It turns ut that Chris' sugestion is effective in sympy 1.9 (current), but not in sympy 1.8 (still used by Sage 9.5.beta7). I still think that this equation preprocessing shouldn't be necessary...
Sorry for the noise. Le lundi 6 décembre 2021 à 11:10:16 UTC+1, emanuel.c...@gmail.com a écrit : > It’s a bit more involved than that; see the parallel discussion of the > same problem in this post > <https://groups.google.com/g/sage-support/c/ZnKV3D-i9t0> on Sagemath > support list. > > Le lundi 6 décembre 2021 à 02:28:31 UTC+1, smi...@gmail.com a écrit : > >> It just looks like the OP issue has two extra solutions (when the initial >> equations are not factored) which couldn't be eliminated (easily) and the >> solver sends them back to be safe. >> >> NOTE: `fsol` in the previous was obtained with `solve([factor(i) for i in >> Sys], Unk, simplify=False)`. >> >> If checking is turned off then two invalid solutions are included, but >> the solution involving `sqrt(303)` is missing: >> ```python >> >>> s=solve([factor(i) for i in Sys], Unk, check=0,dict=1); s >> [{a2: 0, a1: 0}, {a2: 0, a1: 0}, {a1: 0, a2: 0}, {a1: 0, b2: 0}, {a2: 0, >> b2: 0}, {a2: a1**2, b1: b2*(a1**3 + 2*a1**2 - 1)/(3*a1**3)}] >> >>> [[e.subs(si).simplify() for e in Sys] for si in s] >> [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, a2**3, -2*a2**2*b1], >> [0, 0, 0, 0], [0, 0, 0, a1*b2*(a1**3 - 4*a1**2 + 2)/3]] >> ``` >> >> /c >> On Sunday, December 5, 2021 at 8:50:27 AM UTC-6 Chris Smith wrote: >> >>> If you factor the equations before passing them to solve then the >>> solution is >>> ``` >>> >>> fsol >>> [{a2: 0, a1: 0}, {a2: 0, a1: 0}, {a2: 0, b2: 0}, {a2: 0, b1: b2/2, a1: >>> 0}, {a2: (64 + (-8 + (1 - sqrt(3)*I)*(37 + 3*sqrt(303)*I)**(1/3))*(1 - >>> sqrt(3)*I)*(37 + 3*sqrt(303)*I)**(1/3))**2/(36*(1 - sqrt(3)*I)**2*(37 >>> + 3*sqrt(303)*I)**(2/3)), b1: b2/2, a1: (-64 + (1 - sqrt(3)*I)*(8 + >>> (-1 + sqrt(3)*I)*(37 + 3*sqrt(303)*I)**(1/3))*(37 + >>> 3*sqrt(303)*I)**(1/3))/(6*(1 - sqrt(3)*I)*(37 + >>> 3*sqrt(303)*I)**(1/3))}, {a2: (64 + (-8 + (1 + sqrt(3)*I)*(37 + >>> 3*sqrt(303)*I)**(1/3))*(1 + sqrt(3)*I)*(37 + >>> 3*sqrt(303)*I)**(1/3))**2/(36*(1 + sqrt(3)*I)**2*(37 + >>> 3*sqrt(303)*I)**(2/3)), b1: b2/2, a1: (-64 + (1 + sqrt(3)*I)*(8 - (1 + >>> sqrt(3)*I)*(37 + 3*sqrt(303)*I)**(1/3))*(37 + >>> 3*sqrt(303)*I)**(1/3))/(6*(1 + sqrt(3)*I)*(37 + >>> 3*sqrt(303)*I)**(1/3))}, {a2: (16 + (4 + (37 + >>> 3*sqrt(303)*I)**(1/3))*(37 + 3*sqrt(303)*I)**(1/3))**2/(9*(37 + >>> 3*sqrt(303)*I)**(2/3)), b1: b2/2, a1: 4/3 + 16/(3*(37 + >>> 3*sqrt(303)*I)**(1/3)) + (37 + 3*sqrt(303)*I)**(1/3)/3}, {a2: a1**2, >>> b1: 0, b2: 0}] >>> ``` >>> and all of them are true solutions: >>> ``` >>> >>> [[e.subs(s).simplify() for e in Sys] for s in fsol] >>> [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], >>> [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]] >>> ``` >>> >>> On Sunday, December 5, 2021 at 8:35:28 AM UTC-6 Chris Smith wrote: >>> >>>> `nonlinsolve(Sys, Unk)` gives >>>> ``` >>>> {(0, 0, b1, b2), (0, 0, b2/2, b2), (a1, 0, b1, 0), (-sqrt(a2), a2, 0, >>>> 0), (sqrt(a2), a2, 0, 0)} >>>> ``` >>>> On Saturday, December 4, 2021 at 4:36:04 PM UTC-6 >>>> emanuel.c...@gmail.com wrote: >>>> >>>>> A bug in Sympy’s solve >>>>> >>>>> Inspired by this ask.sagemath.orgquestion >>>>> <https://ask.sagemath.org/question/59063/weird-c-values-from-solving-system-of-equations/> >>>>> . >>>>> >>>>> This question was about the names of the symbolic variables created by >>>>> Maxima’s solve to denote arbitrary constants. Exploring the example >>>>> used in this questions revealed a non-trivial problem with sympy’s >>>>> solve. >>>>> Problem >>>>> >>>>> Solve the system : >>>>> $$ >>>>> \begin{align*} >>>>> >>>>> - a*{1}^{3} a*{2} + a*{1} a*{2}^{2} &= 0 \ >>>>> - 3 a*{1}^{2} a*{2} b*{1} + 2 a*{1} a*{2} b*{2} - a*{1} b*{2} + >>>>> a*{2}^{2} >>>>> b*{2} &= 0 \ >>>>> - a*{1}^{2} a*{2}^{2} + a_{2}^{3} &= 0 \ >>>>> - 2 a*{1}^{2} a*{2} b*{2} - 2 a*{2}^{2} b*{1} + 3 a*{2}^{2} b_{2} >>>>> &= 0 >>>>> \end{align*} >>>>> $$ >>>>> >>>>> # Set up sympy (brutal version)import sympyfrom sympy import * >>>>> init_session() >>>>> init_printing(pretty_print=False)# System to solve >>>>> a1, a2, b1, b2 = symbols('a1 a2 b1 b2') >>>>> Unk = [a1, a2, b1, b2] >>>>> eq1 = a1 * a2**2 - a2 * a1**3 >>>>> eq2 = 2*a1*a2*b2 + b2*a2**2 - 3*a2*a1**2*b1 - a1*b2 >>>>> eq3 = a2**3 - a2**2*a1**2 >>>>> eq4 = 3*a2**2*b2 - 2*a2*a1**2*b2 - 2*a2**2*b1 >>>>> Sys = [eq1, eq2, eq3, eq4] >>>>> >>>>> IPython console for SymPy 1.9 (Python 3.9.9-64-bit) (ground types: gmpy) >>>>> >>>>> These commands were executed: >>>>> >>> from __future__ import division >>>>> >>> from sympy import * >>>>> >>> x, y, z, t = symbols('x y z