Re: How to do quickest

• To: mathgroup at smc.vnet.net
• Subject: [mg123102] Re: How to do quickest
• From: DrMajorBob <btreat1 at austin.rr.com>
• Date: Wed, 23 Nov 2011 07:07:32 -0500 (EST)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <201111210929.EAA14830@smc.vnet.net>

```Defining poly as before, here's another small improvement:

Clear[x, y, x3, y2]
poly2 = Collect[
Eliminate[{poly == 0, x3 == x^3, y2 == y^2}, {y}] /. {Power[x, 3] ->
x3, Power[x, 6] -> x3^2} /. Equal -> Subtract, m]

3645 m^12 - 2916 m^10 x + x3^2 + m^6 (270 x3 - 270 y2) - 2 x3 y2 + y2^2

Last@Reap[Do[x3 = x^3; y = Round@Sqrt@x3; y2 = y^2;
! IrreduciblePolynomialQ@poly2 && x3 != y2 && Sow@x, {x, 2,
10^6}]] // Timing

{876.934, {{1942, 2878, 3862, 6100, 8380, 11512, 15448, 18694, 31228,
93844, 111382, 117118, 129910, 143950, 186145, 210025, 375376,
445528, 468472, 575800, 844596, 950026}}}

The previous timing was 979 seconds.

Bobby

On Tue, 22 Nov 2011 14:58:11 -0600, Artur <grafix at csl.pl> wrote:

> Congratulations Bob! Very good timing 1174s in range 10^6, 2*10^6 on my
> computer.
> Thank You very much for help!
> Best wishes
> Artur
>
> W dniu 2011-11-22 21:17, DrMajorBob pisze:
>> The Block construction causes Round[Sqrt[x^3]] to be recomputed each
>> time y is mentioned, or so it would seem, and timing bears that out:
>>
>> Collect[poly =
>>   Eliminate[{4*m^2 + 6*m*n + n^2 ==
>>       x, (19*m^2 + 9*m*n + n^2)*Sqrt[m^2 + n^2] == y}, {n}] /.
>>    Equal -> Subtract, m]
>>
>> 3645 m^12 - 2916 m^10 x + x^6 - 2 x^3 y^2 + y^4 +
>>  m^6 (270 x^3 - 270 y^2)
>>
>> Block[{y = Round[Sqrt[x^3]]},
>>   Reap[Do[x^3 - y^2 != 0 &&
>>       Or[OddQ[x^6 - 2*x^3*y^2 + y^4],
>>        Mod[x^6 - 2*x^3*y^2 + y^4, 4] == 0] && !
>>        IrreduciblePolynomialQ[poly] && Sow@x, {x, 2,
>>       10^4}]][[2]]] // Timing
>>
>> {10.98, {{1942, 2878, 3862, 6100, 8380}}}
>>
>> First@Last@
>>    Reap[Do[y = Round[Sqrt[x^3]];
>>      x^3 - y^2 != 0 &&
>>       Or[OddQ[x^6 - 2*x^3*y^2 + y^4],
>>        Mod[x^6 - 2*x^3*y^2 + y^4, 4] == 0] && !
>>        IrreduciblePolynomialQ[poly] && Sow@x, {x, 2, 10^4}]] // Timing
>>
>> {8.36221, {1942, 2878, 3862, 6100, 8380}}
>>
>> This is slightly faster (surprisingly?):
>>
>> First@Last@
>>    Reap[Do[y = Round[Sqrt[x^3]]; ! IrreduciblePolynomialQ[poly] &&
>>       x^3 - y^2 != 0 &&
>>       Or[OddQ[x^6 - 2*x^3*y^2 + y^4],
>>        Mod[x^6 - 2*x^3*y^2 + y^4, 4] == 0] && Sow@x, {x, 2,
>>       10^4}]] // Timing
>>
>> {8.26819, {1942, 2878, 3862, 6100, 8380}}
>>
>> And that suggests the Eisenstein criterion is unhelpful after all, so:
>>
>> First@Last@
>>    Reap[Do[y = Round[Sqrt[x^3]]; ! IrreduciblePolynomialQ[poly] &&
>>       x^3 - y^2 != 0 && Sow@x, {x, 2, 10^6}]] // Timing
>>
>> {978.795, {1942, 2878, 3862, 6100, 8380, 11512, 15448, 18694, 31228,
>>   93844, 111382, 117118, 129910, 143950, 186145, 210025, 375376,
>>   445528, 468472, 575800, 844596, 950026}}
>>
>> Bobby
>>
>> On Tue, 22 Nov 2011 12:40:48 -0600, Andrzej Kozlowski
>> <akoz at mimuw.edu.pl> wrote:
>>
>>> I think the correct use of Eisenstein's criterion is this:
>>>
>>>
>>> Block[{y = Round[Sqrt[x^3]]},
>>>   Reap[Table[
>>>      If[x^3 - y^2 != 0 &&
>>>        Not[Mod[x^6 - 2*x^3*y^2 + y^4, 2] == 0 &&
>>>          Mod[x^6 - 2*x^3*y^2 + y^4, 4] != 0] && !
