I have a system of 8 polynomial equations. I solved the equations in Maple but I'm trying to solve them also in sagemath since it's open source. I've been trying for a long time (weeks) and still couldn't solve the equations in sagemath (Maple finds all the solutions in 37 seconds). I don't need the complex solutions (if exist). only the real solutions.
I have posted before a similar question but with 18 equations and it took Maple 40 minutes to solve them. But since then I was able to reduce the number of equations from 18 to 8. This time Maple solved the equations much faster (37 seconds instead of 40 minutes). But unfortunately, still can't solve the equations with sagemath.
I tried to use groebner basis in sagemath but there are so many different implementations with different order of the polynomials ('lex', 'degrevlex',etc). I tried many of them and none of them could solve the equations.
I will appreciate if anyone can suggest other ways to solve the equations. Maybe not groebner? any method that works and returns all the 24 solutions.
This is the code in Maple that defines the 8 equations and finds all the solutions:
restart;
eq1 := -41.62655310*qb^2-17.84524397*qd*qc-54.90839708*qa*qd+80.88591073*qc*qb-36.57989776*qa*qc-38.37186364*qd*qb+103.9400553*qa*qb-43.34367897*qd^2+3.30995945*qc^2-3.030271028*t13-4.168000558*t12+13.08022556*t11-397.1837038;
eq2 := -2.8905538*qb^2-69.7800994*qd*qc+27.87345418*qa*qd+116.1536456*qc*qb-40.97241364*qa*qc-80.81288558*qd*qb+113.6742097*qa*qb+26.45817378*qd^2-55.32624148*qc^2-5.174596465*t13+8.630792519*t12-4.168000558*t11+126.0503892;
eq3 := -151.1500904*qb^2+13.88726142*qd*qc-62.96764170*qa*qd+23.12716969*qc*qb+68.70359049*qa*qc-73.73067590*qd*qb-70.78770770*qa*qb-21.05107218*qd^2-128.3070522*qc^2+10.28898193*t13-5.174596465*t12-3.030271028*t11+252.2534640;
eq4 := 48643.74394*qa*qd*qb+55369.81474*qa*qd*qc+83823.49379*qa*qb*qc+1068.121378*qc*qb*qd+2*b*qa+57179.40910*qa*qd^2-6744.998133*qc*qb^2-32360.07260*qc^2*qb-7128.296550*qb^2*qd+7808.793570*qd^2*qb+9053.831581*qd^2*qc+60527.22650*qa*qb^2+5696.900092*qd*qc^2+27.87345422*t12*qd+113.6742097*t12*qb-40.97241363*t12*qc+87381.51742*qa*qc^2-54.90839700*t11*qd+103.9400554*t11*qb-36.57989777*t11*qc-62.96764170*t13*qd-70.78770775*t13*qb+68.70359050*t13*qc-20550.50538*qb^3-22371.46454*qd^3+15873.58643*qc^3+16945.27914*qb+22351.84382*qd+15214.00155*qc;
eq5 := 41911.74689*qa^2*qc+103.9400554*t11*qa+60527.22650*qa^2*qb+113.6742097*t12*qa-70.78770775*t13*qa+2*b*qb+24321.87195*qa^2*qd-14256.59311*qa*qd*qb+1068.121383*qa*qd*qc-13489.99629*qa*qb*qc+94398.45549*qc*qb*qd+7808.793561*qa*qd^2+18413.07448*qc*qb^2+190741.1431*qc^2*qb+27014.36446*qb^2*qd+181674.7133*qd^2*qb+16197.41252*qd^2*qc-61651.51619*qa*qb^2+23001.42975*qd*qc^2-80.81288568*t12*qd-5.781107439*t12*qb+116.1536457*t12*qc-32360.07262*qa*qc^2-38.37186355*t11*qd-83.25310616*t11*qb+80.88591081*t11*qc-73.73067575*t13*qd-302.3001806*t13*qb+23.12716970*t13*qc+196624.8969*qb^3+8254.040572*qd^3+11741.93330*qc^3-126998.3098*qb+16945.27914*qa-9234.551898*qd-19188.23950*qc;
