CoCoA » CoCoA-5

On my old MacBook with CoCoA-5.1.5 this computation took more than 1 hour (then I stopped it).

I've added the improvements we talked about at ICMS Berlin.
Now this example is almost instant :-) :-)
There are a few thing to settle (how to extract a RAT out of a TwinFloat which is not rational)
but it ready to be played with!
Use ApproxSolve2.
If it looks satisfactory on other examples we should write this in a paper!!

John Abbott wrote:

I have just tried the examples again (after Anna revised ApproxSolve yesterday or this morning). It still seems to be quite slow... It certainly wasn't "almost instant": I gave up waiting for the result.

:-(

Try ApproxSolve2

OK. ApproxSolve2 solves the problem in about 0.8s (on my old MacBook... it'll be faster on the new computer, if Dell ever decides to let me actually have it!)

I do not like the name ApproxSolve2.

Any idea why ApproxSolve is so slow?

John Abbott wrote:

OK. ApproxSolve2 solves the problem in about 0.8s (on my old MacBook... it'll be faster on the new computer, if Dell ever decides to let me actually have it!)

not bad!

I do not like the name ApproxSolve2.

Needs also refinement and testing ;-)

Any idea why ApproxSolve is so slow?

gbasis lex. I think we should use FGLM!

Do you really need lex? I vaguely recall using several eliminations to compute the various polynomials of the form x[j] - poly(x[n]).

There ought to be a quick(ish) way of getting these polys...

This is another example where ApproxSolve is too slow (in CoCoA-5.2.2).
The example came from "playing with" palindromic sparse squares -- preparation for Cameroon.

