1,t->1},{x->0,y->0,z->2,t->32},{x->0,y->0,z->3,t->243},{x->0,y->0,z->4,t->1024},{x->0,y->0,z->5,t->3125},{x->0,y->0,z->6,t->7776},{x->0,y->0,z->7,t->16807},{x->0,y->0,z->8,t->32768},{x->0,y->0,z->9,t->59049}}}
>>
>> I think for very large numbers PowersRepresentations
>> will give you more satisfactory answers. For example,
>> compare the output
>>
>> Solve[x^10 + y^10 + z^10 == 10^20 && 0 <= x && x <= y &&
>> y <= z, {x,
>> y, z, t}]
>>
>> with
>>
>> PowersRepresentations[10^21, 3, 10^20]
>>
>> {}
>>
>> Andrzej Kozlowski
>>
>>
>>
>>
>> On 12 Jan 2011, at 13:16, Ivan Smirnov wrote:
>>
>> Hello, Andrzej.
>> Many thanks for reply.
>> What PC do you use (OS, CPU & RAM) and how many
>> minutes did it take to compute, what is 0.872...?
>> What is the upper margin for t which can cause
>> overflow?
>> Do you have any other ideas how to increase
>> performance for my task?
>> Will be very glad for help
>>
>> 2011/1/12 Andrzej Kozlowski <akoz at mimuw.edu.pl
>> <mailto:akoz at mimuw.edu.pl>>
>>
>> This seems to show that there are only trivial
>> solutions for 1<=t<=90000
>>
>> Timing[
>> Select[Table[
>> PowersRepresentations[t^2, 3, 10], {t, 1, 90000}],
>> #1 != {} & ]]
>>
>> {0.8722430000000259, {{{0, 0, 1}}, {{0, 0, 2}}, {{0, 0,
>> 3}}, {{0, 0, 4}}, {{0, 0, 5}}, {{0, 0, 6}}, {{0, 0,
>> 7}},
>> {{0, 0, 8}}, {{0, 0, 9}}}}
>>
>> The algorithm basically uses "brute force" so you
>> will start getting overflows for very large t.
>>
>> Andrzej Kozlowski
>>
>>
>>
>> On 12 Jan 2011, at 01:25, Ivan Smirnov wrote:
>>
>> Hi all,
>> I've installed trial of Mathematica 8.
>> I would like to search for possible solutions of
>> diophantine equation
>> x^10+y^10+z^10=t^2.
>> How to do this efficiently?
>> FindInstance seems to be VERY slow! And indeed
>> it doesn't always find every
>> solution of diophantine equations. For example
>> I've tried it with
>> x^4+y^4+z^4=t^4 and it didn't find anything (but
>> there are solutions!).
>> And Solve command just don't want to search!
>> With some seconds it gives
>> During evaluation of In[1]:= Solve::svars:
>> Equations may not give solutions
>> for all "solve" variables. >>
>> I will be very glad if someone make INDEED FAST
>> algorithm for searching.
>>
>> Ivan
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>
Prev by Date:
**Re: question on diophantine equations in Mathematica**
Next by Date:
**Re: question on diophantine equations in Mathematica**
Previous by thread:
**Re: question on diophantine equations in Mathematica**
Next by thread:
**Re: question on diophantine equations in Mathematica**
| |