t') >>>>> >>> k, m, n = symbols('k m n', integer=True) >>>>> >>> f, g, h = symbols('f g h', cls=Function) >>>>> >>> init_printing() >>>>> >>>>> Documentation can be found at https://docs.sympy.org/1.9/ >>>>> >>>>> Attempt to use the “automatic” Sympy solver: >>>>> >>>>> Sol = solve(Sys, Unk) >>>>> DSol = [dict(zip(Unk, u)) for u in Sol] >>>>> DSol >>>>> >>>>> [{a1: 0, a2: 0, b1: b1, b2: b2}, {a1: 0, a2: 0, b1: b1, b2: b2}, {a1: 0, >>>>> a2: 0, b1: b1, b2: b2}, {a1: 0, a2: 0, b1: b2/2, b2: b2}, {a1: a1, a2: 0, >>>>> b1: b1, b2: 0}, {a1: -sqrt(a2), a2: a2, b1: 0, b2: 0}, {a1: >>>>> -2**(5/6)*sqrt(48*2**(1/3) + 624/(1499 + 3*sqrt(303)*I)**(1/3) + >>>>> 3*2**(2/3)*(1499 + 3*sqrt(303)*I)**(1/3))/6, a2: 16/3 + >>>>> 104*2**(2/3)/(3*(1499 + 3*sqrt(303)*I)**(1/3)) + 2**(1/3)*(1499 + >>>>> 3*sqrt(303)*I)**(1/3)/3, b1: b2/2, b2: b2}, {a1: -sqrt(6)*sqrt(32 - >>>>> 2**(1/3)*(1 + sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3) - 416*2**(2/3)/((1 >>>>> + sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3)))/6, a2: 2**(2/3)*(-208/3 + (1 >>>>> + sqrt(3)*I)*(32 + (-1 - sqrt(3)*I)*(2998 + 6*sqrt(303)*I)**(1/3))*(2998 >>>>> + 6*sqrt(303)*I)**(1/3)/12)/((1 + sqrt(3)*I)*(1499 + >>>>> 3*sqrt(303)*I)**(1/3)), b1: b2/2, b2: b2}, {a1: -sqrt(6)*sqrt(32 - >>>>> 416*2**(2/3)/((1 - sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3)) - >>>>> 2**(1/3)*(1 - sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3))/6, a2: >>>>> 2**(2/3)*(-208/3 + (1 - sqrt(3)*I)*(32 + (-1 + sqrt(3)*I)*(2998 + >>>>> 6*sqrt(303)*I)**(1/3))*(2998 + 6*sqrt(303)*I)**(1/3)/12)/((1 - >>>>> sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3)), b1: b2/2, b2: b2}] >>>>> >>>>> Something went sideways: the six first solutions are okay,but the last >>>>> three use expressions, some of them being polynomials in b2. >>>>> >>>>> Attempt to check them formally : >>>>> >>>>> Chk=[[u.subs(s) for u in Sys] for s in DSol] >>>>> Chk >>>>> >>>>> [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], >>>>> [0, 0, 0, 0], [-2**(5/6)*(16/3 + 104*2**(2/3)/(3*(1499 + >>>>> 3*sqrt(303)*I)**(1/3)) + 2**(1/3)*(1499 + >>>>> 3*sqrt(303)*I)**(1/3)/3)**2*sqrt(48*2**(1/3) + 624/(1499 + >>>>> 3*sqrt(303)*I)**(1/3) + 3*2**(2/3)*(1499 + 3*sqrt(303)*I)**(1/3))/6 + >>>>> sqrt(2)*(16/3 + 104*2**(2/3)/(3*(1499 + 3*sqrt(303)*I)**(1/3)) + >>>>> 2**(1/3)*(1499 + 3*sqrt(303)*I)**(1/3)/3)*(48*2**(1/3) + 