use QQ[a[1..5],nz];
//SetVerbosityLevel(100);
L := [a[3]^2 +2*a[2]*a[4] +4*a[1]*a[5] +2*a[4],
  a[1]*a[3] +a[2]*a[4] +(1/2)*a[4]^2 +2*a[3]*a[5] +a[2],
  a[2]^2 -2*a[2]*a[4] -a[4]^2 -4*a[3]*a[5] -2*a[2] +2*a[4],
  a[1]*a[2] +a[3],
  a[2]*a[3] +a[3]*a[4] +2*a[4]*a[5] +a[1] -a[3],
  a[2]*a[4]^2 +120*a[2]*a[4] +14*a[4]^2 -120*a[1]*a[5] +152*a[3]*a[5] +16*a[5]^2 -96*a[1]*nz +96*a[5]*nz +128*a[2] +104*a[4] -88,
  a[3]*a[4]^2 -528*nz^3 +663*a[1]*a[4] +511*a[3]*a[4] +1748*a[2]*a[5] +1156*a[4]*a[5] -658*a[2]*nz +2362*a[4]*nz -3888*a[1] -1626*a[3] +3172*a[5] +4429*nz,
  a[4]^3 +(-11054/17)*a[2]*a[4] +(-1381/17)*a[4]^2 +(10104/17)*a[1]*a[5] +(-14324/17)*a[3]*a[5] +(-1472/17)*a[5]^2 +(8240/17)*a[1]*nz +(-8224/17)*a[5]*nz +(-96/17)*nz^2 +(-11602/17)*a[2] +(-8844/17)*a[4] +8386/17,
  a[1]*a[4]*a[5] +(1/34)*a[2]*a[4] +(53/68)*a[4]^2 +(148/17)*a[1]*a[5] +(-3/17)*a[3]*a[5] +(-20/17)*a[5]^2 +(92/17)*a[1]*nz +(-94/17)*a[5]*nz +(12/17)*nz^2 +(-149/68)*a[2] +(-203/34)*a[4] -31/34,
  a[1]*a[4]^2 +24*nz^3 +(-69/2)*a[1]*a[4] +(-89/4)*a[3]*a[4] -70*a[2]*a[5] +(-107/2)*a[4]*a[5] +25*a[2]*nz -99*a[4]*nz +(697/4)*a[1] +(271/4)*a[3] -129*a[5] +(-403/2)*nz,
  a[3]*nz +1,
  a[2]*a[4]*nz +(2/5)*a[1]*a[4] +(1/10)*a[3]*a[4] +(1/5)*a[4]*a[5] +(4/5)*a[2]*nz +(-7/10)*a[1] +(-1/10)*a[3] +(-6/5)*a[5],
  a[4]*a[5]*nz +(1/2)*a[1]*nz +(-1/2)*a[2] +(-1/2)*a[4] +1/2,
  a[4]^2*nz +(-4/5)*a[1]*a[4] +(-1/5)*a[3]*a[4] +(-2/5)*a[4]*a[5] +(2/5)*a[2]*nz +(-3/5)*a[1] +(1/5)*a[3] +(-8/5)*a[5],
  a[1]*a[4]*nz +(-137/17)*a[2]*a[4] +(-52/17)*a[4]^2 +(-92/17)*a[1]*a[5] +(-164/17)*a[3]*a[5] +(40/17)*a[5]^2 +(-48/17)*a[1]*nz +(52/17)*a[5]*nz +(-24/17)*nz^2 +(-87/17)*a[2] +(67/17)*a[4] +150/17,
  a[1]*a[5]*nz +(-1/5)*a[1]*a[4] +(-1/20)*a[3]*a[4] +(-1/10)*a[4]*a[5] +(-2/5)*a[2]*nz +(1/2)*a[4]*nz +(7/20)*a[1] +(-1/5)*a[3] +(3/5)*a[5],
  a[2]*a[5]*nz +(-9/34)*a[2]*a[4] +(-1/68)*a[4]^2 +(-40/17)*a[1]*a[5] +(-7/17)*a[3]*a[5] +(-24/17)*a[5]^2 +(-12/17)*a[1]*nz +(13/17)*a[5]*nz +(-6/17)*nz^2 +(-53/68)*a[2] +(-9/34)*a[4] +29/17,
  a[3]*a[4]*a[5] +(3317/34)*a[2]*a[4] +(837/68)*a[4]^2 +(-1438/17)*a[1]*a[5] +(2187/17)*a[3]*a[5] +(232/17)*a[5]^2 +(-1176/17)*a[1]*nz +(1172/17)*a[5]*nz +(24/17)*nz^2 +(1719/17)*a[2] +(1242/17)*a[4] -1289/17,
  a[2]*a[4]*a[5] -48*nz^3 +(307/5)*a[1]*a[4] +(223/5)*a[3]*a[4] +154*a[2]*a[5] +(526/5)*a[4]*a[5] +(-286/5)*a[2]*nz +210*a[4]*nz +(-1756/5)*a[1] +(-1441/10)*a[3] +(1394/5)*a[5] +403*nz,
  a[4]^2*a[5] +216*nz^3 +(-1358/5)*a[1]*a[4] +(-2079/10)*a[3]*a[4] -714*a[2]*a[5] +(-2349/5)*a[4]*a[5] +(1339/5)*a[2]*nz -963*a[4]*nz +(15903/10)*a[1] +(6639/10)*a[3] +(-6456/5)*a[5] +(-3623/2)*nz,