eq6 := 87381.51743*qa^2*qc-36.57989776*t11*qa+41911.74688*qa^2*qb-40.97241364*t12*qa+68.70359049*t13*qa+27684.90736*qa^2*qd+1068.121370*qa*qd*qb+11393.80020*qa*qd*qc-64720.14523*qa*qb*qc+46002.85945*qc*qb*qd+9053.831568*qa*qd^2+190741.1430*qc*qb^2+35225.79992*qc^2*qb+47199.22777*qb^2*qd+16197.41253*qd^2*qb+190164.0459*qd^2*qc-6744.998130*qa*qb^2+94531.22698*qd*qc^2-69.7800994*t12*qd+116.1536456*t12*qb-110.6524831*t12*qc+47620.75929*qa*qc^2-17.84524397*t11*qd+80.88591073*t11*qb+6.61991877*t11*qc+13.88726142*t13*qd+23.12716969*t13*qb-256.6141048*t13*qc+6137.691497*qb^3+43885.84331*qd^3+135209.5818*qc^3-19188.23950*qb+15214.00155*qa-46604.79008*qd-78396.52112*qc+2*b*qc;
eq7 := 27684.90736*qa^2*qc-54.90839710*t11*qa+24321.87196*qa^2*qb+27.87345416*t12*qa-62.96764170*t13*qa+57179.40910*qa^2*qd+15617.58713*qa*qd*qb+18107.66316*qa*qd*qc+1068.121368*qa*qb*qc+32394.82511*qc*qb*qd-67114.39361*qa*qd^2+47199.22776*qc*qb^2+23001.42974*qc^2*qb+181674.7134*qb^2*qd+24762.12165*qd^2*qb+131657.5299*qd^2*qc-7128.296555*qa*qb^2+190164.0459*qd*qc^2+52.91634761*t12*qd-80.81288562*t12*qb-69.78009942*t12*qc+5696.900066*qa*qc^2-86.68735798*t11*qd-38.37186373*t11*qb-17.84524400*t11*qc-42.10214425*t13*qd-73.73067582*t13*qb+13.88726138*t13*qc+9004.788156*qb^3+133077.8789*qd^3+31510.40898*qc^3-9234.551898*qb+22351.84381*qa-92497.15079*qd-46604.79008*qc+2*b*qd;
eq8 := qa^2+qb^2+qc^2+qd^2-1;
vars := [b, qa, qb, qc, qd, t11, t12, t13];
polysys:={eq1, eq2, eq3, eq4, eq5, eq6, eq7, eq8}:
sols := CodeTools:-Usage(RootFinding:-Isolate(polysys, vars, output = numeric, method = RS));
These are all the solutions that Maple found (24 solutions):
sols :=
[[b = -23289.41591, qa = .1029740229, qb = 0.2420731726e-1, qc = -0.4970119805e-1, qd = -.9931465890, t11 = 16.14238504, t12 = -29.28513588, t13 = -33.24421757],
[b = -9080.512857, qa = -.2519640476, qb = .3367303333, qc = 0.6074726399e-1, qd = -.9052273589, t11 = 14.84913687, t12 = -34.27497609, t13 = -35.23476883],
[b = -22469.90836, qa = .2258225797, qb = .6051413616, qc = 0.7344035683e-1, qd = -.7598780224, t11 = 5.465349244, t12 = -45.78980429, t13 = -42.84832688],
[b = -14992.17057, qa = .6769861076, qb = -.1826569123, qc = -.3413288441, qd = -.6259559750, t11 = 18.19534486, t12 = -18.82393484, t13 = -27.39078848],
[b = -11859.14621, qa = -.5268421637, qb = -.2338892114, qc = .6027505370, qd = -.5517471898, t11 = 26.44677831, t12 = -9.548492453, t13 = -9.135314959],
[b = -15721.13615, qa = .6439998765, qb = -.5296654469, qc = -.1435337526, qd = -.5330260175, t11 = 30.10702525, t12 = -1.383387852, t13 = -13.46736812],
[b = -10200.41071, qa = .6272125842, qb = .1077303535, qc = -.5834210152, qd = -.5045973287, t11 = 18.65647237, t12 = -16.21794687, t13 = -21.91673231],
[b = -12355.11404, qa = -.5206156206, qb = .7022865053, qc = 0.5801337392e-1, qd = -.4820658550, t11 = 26.31272494, t12 = -14.11357284, t13 = -19.36027209],
[b = -12845.72667, qa = .4807602059, qb = .7880310413, qc = -0.1338740602e-1, qd = -.3843142980, t11 = 9.666002146, t12 = -35.75962940, t13 = -30.86212126],