624/(1499 + >>>>> 3*sqrt(303)*I)**(1/3) + 3*2**(2/3)*(1499 + >>>>> 3*sqrt(303)*I)**(1/3))**(3/2)/54, -2**(2/3)*b2*(16/3 + >>>>> 104*2**(2/3)/(3*(1499 + 3*sqrt(303)*I)**(1/3)) + 2**(1/3)*(1499 + >>>>> 3*sqrt(303)*I)**(1/3)/3)*(48*2**(1/3) + 624/(1499 + 3*sqrt(303)*I)**(1/3) >>>>> + 3*2**(2/3)*(1499 + 3*sqrt(303)*I)**(1/3))/12 - 2**(5/6)*b2*(16/3 + >>>>> 104*2**(2/3)/(3*(1499 + 3*sqrt(303)*I)**(1/3)) + 2**(1/3)*(1499 + >>>>> 3*sqrt(303)*I)**(1/3)/3)*sqrt(48*2**(1/3) + 624/(1499 + >>>>> 3*sqrt(303)*I)**(1/3) + 3*2**(2/3)*(1499 + 3*sqrt(303)*I)**(1/3))/3 + >>>>> b2*(16/3 + 104*2**(2/3)/(3*(1499 + 3*sqrt(303)*I)**(1/3)) + >>>>> 2**(1/3)*(1499 + 3*sqrt(303)*I)**(1/3)/3)**2 + >>>>> 2**(5/6)*b2*sqrt(48*2**(1/3) + 624/(1499 + 3*sqrt(303)*I)**(1/3) + >>>>> 3*2**(2/3)*(1499 + 3*sqrt(303)*I)**(1/3))/6, -2**(2/3)*(16/3 + >>>>> 104*2**(2/3)/(3*(1499 + 3*sqrt(303)*I)**(1/3)) + 2**(1/3)*(1499 + >>>>> 3*sqrt(303)*I)**(1/3)/3)**2*(48*2**(1/3) + 624/(1499 + >>>>> 3*sqrt(303)*I)**(1/3) + 3*2**(2/3)*(1499 + 3*sqrt(303)*I)**(1/3))/18 + >>>>> (16/3 + 104*2**(2/3)/(3*(1499 + 3*sqrt(303)*I)**(1/3)) + 2**(1/3)*(1499 + >>>>> 3*sqrt(303)*I)**(1/3)/3)**3, -2**(2/3)*b2*(16/3 + 104*2**(2/3)/(3*(1499 + >>>>> 3*sqrt(303)*I)**(1/3)) + 2**(1/3)*(1499 + >>>>> 3*sqrt(303)*I)**(1/3)/3)*(48*2**(1/3) + 624/(1499 + 3*sqrt(303)*I)**(1/3) >>>>> + 3*2**(2/3)*(1499 + 3*sqrt(303)*I)**(1/3))/9 + 2*b2*(16/3 + >>>>> 104*2**(2/3)/(3*(1499 + 3*sqrt(303)*I)**(1/3)) + 2**(1/3)*(1499 + >>>>> 3*sqrt(303)*I)**(1/3)/3)**2], [2**(1/6)*sqrt(3)*(-208/3 + (1 + >>>>> sqrt(3)*I)*(32 + (-1 - sqrt(3)*I)*(2998 + 6*sqrt(303)*I)**(1/3))*(2998 + >>>>> 6*sqrt(303)*I)**(1/3)/12)*(32 - 2**(1/3)*(1 + sqrt(3)*I)*(1499 + >>>>> 3*sqrt(303)*I)**(1/3) - 416*2**(2/3)/((1 + sqrt(3)*I)*(1499 + >>>>> 3*sqrt(303)*I)**(1/3)))**(3/2)/(18*(1 + sqrt(3)*I)*(1499 + >>>>> 3*sqrt(303)*I)**(1/3)) - 2**(5/6)*sqrt(3)*(-208/3 + (1 + sqrt(3)*I)*(32 + >>>>> (-1 - sqrt(3)*I)*(2998 + 6*sqrt(303)*I)**(1/3))*(2998 + >>>>> 6*sqrt(303)*I)**(1/3)/12)**2*sqrt(32 - 2**(1/3)*(1 + sqrt(3)*I)*(1499 + >>>>> 3*sqrt(303)*I)**(1/3) - 416*2**(2/3)/((1 + sqrt(3)*I)*(1499 + >>>>> 3*sqrt(303)*I)**(1/3)))/(3*(1 + sqrt(3)*I)**2*(1499 + >>>>> 3*sqrt(303)*I)**(2/3)), -2*2**(1/6)*sqrt(3)*b2*(-208/3 + (1 + >>>>> sqrt(3)*I)*(32 + (-1 - sqrt(3)*I)*(2998 + 6*sqrt(303)*I)**(1/3))*(2998 + >>>>> 6*sqrt(303)*I)**(1/3)/12)*sqrt(32 - 