>>>         IrreduciblePolynomialQ[poly], Sow[{x, y}]], {x, 2,
>>>       1000000}]][[2]]] // Timing
>>>
>>> This is because 3645 is not divisible by 2, but  2916 and 270 both
>>> are. So if the term x^6-2 x^3 y^2+y^4 is divisible by 2 but not
>>> divisible by 4, the polynomial is irreducible and there is no need to
>>> test it further. Only when this isn't the case, we need to use
>>> IrreduciblePolynomialQ.
>>>
>>> I am not sure if adding this will speed up the code. I only have one
>>> copy of Mathematica and I need to use it so I can't afford the time to
>>> run these programs now, when I am a hurry.
>>>
>>> Andrzej
>>>
>>>
>>>
>>> On 22 Nov 2011, at 19:30, Andrzej Kozlowski wrote:
>>>
>>>> My memory of Eisenstein's criterion was wrong (also, I was too much
>>>> in a hurry to look it up).  Rather than correcting it I got rid of it
>>>> altogether since I think Mathematica probably uses it anyway. I then
>>>> get:
>>>>
>>>> In[1]:= Collect[
>>>> poly = Eliminate[{4*m^2 + 6*m*n + n^2 ==
>>>>      x, (19*m^2 + 9*m*n + n^2)*Sqrt[m^2 + n^2] == y}, {n}] /.
>>>>   Equal -> Subtract, m]
>>>>
>>>> Out[1]= 3645 m^12-2916 m^10 x+m^6 (270 x^3-270 y^2)+x^6-2 x^3 y^2+y^4
>>>>
>>>> In[2]:= Block[{y = Round[Sqrt[x^3]]},
>>>>  Reap[Table[
>>>>     If[x^3 - y^2 != 0 && Not[IrreduciblePolynomialQ[poly]],
>>>>      Sow[{x, y}]], {x, 2, 1000000}]][[2]]] // Timing
>>>>
>>>>
>>>> Out[2]= {1089.54,({1942,85580}    {2878,154396}    {3862,240004}
>>>> {6100,476425}    {8380,767125}    {11512,1235168}
>>>> {15448,1920032}    {18694,2555956}    {31228,5518439}
>>>> {93844,28748141}    {111382,37172564}    {117118,40080716}
>>>> {129910,46823500}    {143950,54615700}    {186145,80311375}
>>>> {210025,96251275}    {375376,229985128}    {445528,297380512}
>>>> {468472,320645728}    {575800,436925600}    {844596,776199807}
>>>> {950026,925983476}
>>>>
>>>> )}
>>>>
>>>> This gets all the numbers but is much slower (I guess it will be
>>>> better to add the Eisenstein criterion after all, but of course, in
>>>> correct form).
>>>>
>>>> Andrzej Kozlowski
>>>>
>>>>
>>>>
>>>>
>>>> On 22 Nov 2011, at 18:02, Artur wrote:
>>>>
>>>>> Dear Andrzej,
>>>>> Your procedure omiited some points
>>>>> 6100
>>>>> 8380 Best wishes
>>>>> Artur
>>>>>
>>>>> W dniu 2011-11-22 13:23, Andrzej Kozlowski pisze:
>>>>>> On 22 Nov 2011, at 10:07, Andrzej Kozlowski wrote:
>>>>>>
>>>>>>
>>>>>>> On 22 Nov 2011, at 10:06, Andrzej Kozlowski wrote:
>>>>>>>
>>>>>>>
>>>>>>>> On 21 Nov 2011, at 10:29, Artur wrote:
>>>>>>>>
>>>>>>>>
>>>>>>>>> Dear Mathematica Gurus,
>>>>>>>>> How to do quickest following procedure (which is very slowly):
>>>>>>>>>
>>>>>>>>> qq = {}; Do[y = Round[Sqrt[x^3]];
>>>>>>>>> If[(x^3 - y^2) != 0,
>>>>>>>>> kk = m /. Solve[{4 m^2 + 6 m n + n^2 ==
>>>>>>>>>     x, (19 m^2 + 9 m n + n^2) Sqrt[m^2 + n^2] == y}, {m,
>>>>>>>>> n}][[1]];
>>>>>>>>> ll = CoefficientList[MinimalPolynomial[kk][[1]], #1];
>>>>>>>>> lll = Length[ll];
>>>>>>>>> If[lll < 12, Print[{x/(x^3 - y^2)^2, kk, x, y, x^3 - y^2}];
>>>>>>>>> If[Length[ll] == 3, Print[{kk, x, y}]]]], {x, 2, 1000000}];
>>>>>>>>> qq
>>>>>>>>>
>>>>>>>>>
>>>>>>>>> (*Best wishes Artur*)
>>>>>>>>>
>>>>>>>>>
>>>>>>>> I think it would be better to send not only the code but also the
>>>>>>>> mathematical problem, as there may be a way to do it in a
>>>>>>>> different way. Unless I am misunderstanding something, what you
>>>>>>>> are trying to do is the same as this:
>>>>>>>>
>>>>>>>> In[31]:= Block[{y = Round[Sqrt[x^3]]},
>>>>>>>> Reap[Table[
>>>>>>>>  If[(x^3 - y^2) != 0 && Not[IrreduciblePolynomialQ[poly]],
>>>>>>>>   Sow[{x, y}]], {x, 2, 1000000}]][[2]]] // Timing
>>>>>>>>
>>>>>>>> Out[31]= {721.327,{}}
>>>>>>>>
>>>>>>>> This ought to be a lot faster than your code, but I have not
>>>>>>>> tried to run yours to the end. Also, it is possible that using
>>>>>>>> the Eisenstein Test explicitly may be somewhat faster:
>>>>>>>>
>>>>>>>> Block[{y = Round[Sqrt[x^3]]},
>>>>>>>> Reap[Table[
>>>>>>>> If[x^3 - y^2 != 0 && Mod[x^6 - 2*x^3*y^2 + y^4, 4] == 0 &&
>>>>>>>>           ! IrreduciblePolynomialQ[poly], Sow[{x, y}]], {x, 2,
>>>>>>>>  1000000}]][[2]]]
>>>>>>>>
>>>>>>>> {}
>>>>>>>>
>>>>>>>> but I forgot to use Timing and don't want to wait again,
>>>>>>>> particularly that the answer is the empty set.