[b = -42154.93615, qa = -.7189190835, qb = -.1549310524, qc = -.6415225544, qd = -.2181754628, t11 = 22.10561113, t12 = -13.55120889, t13 = -20.76296353],
[b = -28524.04913, qa = .1379870648, qb = .9524424961, qc = .2351876296, qd = -.1360133831, t11 = 21.44096587, t12 = -18.07878760, t13 = -14.05639922],
[b = -33340.87135, qa = 0.2194307551e-1, qb = .9715259283, qc = .2335069231, qd = -0.3362125582e-1, t11 = 27.45828712, t12 = -8.392733724, t13 = -6.728265559],
[b = -33340.87135, qa = -0.2194307551e-1, qb = -.9715259283, qc = -.2335069231, qd = 0.3362125582e-1, t11 = 27.45828712, t12 = -8.392733724, t13 = -6.728265559],
[b = -28524.04913, qa = -.1379870648, qb = -.9524424961, qc = -.2351876296, qd = .1360133831, t11 = 21.44096587, t12 = -18.07878760, t13 = -14.05639922],
[b = -42154.93615, qa = .7189190835, qb = .1549310524, qc = .6415225544, qd = .2181754628, t11 = 22.10561113, t12 = -13.55120889, t13 = -20.76296353],
[b = -12845.72667, qa = -.4807602059, qb = -.7880310413, qc = 0.1338740602e-1, qd = .3843142980, t11 = 9.666002146, t12 = -35.75962940, t13 = -30.86212126],
[b = -12355.11404, qa = .5206156206, qb = -.7022865053, qc = -0.5801337392e-1, qd = .4820658550, t11 = 26.31272494, t12 = -14.11357284, t13 = -19.36027209],
[b = -10200.41071, qa = -.6272125842, qb = -.1077303535, qc = .5834210152, qd = .5045973287, t11 = 18.65647237, t12 = -16.21794687, t13 = -21.91673231],
[b = -15721.13615, qa = -.6439998765, qb = .5296654469, qc = .1435337526, qd = .5330260175, t11 = 30.10702525, t12 = -1.383387852, t13 = -13.46736812],
[b = -11859.14621, qa = .5268421637, qb = .2338892114, qc = -.6027505370, qd = .5517471898, t11 = 26.44677831, t12 = -9.548492453, t13 = -9.135314959],
[b = -14992.17057, qa = -.6769861076, qb = .1826569123, qc = .3413288441, qd = .6259559750, t11 = 18.19534486, t12 = -18.82393484, t13 = -27.39078848],
[b = -22469.90836, qa = -.2258225797, qb = -.6051413616, qc = -0.7344035683e-1, qd = .7598780224, t11 = 5.465349244, t12 = -45.78980429, t13 = -42.84832688],
[b = -9080.512857, qa = .2519640476, qb = -.3367303333, qc = -0.6074726399e-1, qd = .9052273589, t11 = 14.84913687, t12 = -34.27497609, t13 = -35.23476883],
[b = -23289.41591, qa = -.1029740229, qb = -0.2420731726e-1, qc = 0.4970119805e-1, qd = .9931465890, t11 = 16.14238504, t12 = -29.28513588, t13 = -33.24421757]]
And this is the code in sagemath that I couldn't solve. Same equations as before, only converted them to rational coefficients using eq_i=P(eq_i). It is stuck in I.groebner_basis.
P.<b, qa, qb, qc, qd, t11, t12, t13>=PolynomialRing(QQ,order='degrevlex')
eq1 = -41.62655310*qb^2-17.84524397*qd*qc-54.90839708*qa*qd+80.88591073*qc*qb-36.57989776*qa*qc-38.37186364*qd*qb+103.9400553*qa*qb-43.34367897*qd^2+3.30995945*qc^2-3.030271028*t13-4.168000558*t12+13.08022556*t11-397.1837038
eq2 = -2.8905538*qb^2-69.7800994*qd*qc+27.87345418*qa*qd+116.1536456*qc*qb-40.97241364*qa*qc-80.81288558*qd*qb+113.6742097*qa*qb+26.45817378*qd^2-55.32624148*qc^2-5.174596465*t13+8.630792519*t12-4.168000558*t11+126.0503892