2**(1/3)*(1 + sqrt(3)*I)*(1499 + >>>>> 3*sqrt(303)*I)**(1/3) - 416*2**(2/3)/((1 + sqrt(3)*I)*(1499 + >>>>> 3*sqrt(303)*I)**(1/3)))/(3*(1 + sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3)) >>>>> - 3*2**(2/3)*b2*(-208/3 + (1 + sqrt(3)*I)*(32 + (-1 - sqrt(3)*I)*(2998 + >>>>> 6*sqrt(303)*I)**(1/3))*(2998 + 6*sqrt(303)*I)**(1/3)/12)*(16/3 - >>>>> 2**(1/3)*(1 + sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3)/6 - >>>>> 208*2**(2/3)/(3*(1 + sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3)))/(2*(1 + >>>>> sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3)) + sqrt(6)*b2*sqrt(32 - >>>>> 2**(1/3)*(1 + sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3) - 416*2**(2/3)/((1 >>>>> + sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3)))/6 + 2*2**(1/3)*b2*(-208/3 + >>>>> (1 + sqrt(3)*I)*(32 + (-1 - sqrt(3)*I)*(2998 + >>>>> 6*sqrt(303)*I)**(1/3))*(2998 + 6*sqrt(303)*I)**(1/3)/12)**2/((1 + >>>>> sqrt(3)*I)**2*(1499 + 3*sqrt(303)*I)**(2/3)), -2*2**(1/3)*(-208/3 + (1 + >>>>> sqrt(3)*I)*(32 + (-1 - sqrt(3)*I)*(2998 + 6*sqrt(303)*I)**(1/3))*(2998 + >>>>> 6*sqrt(303)*I)**(1/3)/12)**2*(16/3 - 2**(1/3)*(1 + sqrt(3)*I)*(1499 + >>>>> 3*sqrt(303)*I)**(1/3)/6 - 208*2**(2/3)/(3*(1 + sqrt(3)*I)*(1499 + >>>>> 3*sqrt(303)*I)**(1/3)))/((1 + sqrt(3)*I)**2*(1499 + >>>>> 3*sqrt(303)*I)**(2/3)) + 4*(-208/3 + (1 + sqrt(3)*I)*(32 + (-1 - >>>>> sqrt(3)*I)*(2998 + 6*sqrt(303)*I)**(1/3))*(2998 + >>>>> 6*sqrt(303)*I)**(1/3)/12)**3/((1 + sqrt(3)*I)**3*(1499 + 3*sqrt(303)*I)), >>>>> -2*2**(2/3)*b2*(-208/3 + (1 + sqrt(3)*I)*(32 + (-1 - sqrt(3)*I)*(2998 + >>>>> 6*sqrt(303)*I)**(1/3))*(2998 + 6*sqrt(303)*I)**(1/3)/12)*(16/3 - >>>>> 2**(1/3)*(1 + sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3)/6 - >>>>> 208*2**(2/3)/(3*(1 + sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3)))/((1 + >>>>> sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3)) + 4*2**(1/3)*b2*(-208/3 + (1 + >>>>> sqrt(3)*I)*(32 + (-1 - sqrt(3)*I)*(2998 + 6*sqrt(303)*I)**(1/3))*(2998 + >>>>> 6*sqrt(303)*I)**(1/3)/12)**2/((1 + sqrt(3)*I)**2*(1499 + >>>>> 3*sqrt(303)*I)**(2/3))], [-2**(5/6)*sqrt(3)*(-208/3 + (1 - sqrt(3)*I)*(32 >>>>> + (-1 + sqrt(3)*I)*(2998 + 6*sqrt(303)*I)**(1/3))*(2998 + >>>>> 6*sqrt(303)*I)**(1/3)/12)**2*sqrt(32 - 416*2**(2/3)/((1 - >>>>> sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3)) - 2**(1/3)*(1 - >>>>> sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3))/(3*(1 - sqrt(3)*I)**2*(1499 + >>>>> 3*sqrt(303)*I)**(2/3)) + 2**(1/6)*sqrt(3)*(-208/3 + (1 - sqrt(3)*I)*(32 + >>>>> (-1 + sqrt(3)*I)*(2998 + 6*sqrt(303)*I)**(1/3))*(2998 + >>>>> 6*sqrt(303)*I)**(1/3)/12)*(32 - 416*2**(2/3)/((1 - sqrt(3)*I)*(1499 + >>>>> 3*sqrt(303)*I)**(1/3)) - 2**(1/3)*(1 - sqrt(3)*I)*(1499 + >>>>> 3*sqrt(303)*I)**(1/3))**(3/2)/(18*(1 - sqrt(3)*I)*(1499 + >>>>> 3*sqrt(303)*I)**(1/3)), sqrt(6)*b2*sqrt(32 - 416*2**(2/3)/((1 - >>>>> sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3)) - 2**(1/3)*(1 - >>>>> sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3))/6 + 2*2**(1/3)*b2*(-208/3 + (1 >>>>> - sqrt(3)*I)*(32 + (-1 + sqrt(3)*I)*(2998 + 6*sqrt(303)*I)**(1/3))*(2998 >>>>> + 6*sqrt(303)*I)**(1/3)/12)**2/((1 - sqrt(3)*I)**2*(1499 + >>>>> 3*sqrt(303)*I)**(2/3)) - 3*2**(2/3)*b2*(-208/3 + (1 - sqrt(3)*I)*(32 + >>>>> (-1 + sqrt(3)*I)*(2998 + 6*sqrt(303)*I)**(1/3))*(2998 + >>>>> 6*sqrt(303)*I)**(1/3)/12)*(16/3 - 208*2**(2/3)/(3*(1 - sqrt(3)*I)*(1499 + >>>>> 3*sqrt(303)*I)**(1/3)) - 2**(1/3)*(1 - sqrt(3)*I)*(1499 + >>>>> 3*sqrt(303)*I)**(1/3)/6)/(2*(1 - sqrt(3)*I)*(1499 + >>>>> 3*sqrt(303)*I)**(1/3)) - 2*2**(1/6)*sqrt(3)*b2*(-208/3 + (1 - >>>>> sqrt(3)*I)*(32 + (-1 + sqrt(3)*I)*(2998 + 6*sqrt(303)*I)**(1/3))*(2998 + >>>>> 6*sqrt(303)*I)**(1/3)/12)*sqrt(32 - 416*2**(2/3)/((1 - sqrt(3)*I)*(1499 + >>>>> 3*sqrt(303)*I)**(1/3)) - 2**(1/3)*(1 - sqrt(3)*I)*(1499 + >>>>> 3*sqrt(303)*I)**(1/3))/(3*(1 - sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3)), >>>>> 4*(-208/3 + (1 - sqrt(3)*I)*(32 + (-1 + sqrt(3)*I)*(2998 + >>>>> 6*sqrt(303)*I)**(1/3))*(2998 + 6*sqrt(303)*I)**(1/3)/12)**3/((1 - >>>>> sqrt(3)*I)**3*(1499 + 3*sqrt(303)*I)) - 2*2**(1/3)*(-208/3 + (1 - >>>>> sqrt(3)*I)*(32 + (-1 + sqrt(3)*I)*(2998 + 6*sqrt(303)*I)**(1/3))*(2998 + >>>>> 6*sqrt(303)*I)**(1/3)/12)**2*(16/3 - 208*2**(2/3)/(3*(1 - >>>>> sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3)) - 2**(1/3)*(1 - >>>>> sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3)/6)/((1 - sqrt(3)*I)**2*(1499 + >>>>> 3*sqrt(303)*I)**(2/3)), -2*2**(2/3)*b2*(-208/3 + (1 - sqrt(3)*I)*(32 + >>>>> (-1 + sqrt(3)*I)*(2998 + 6*sqrt(303)*I)**(1/3))*(2998 + >>>>> 6*sqrt(303)*I)**(1/3)/12)*(16/3 - 208*2**(2/3)/(3*(1 - sqrt(3)*I)*(1499 + >>>>> 3*sqrt(303)*I)**(1/3)) - 2**(1/3)*(1 - sqrt(3)*I)*(1499 + >>>>> 