>>>>>>>>
>>>>>>>> Andrzej Kozlowski
>>>>>>>>
>>>>>>> I forgot to include the definition of poly:
>>>>>>>
>>>>>>> Collect[poly = Eliminate[{4*m^2 + 6*m*n + n^2 == x,
>>>>>>>    (19*m^2 + 9*m*n + n^2)*Sqrt[m^2 + n^2] == y}, {n}] /. Equal ->
>>>>>>> Subtract, m]
>>>>>>>
>>>>>>> 3645*m^12 - 2916*m^10*x + m^6*(270*x^3 - 270*y^2) + x^6 -
>>>>>>> 2*x^3*y^2 + y^4
>>>>>>>
>>>>>>> Andrzej Kozlowski
>>>>>>>
>>>>>>
>>>>>> Strange but I run this code with a fresh kernel and got the
>>>>>>
>>>>>> In[1]:=
>>>>>> Collect[poly=Eliminate[{4*m^2+6*m*n+n^2==x,(19*m^2+9*m*n+n^2)*Sqrt[m^2+n^2]==y},{n}]/.Equal->Subtract,m]
>>>>>> Out[1]= 3645 m^12-2916 m^10 x+m^6 (270 x^3-270 y^2)+x^6-2 x^3
>>>>>> y^2+y^4
>>>>>>
>>>>>> In[2]:=
>>>>>> Block[{y=Round[Sqrt[x^3]]},Reap[Table[If[x^3-y^2!=0&&Mod[x^6-2*x^3*y^2+y^4,4]==0&&!IrreduciblePolynomialQ[poly],Sow[{x,y}]],{x,2,1000000}]][[2]]]//Timing
>>>>>>
>>>>>> Out[2]= {766.05,({1942,85580}    {2878,154396}    {3862,240004}
>>>>>> {11512,1235168}    {15448,1920032}    {18694,2555956}
>>>>>> {111382,37172564}    {117118,40080716}    {129910,46823500}
>>>>>> {143950,54615700}    {186145,80311375}    {210025,96251275}
>>>>>> {375376,229985128}    {445528,297380512}    {468472,320645728}
>>>>>> {575800,436925600}    {950026,925983476}
>>>>>>
>>>>>> )}
>>>>>>
>>>>>>
>>>>>> I tested the first one and it does seem to be a solution to your
>>>>>> problem.
>>>>>>
>>>>>> {x, y} = {950026, 925983476};
>>>>>>
>>>>>> y == Round[Sqrt[x^3]]
>>>>>>
>>>>>> True
>>>>>>
>>>>>> x^3 - y^2 != 0
>>>>>>
>>>>>> True
>>>>>>
>>>>>> kk =
>>>>>> m /. Solve[{4 m^2 + 6 m n + n^2 ==
>>>>>>     x, (19 m^2 + 9 m n + n^2) Sqrt[m^2 + n^2] == y}, {m, n}][[1]]
>>>>>>
>>>>>> Out[12]= -Sqrt[-(198/5)-(44 I Sqrt[11])/5]
>>>>>>
>>>>>> ll = CoefficientList[MinimalPolynomial[kk][[1]], #1];
>>>>>>
>>>>>> Length[ll]
>>>>>>
>>>>>> 5
>>>>>>
>>>>>> I don't know why I got no answers the first time round, perhaps one
>>>>>> of the variables had values assigned.
>>>>>>
>>>>>> Andrzej
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>
>>>
>>
>>

--
DrMajorBob at yahoo.com

```

• Prev by Date: Re: How to do quickest
• Next by Date: Re: How to do quickest
• Previous by thread: Re: How to do quickest
• Next by thread: Re: How to do quickest