  a[2]*a[5]^2 +(71/8)*a[2]*a[4] +(5/4)*a[4]^2 -8*a[1]*a[5] +(21/2)*a[3]*a[5] +2*a[5]^2 +(-13/2)*a[1]*nz +6*a[5]*nz +9*a[2] +(53/8)*a[4] -25/4,
  a[1]*a[5]^2 +3*nz^3 +(-309/80)*a[1]*a[4] +(-211/80)*a[3]*a[4] +(-17/2)*a[2]*a[5] +(-143/20)*a[4]*a[5] +(131/40)*a[2]*nz +(-103/8)*a[4]*nz +(107/5)*a[1] +(333/40)*a[3] +(-337/20)*a[5] +(-407/16)*nz,
  a[3]*a[5]^2 -27*nz^3 +(2651/80)*a[1]*a[4] +(4363/160)*a[3]*a[4] +(373/4)*a[2]*a[5] +(4673/80)*a[4]*a[5] +(-1409/40)*a[2]*nz +(987/8)*a[4]*nz +(-32011/160)*a[1] +(-13653/160)*a[3] +(6691/40)*a[5] +(3619/16)*nz,
  a[2]*nz^2 +(29/68)*a[2]*a[4] +(29/68)*a[4]^2 +(-234/17)*a[1]*a[5] +(-18/17)*a[3]*a[5] +(-120/17)*a[5]^2 +(-171/34)*a[1]*nz +(147/34)*a[5]*nz +(-13/17)*nz^2 +(-105/136)*a[2] +(23/34)*a[4] +60/17,
  a[5]^2*nz +(1/5)*a[1]*a[4] +(-3/40)*a[3]*a[4] +(1/4)*a[2]*a[5] +(-3/20)*a[4]*a[5] +(-1/10)*a[2]*nz +(-19/40)*a[1] +(-17/40)*a[3] +(-11/10)*a[5] +(-1/4)*nz,
  a[4]*nz^2 +(-235/136)*a[2]*a[4] +(-41/68)*a[4]^2 +(-110/17)*a[1]*a[5] +(-49/17)*a[3]*a[5] +(-32/17)*a[5]^2 +(-217/68)*a[1]*nz +(74/17)*a[5]*nz +(-8/17)*nz^2 +(-65/68)*a[2] +(95/68)*a[4] +217/68,
  a[5]^3 +9*nz^3 +(-87/8)*a[1]*a[4] +(-597/64)*a[3]*a[4] +(-261/8)*a[2]*a[5] +(-591/32)*a[4]*a[5] +(49/4)*a[2]*nz +(-337/8)*a[4]*nz +(4333/64)*a[1] +(1899/64)*a[3] +(-927/16)*a[5] +(-1203/16)*nz,
  a[4]*a[5]^2 +(-2739/68)*a[2]*a[4] +(-701/136)*a[4]^2 +(1047/34)*a[1]*a[5] +(-898/17)*a[3]*a[5] +(-82/17)*a[5]^2 +(452/17)*a[1]*nz +(-450/17)*a[5]*nz +(-12/17)*nz^2 +(-1379/34)*a[2] +(-485/17)*a[4] +517/17,
  a[1]*nz^2 -6*nz^3 +(389/40)*a[1]*a[4] +(347/80)*a[3]*a[4] +(29/2)*a[2]*a[5] +(517/40)*a[4]*a[5] +(-101/20)*a[2]*nz +(45/2)*a[4]*nz +(-3719/80)*a[1] +(-1337/80)*a[3] +(459/20)*a[5] +(411/8)*nz,
  a[1]^2 +(-137/17)*a[2]*a[4] +(-52/17)*a[4]^2 +(-92/17)*a[1]*a[5] +(-164/17)*a[3]*a[5] +(40/17)*a[5]^2 +(-48/17)*a[1]*nz +(52/17)*a[5]*nz +(-24/17)*nz^2 +(-191/34)*a[2] +(67/17)*a[4] +150/17,
  a[5]*nz^2 -6*nz^3 +(181/20)*a[1]*a[4] +(21/5)*a[3]*a[4] +(29/2)*a[2]*a[5] +(253/20)*a[4]*a[5] +(-211/40)*a[2]*nz +(97/4)*a[4]*nz +(-1801/40)*a[1] +(-673/40)*a[3] +(241/10)*a[5] +(101/2)*nz,
  nz^4 +(-62855/3264)*a[2]*a[4] +(-5807/816)*a[4]^2 +(-5381/204)*a[1]*a[5] +(-3407/136)*a[3]*a[5] +(-137/102)*a[5]^2 +(-2955/544)*a[1]*nz +(11791/816)*a[5]*nz +(-3395/272)*nz^2 +(-43787/3264)*a[2] +(15263/1632)*a[4] +13393/544];
I := ideal(L);
IsZeroDim(I);
mp1 := MinPolyQuot(a[1],I,a[1]);
mp2 :=MinPolyQuot(a[2],I,a[2]);
mp3 := MinPolyQuot(a[3],I,a[3]);
mp4 := MinPolyQuot(a[4],I,a[4]);
mp5 := MinPolyQuot(a[5],I,a[5]);

ApproxSolns := ApproxSolve(L); --> takes too long!