3*sqrt(303)*I)**(1/3)/6)/((1 - sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3)) >>>>> + 4*2**(1/3)*b2*(-208/3 + (1 - sqrt(3)*I)*(32 + (-1 + sqrt(3)*I)*(2998 + >>>>> 6*sqrt(303)*I)**(1/3))*(2998 + 6*sqrt(303)*I)**(1/3)/12)**2/((1 - >>>>> sqrt(3)*I)**2*(1499 + 3*sqrt(303)*I)**(2/3))]] >>>>> >>>>> Some of these expressions are polynomial in b2 whose coefficients of >>>>> not-null dregree are nort *obviously* null. >>>>> A bit of hit-and-miss trials leads to this attempt at *numerical* >>>>> check : >>>>> >>>>> def chz(x): >>>>> r = x.factor().is_zero >>>>> if r is None: return x.coeff(b2).n() >>>>> return r >>>>> [[chz(u) for u in v] for v in Chk] >>>>> >>>>> [[True, True, True, True], >>>>> [True, True, True, True], >>>>> [True, True, True, True], >>>>> [True, True, True, True], >>>>> [True, True, True, True], >>>>> [True, True, True, True], >>>>> [True, -223.427427555427 + 0.e-25*I, True, True], >>>>> [True, -0.388012232966408 - 5.25038392589403e-29*I, True, True], >>>>> [True, -0.e-137 + 1.12445821450677e-139*I, True, True]] >>>>> >>>>> Solution #8 may be exactr, but #6 and #7 cannot. >>>>> Manual solution. >>>>> >>>>> eq1 and eq3 give us solutions for a1 and a2 : >>>>> >>>>> S13=solve([eq3, eq1], [a2, a1], dict=True) ; S13 >>>>> >>>>> [{a1: -sqrt(a2)}, {a1: sqrt(a2)}, {a2: 0}] >>>>> >>>>> which we prefer to rewrite as : >>>>> >>>>> S13 = [{a2:a1**2}, {a2:0}]; S13 >>>>> >>>>> [{a2: a1**2}, {a2: 0}] >>>>> >>>>> Substituting these values in eq4 gives us : >>>>> >>>>> E4 = [eq4.subs(s) for s in S13] ; E4 >>>>> >>>>> [-2*a1**4*b1 + a1**4*b2, 0] >>>>> >>>>> The second solution tells us that S13[1] is also,a solution to [eq1, >>>>> eq3, eq4]. >>>>> >>>>> S134=S13[1:] ; S134 >>>>> >>>>> [{a2: 0}] >>>>> >>>>> Substituting S13[0] ineq4and solving for the variables of the >>>>> resulting polynomial augments the setS134of solutions of[eq1, eq3`, >>>>> eq4]’ : >>>>> >>>>> V4=E4[0].free_symbols >>>>> S4=[solve(E4[0], v, dict=True) for v in V4] ; S4for S in S4: >>>>> for s in S: >>>>> S0={u:S13[0][u].subs(s) for u in S13[0].keys()} >>>>> S134 += [S0.copy()|s] >>>>> S134 >>>>> >>>>> [{a2: 0}, {a2: a1**2, b1: b2/2}, {a1: 0, a2: 0}, {a2: a1**2, b2: 2*b1}] >>>>> >>>>> Again, we prefer to rewrite it in a simpler (and shorter) fashion : >>>>> >>>>> S134=[{a2:0}, {a2:a1**2, b2:2*b1}] ; S134 >>>>> >>>>> [{a2: 0}, {a2: a1**2, b2: 