eq3 = -151.1500904*qb^2+13.88726142*qd*qc-62.96764170*qa*qd+23.12716969*qc*qb+68.70359049*qa*qc-73.73067590*qd*qb-70.78770770*qa*qb-21.05107218*qd^2-128.3070522*qc^2+10.28898193*t13-5.174596465*t12-3.030271028*t11+252.2534640
eq4 = 48643.74394*qa*qd*qb+55369.81474*qa*qd*qc+83823.49379*qa*qb*qc+1068.121378*qc*qb*qd+2*b*qa+57179.40910*qa*qd^2-6744.998133*qc*qb^2-32360.07260*qc^2*qb-7128.296550*qb^2*qd+7808.793570*qd^2*qb+9053.831581*qd^2*qc+60527.22650*qa*qb^2+5696.900092*qd*qc^2+27.87345422*t12*qd+113.6742097*t12*qb-40.97241363*t12*qc+87381.51742*qa*qc^2-54.90839700*t11*qd+103.9400554*t11*qb-36.57989777*t11*qc-62.96764170*t13*qd-70.78770775*t13*qb+68.70359050*t13*qc-20550.50538*qb^3-22371.46454*qd^3+15873.58643*qc^3+16945.27914*qb+22351.84382*qd+15214.00155*qc
eq5 = 41911.74689*qa^2*qc+103.9400554*t11*qa+60527.22650*qa^2*qb+113.6742097*t12*qa-70.78770775*t13*qa+2*b*qb+24321.87195*qa^2*qd-14256.59311*qa*qd*qb+1068.121383*qa*qd*qc-13489.99629*qa*qb*qc+94398.45549*qc*qb*qd+7808.793561*qa*qd^2+18413.07448*qc*qb^2+190741.1431*qc^2*qb+27014.36446*qb^2*qd+181674.7133*qd^2*qb+16197.41252*qd^2*qc-61651.51619*qa*qb^2+23001.42975*qd*qc^2-80.81288568*t12*qd-5.781107439*t12*qb+116.1536457*t12*qc-32360.07262*qa*qc^2-38.37186355*t11*qd-83.25310616*t11*qb+80.88591081*t11*qc-73.73067575*t13*qd-302.3001806*t13*qb+23.12716970*t13*qc+196624.8969*qb^3+8254.040572*qd^3+11741.93330*qc^3-126998.3098*qb+16945.27914*qa-9234.551898*qd-19188.23950*qc
eq6 = 87381.51743*qa^2*qc-36.57989776*t11*qa+41911.74688*qa^2*qb-40.97241364*t12*qa+68.70359049*t13*qa+27684.90736*qa^2*qd+1068.121370*qa*qd*qb+11393.80020*qa*qd*qc-64720.14523*qa*qb*qc+46002.85945*qc*qb*qd+9053.831568*qa*qd^2+190741.1430*qc*qb^2+35225.79992*qc^2*qb+47199.22777*qb^2*qd+16197.41253*qd^2*qb+190164.0459*qd^2*qc-6744.998130*qa*qb^2+94531.22698*qd*qc^2-69.7800994*t12*qd+116.1536456*t12*qb-110.6524831*t12*qc+47620.75929*qa*qc^2-17.84524397*t11*qd+80.88591073*t11*qb+6.61991877*t11*qc+13.88726142*t13*qd+23.12716969*t13*qb-256.6141048*t13*qc+6137.691497*qb^3+43885.84331*qd^3+135209.5818*qc^3-19188.23950*qb+15214.00155*qa-46604.79008*qd-78396.52112*qc+2*b*qc
eq7 = 27684.90736*qa^2*qc-54.90839710*t11*qa+24321.87196*qa^2*qb+27.87345416*t12*qa-62.96764170*t13*qa+57179.40910*qa^2*qd+15617.58713*qa*qd*qb+18107.66316*qa*qd*qc+1068.121368*qa*qb*qc+32394.82511*qc*qb*qd-67114.39361*qa*qd^2+47199.22776*qc*qb^2+23001.42974*qc^2*qb+181674.7134*qb^2*qd+24762.12165*qd^2*qb+131657.5299*qd^2*qc-7128.296555*qa*qb^2+190164.0459*qd*qc^2+52.91634761*t12*qd-80.81288562*t12*qb-69.78009942*t12*qc+5696.900066*qa*qc^2-86.68735798*t11*qd-38.37186373*t11*qb-17.84524400*t11*qc-42.10214425*t13*qd-73.73067582*t13*qb+13.88726138*t13*qc+9004.788156*qb^3+133077.8789*qd^3+31510.40898*qc^3-9234.551898*qb+22351.84381*qa-92497.15079*qd-46604.79008*qc+2*b*qd
eq8 = qa^2+qb^2+qc^2+qd^2-1
eq1=P(eq1)
eq2=P(eq2)
eq3=P(eq3)
eq4=P(eq4)
eq5=P(eq5)
eq6=P(eq6)
eq7=P(eq7)
eq8=P(eq8)
I=Ideal(eq1, eq2, eq3, eq4, eq5, eq6, eq7, eq8)
I.groebner_basis()
I.variety(RR)
Can anyone suggest a different way for solving the equations? It's really hard to believe that no function works when Maple can solve them in 37 seconds. I will appreciate any help.
Thanks
