Thank you very much, Oscar. That is excellent feedback. You are right. The solver didn't find a solution after about 18h of running. For the solver to work, I had broken up my vector equation into vector component equations, and while doing that, I chose slightly different variables to solve for, too. The equations were still monstrous, though, and it is little wonder that the solver found no solution. I will give your suggestion to use quaternions a try once I understand what they are. :-)
I don't care if the solution I get is too complicated to look at, as long as I can implement it and feed it with my known values. I need it to spit out the one result I want - that vector CM0 (or CM1, CM2, or CM3 - in real life, those will be roughly similar), but even the value c02, c12, c22, or c32 would be equally good to know. I still refuse to throw a numerical search algorithm at this problem, even though that most likely would work, because I hope to better understand the problem (and solution) by solving it symbolically. Oscar schrieb am Sonntag, 31. Oktober 2021 um 12:45:36 UTC+1: > On Sat, 30 Oct 2021 at 15:41, Andreas Schuldei <and...@schuldei.org> > wrote: > > > > > > Thank you, that seems to have been the issue. I split up the vectors > into their coefficients and entered those vector components as separate > equations. Now the python process is humming along at 1.8Gbyte memory and > low to medium CPU utilization. How long is realistic to wait for a positive > result? > > Symbolic algorithms often have extremely bad algorithmic complexity so > it's hard to say. > > I took a look at the equations you are solving from the code you > showed and I don't quite understand what the unknowns are e.g. the > first 4 equations are: > > In [24]: equations[0] > Out[24]: (c₀₁ - c₁₁)⋅(c₀₁ - m₀₁) + (c₀₂ - c₁₂)⋅(c₀₂ - m₀₂) + (c₀₃ - > c₁₃)⋅(c₀₃ - m₀₃) = 0 > > In [25]: equations[1] > Out[25]: (c₀₁ - c₁₁)⋅(c₁₁ - m₁₁) + (c₀₂ - c₁₂)⋅(c₁₂ - m₁₂) + (c₀₃ - > c₁₃)⋅(c₁₃ - m₁₃) = 0 > > In [26]: equations[2] > Out[26]: (c₀₁ - c₂₁)⋅(c₂₁ - m₂₁) + (c₀₂ - c₂₂)⋅(c₂₂ - m₂₂) + (c₀₃ - > c₂₃)⋅(c₂₃ - m₂₃) = 0 > > In [46]: equations[3] > Out[46]: (c₀₁ - c₃₁)⋅(c₃₁ - m₃₁) + (c₀₂ - c₃₂)⋅(c₃₂ - m₃₂) + (c₀₃ - > c₃₃)⋅(c₃₃ - m₃₃) = 0 > > The first unknown listed in the call to solve is CM0: > > In [27]: CM0 > Out[27]: (c₀₁ - m₀₁) i_Sys_sensors + (c₀₂ - m₀₂) j_Sys_sensors + (c₀₃ > - m₀₃) k_Sys_sensors > > I don't know what it means to solve for this vector quantity. Which of > the symbols here are the unknowns (the c0i or the m0i)? > > If you wanted the m0i then these equations are linear but if you > wanted the c0i then these first equations are a quadratic polynomial > system. Either way the complexity of the solutions is going to grow > quicker than you might expect. > > Your next equation 4 equations are monstrous. Here's the first of them > (I guess this is actually 3 scalar equations once you take > components): > > In [48]: print(equations[4]) > Eq((-B01 + B_x + 2*a*i*mu_0*((c01 - > m01)*(sin(theta_i)*sin(theta_j)*cos(theta_k) - > sin(theta_k)*cos(theta_i)) + (c02 - > m02)*(sin(theta_i)*sin(theta_j)*sin(theta_k) + > cos(theta_i)*cos(theta_k)) + (c03 - > m03)*sin(theta_i)*cos(theta_j))*((c01 - m01)*cos(theta_j)*cos(theta_k) > + (c02 - m02)*sin(theta_k)*cos(theta_j) - (c03 - > m03)*sin(theta_j))*cos(theta_j)*cos(theta_k)/(pi*(((c01 - > m01)*(sin(theta_i)*sin(theta_j)*cos(theta_k) - > sin(theta_k)*cos(theta_i)) + (c02 - > m02)*(sin(theta_i)*sin(theta_j)*sin(theta_k) + > cos(theta_i)*cos(theta_k)) + (c03 - m03)*sin(theta_i)*cos(theta_j))**2 > + ((c01 - m01)*cos(theta_j)*cos(theta_k) + (c02 - > m02)*sin(theta_k)*cos(theta_j) - (c03 - m03)*sin(theta_j))**2)**2) + > (sin(theta_i)*sin(theta_j)*cos(theta_k) - > sin(theta_k)*cos(theta_i))*(a*i*mu_0/(pi*(((c01 - > m01)*(sin(theta_i)*sin(theta_j)*cos(theta_k) - > sin(theta_k)*cos(theta_i)) + (c02 - > m02)*(sin(theta_i)*sin(theta_j)*sin(theta_k) + > cos(theta_i)*cos(theta_k)) + (c03 - m03)*sin(theta_i)*cos(theta_j))**2 > + ((c01 - m01)*cos(theta_j)*cos(theta_k) + (c02 - > m02)*sin(theta_k)*cos(theta_j) - (c03 - m03)*sin(theta_j))**2)) - > 2*a*i*mu_0*((c01 - m01)*cos(theta_j)*cos(theta_k) + (c02 - > m02)*sin(theta_k)*cos(theta_j) - (c03 - > m03)*sin(theta_j))**2/(pi*(((c01 - > m01)*(sin(theta_i)*sin(theta_j)*cos(theta_k) - > sin(theta_k)*cos(theta_i)) + (c02 - > m02)*(sin(theta_i)*sin(theta_j)*sin(theta_k) + > cos(theta_i)*cos(theta_k)) + (c03 - m03)*sin(theta_i)*cos(theta_j))**2 > + ((c01 - m01)*cos(theta_j)*cos(theta_k) + (c02 - > m02)*sin(theta_k)*cos(theta_j) - (c03 - > m03)*sin(theta_j))**2)**2)))*Sys_sensors.i + (-B02 + B_y + > 2*a*i*mu_0*((c01 - m01)*(sin(theta_i)*sin(theta_j)*cos(theta_k) - > sin(theta_k)*cos(theta_i)) + (c02 - > m02)*(sin(theta_i)*sin(theta_j)*sin(theta_k) + > cos(theta_i)*cos(theta_k)) + (c03 - > m03)*sin(theta_i)*cos(theta_j))*((c01 - m01)*cos(theta_j)*cos(theta_k) > + (c02 - m02)*sin(theta_k)*cos(theta_j) - (c03 - > m03)*sin(theta_j))*sin(theta_k)*cos(theta_j)/(pi*(((c01 - > m01)*(sin(theta_i)*sin(theta_j)*cos(theta_k) - > sin(theta_k)*cos(theta_i)) + (c02 - > m02)*(sin(theta_i)*sin(theta_j)*sin(theta_k) + > cos(theta_i)*cos(theta_k)) + (c03 - m03)*sin(theta_i)*cos(theta_j))**2 > + ((c01 - m01)*cos(theta_j)*cos(theta_k) + (c02 - > m02)*sin(theta_k)*cos(theta_j) - (c03 - m03)*sin(theta_j))**2)**2) + > (sin(theta_i)*sin(theta_j)*sin(theta_k) + > cos(theta_i)*cos(theta_k))*(a*i*mu_0/(pi*(((c01 - > m01)*(sin(theta_i)*sin(theta_j)*cos(theta_k) - > sin(theta_k)*cos(theta_i)) + (c02 - > m02)*(sin(theta_i)*sin(theta_j)*sin(theta_k) + > cos(theta_i)*cos(theta_k)) + (c03 - m03)*sin(theta_i)*cos(theta_j))**2 > + ((c01 - m01)*cos(theta_j)*cos(theta_k) + (c02 - > m02)*sin(theta_k)*cos(theta_j) - (c03 - m03)*sin(theta_j))**2)) - > 2*a*i*mu_0*((c01 - m01)*cos(theta_j)*cos(theta_k) + (c02 - > m02)*sin(theta_k)*cos(theta_j) - (c03 - > m03)*sin(theta_j))**2/(pi*(((c01 - > m01)*(sin(theta_i)*sin(theta_j)*cos(theta_k) - > sin(theta_k)*cos(theta_i)) + (c02 - > m02)*(sin(theta_i)*sin(theta_j)*sin(theta_k) + > cos(theta_i)*cos(theta_k)) + (c03 - m03)*sin(theta_i)*cos(theta_j))**2 > + ((c01 - m01)*cos(theta_j)*cos(theta_k) + (c02 - > m02)*sin(theta_k)*cos(theta_j) - (c03 - > m03)*sin(theta_j))**2)**2)))*Sys_sensors.j + (-B03 + B_z - > 2*a*i*mu_0*((c01 - m01)*(sin(theta_i)*sin(theta_j)*cos(theta_k) - > sin(theta_k)*cos(theta_i)) + (c02 - > m02)*(sin(theta_i)*sin(theta_j)*sin(theta_k) + > cos(theta_i)*cos(theta_k)) + (c03 - > m03)*sin(theta_i)*cos(theta_j))*((c01 - m01)*cos(theta_j)*cos(theta_k) > + (c02 - m02)*sin(theta_k)*cos(theta_j) - (c03 - > m03)*sin(theta_j))*sin(theta_j)/(pi*(((c01 - > m01)*(sin(theta_i)*sin(theta_j)*cos(theta_k) - > sin(theta_k)*cos(theta_i)) + (c02 - > m02)*(sin(theta_i)*sin(theta_j)*sin(theta_k) + > cos(theta_i)*cos(theta_k)) + (c03 - m03)*sin(theta_i)*cos(theta_j))**2 > + ((c01 - m01)*cos(theta_j)*cos(theta_k) + (c02 - > m02)*sin(theta_k)*cos(theta_j) - (c03 - m03)*sin(theta_j))**2)**2) + > (a*i*mu_0/(pi*(((c01 - m01)*(sin(theta_i)*sin(theta_j)*cos(theta_k) - > sin(theta_k)*cos(theta_i)) + (c02 - > m02)*(sin(theta_i)*sin(theta_j)*sin(theta_k) + > cos(theta_i)*cos(theta_k)) + (c03 - m03)*sin(theta_i)*cos(theta_j))**2 > + ((c01 - m01)*cos(theta_j)*cos(theta_k) + (c02 - > m02)*sin(theta_k)*cos(theta_j) - (c03 - m03)*sin(theta_j))**2)) - > 2*a*i*mu_0*((c01 - m01)*cos(theta_j)*cos(theta_k) + (c02 - > m02)*sin(theta_k)*cos(theta_j) - (c03 - > m03)*sin(theta_j))**2/(pi*(((c01 - > m01)*(sin(theta_i)*sin(theta_j)*cos(theta_k) - > sin(theta_k)*cos(theta_i)) + (c02 - > m02)*(sin(theta_i)*sin(theta_j)*sin(theta_k) + > cos(theta_i)*cos(theta_k)) + (c03 - m03)*sin(theta_i)*cos(theta_j))**2 > + ((c01 - m01)*cos(theta_j)*cos(theta_k) + (c02 - > m02)*sin(theta_k)*cos(theta_j) - (c03 - > m03)*sin(theta_j))**2)**2))*sin(theta_i)*cos(theta_j))*Sys_sensors.k, > 0) > > If you do get an answer from solve then it will be extremely > complicated. I doubt though that solve will be able to give an answer > because it tries to solve polynomial systems in radicals and for a > fully symbolic system of equations like this you will hit up against > the Abel-Ruffini limit. > > Probably you need to try a different basic approach to your problem > from the outset like maybe it's better to use quaternions rather than > rotation angles or something or maybe you shouldn't try to fully solve > the system algebraically. > > The question is what did you want the solutions for? Presumably you > wanted to use them for some other purpose in which case maybe there's > another way to achieve that other purpose. If you just want to look at > the solutions and see what they are then I think that if you do manage > to get expressions for the solution of this system (from any CAS) they > are going to be too complicated to just "look at". > > Oscar > -- You received this message because you are subscribed to the Google Groups "sympy" group. To unsubscribe from this group and stop receiving emails from it, send an email to sympy+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/sympy/e36b540d-6b4c-4471-9f9c-c09bdddf3ecfn%40googlegroups.com.