2*b1}] >>>>> >>>>> Substituting in eq3 gives : >>>>> >>>>> [eq2.subs(s) for s in S134] >>>>> >>>>> [-a1*b2, -a1**4*b1 + 4*a1**3*b1 - 2*a1*b1] >>>>> >>>>> Solving these equations for their free symbols a,d merging with the >>>>> previous partial solutions gives us the solutions of the full system : >>>>> >>>>> S1234=[]for S in S134: >>>>> # print("S=",S) >>>>> E=eq2.subs(S) >>>>> # print("E=",E) >>>>> S1=[solve(E, v, dict=True) for v in E.free_symbols] >>>>> # print("S1=",S1) >>>>> for s in flatten(S1): >>>>> # print("s=",s) >>>>> S0={u:S[u].subs(s) if "subs" in dir(S[u]) else S[u] for u in >>>>> S.keys()} >>>>> S1234+=[S0.copy()|s] >>>>> S1234 >>>>> >>>>> [{a1: 0, a2: 0}, {a2: 0, b2: 0}, {a2: a1**2, b1: 0, b2: 0}, {a1: 0, a2: >>>>> 0, b2: 2*b1}, {a1: 4/3 + (-1/2 - sqrt(3)*I/2)*(37/27 + >>>>> sqrt(303)*I/9)**(1/3) + 16/(9*(-1/2 - sqrt(3)*I/2)*(37/27 + >>>>> sqrt(303)*I/9)**(1/3)), a2: (4/3 + (-1/2 - sqrt(3)*I/2)*(37/27 + >>>>> sqrt(303)*I/9)**(1/3) + 16/(9*(-1/2 - sqrt(3)*I/2)*(37/27 + >>>>> sqrt(303)*I/9)**(1/3)))**2, b2: 2*b1}, {a1: 4/3 + 16/(9*(-1/2 + >>>>> sqrt(3)*I/2)*(37/27 + sqrt(303)*I/9)**(1/3)) + (-1/2 + >>>>> sqrt(3)*I/2)*(37/27 + sqrt(303)*I/9)**(1/3), a2: (4/3 + 16/(9*(-1/2 + >>>>> sqrt(3)*I/2)*(37/27 + sqrt(303)*I/9)**(1/3)) + (-1/2 + >>>>> sqrt(3)*I/2)*(37/27 + sqrt(303)*I/9)**(1/3))**2, b2: 2*b1}, {a1: 4/3 + >>>>> 16/(9*(37/27 + sqrt(303)*I/9)**(1/3)) + (37/27 + sqrt(303)*I/9)**(1/3), >>>>> a2: (4/3 + 16/(9*(37/27 + sqrt(303)*I/9)**(1/3)) + (37/27 + >>>>> sqrt(303)*I/9)**(1/3))**2, b2: 2*b1}] >>>>> >>>>> Again, some solutions, substituted in eq2, give first-degree >>>>> monomials in b1 whose oefficient cannot be shownt to be null byis_zero >>>>> : >>>>> >>>>> [[e.subs(s).is_zero for e in Sys] for s in S1234] >>>>> >>>>> [[True, True, True, True], >>>>> [True, True, True, True], >>>>> [True, True, True, True], >>>>> [True, True, True, True], >>>>> [True, None, True, True], >>>>> [True, None, True, True], >>>>> [True, None, True, True]] >>>>> >>>>> But, this time, the numerical check points to a probably null result : >>>>> >>>>> [Sys[1].subs(S1234[u]).coeff(b1).n() for u in range(3,6)] >>>>> >>>>> [0, 0.e-125 + 0.e-127*I, 0.e-125 - 0.e-127*I] >>>>> >>>>> >>>>> >>>> -- You received this message because you are subscribed to the Google Groups "sympy" group. 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