Re: How to do quickest

*To*: mathgroup at smc.vnet.net*Subject*: [mg123093] Re: How to do quickest*From*: DrMajorBob <btreat1 at austin.rr.com>*Date*: Wed, 23 Nov 2011 07:05:53 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <201111210929.EAA14830@smc.vnet.net>*Reply-to*: drmajorbob at yahoo.com

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 following >>>> answers: >>>> >>>> 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

**References**:**How to do quickest***From:*Artur <grafix@csl.pl>

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