How to determine eigenvalues of symbolic matrix?

2 ビュー (過去 30 日間)
Stepan Podhorsky
Stepan Podhorsky 2020 年 8 月 11 日
コメント済み: Walter Roberson 2020 年 8 月 14 日
Hello,
I have state space model of the three phase induction motor. The system matrix of the state space model has following form:
where ,,,, are the stator resistance, rotor resistance, total leakage inductance, magnetizing inductance and rotor mechanical speed.
My goal is to determine the eigenvalues in symbolic form of this matrix. I have used following Matlab code for that purpose:
syms RS RR LL LM Omg
A = [-(RS+RR)/LL, 0, RR/(LM*LL), Omg/LL; 0, -(RS+RR)/LL, -Omg/LL, RR/(LM*LL); RR, 0, -RR/LM, -Omg; 0, RR, Omg, -RR/LM]
A =
[ -(RR + RS)/LL, 0, RR/(LL*LM), Omg/LL]
[ 0, -(RR + RS)/LL, -Omg/LL, RR/(LL*LM)]
[ RR, 0, -RR/LM, -Omg]
[ 0, RR, Omg, -RR/LM]
eig(A)
Unfortunately the results are in unusable form. Can anybody give me an advice how to get the results in usable form? Thanks in advance for any ideas.

回答 (2 件)

John D'Errico
John D'Errico 2020 年 8 月 11 日
編集済み: John D'Errico 2020 年 8 月 11 日
What form would you expect? Is there a reason why you expect the eigenvalues of a 4x4 symbolic matrix to be some nice pretty thing? Those eigenvalues will be the roots of a rather messy 4th drgree symbolic polynomial.
At first, I was a bit surprised that anything came out at all, but the roots of a quartic polynomial are computable. It is pretty ugly. But computable. And since the polynomial in question is itself a bit of a mess, what do you really expect?
syms lambda
det(A - lambda*eye(4))
ans =
(LL^2*LM^2*Omg^2*lambda^2 + LL^2*LM^2*lambda^4 + 2*LL^2*LM*RR*lambda^3 + LL^2*RR^2*lambda^2 + 2*LL*LM^2*Omg^2*RS*lambda + 2*LL*LM^2*RR*lambda^3 + 2*LL*LM^2*RS*lambda^3 + 2*LL*LM*RR^2*lambda^2 + 4*LL*LM*RR*RS*lambda^2 + 2*LL*RR^2*RS*lambda + LM^2*Omg^2*RS^2 + LM^2*RR^2*lambda^2 + 2*LM^2*RR*RS*lambda^2 + LM^2*RS^2*lambda^2 + 2*LM*RR^2*RS*lambda + 2*LM*RR*RS^2*lambda + RR^2*RS^2)/(LL^2*LM^2)
Gosh. All MATLAB had to do is compute the roots of the symbolic 4th degree polynomial. That is only the polynomial. Now you need to compute the roots thereof. Piece of cake.
If you want something nice and easy to work with, substitute numbers in for all those constants. Then you will get 4 numbers out.
Mathematics is not always pretty.
  1 件のコメント
Stepan Podhorsky
Stepan Podhorsky 2020 年 8 月 14 日
編集済み: Stepan Podhorsky 2020 年 8 月 14 日
Thank you for your reaction. The reason why I expected the "nice" solution was based on fact that the original matrix contains symmetries and can be reduced to 2x2 matrix in case matrix notation of complex numbers is used.

サインインしてコメントする。


Walter Roberson
Walter Roberson 2020 年 8 月 11 日
The results you get are exact solutions.
You have a 4 x 4 matrix. The eigenvalues are going to be the roots of a polynomial of degree 4. Degree 4 is exactly solvable. But the solution is going to be long.
You can simplify() with 'steps', 25 to get a more compact form. For example
(2^(2/3)*3^(1/2)*(2^(1/3)*((2*LL^6*RR^6 + 2*LM^6*RR^6 + 2*LM^6*RS^6 + 12*LL*LM^5*RR^6 + 12*LL^5*LM*RR^6 + 12*LM^6*RR*RS^5 + 12*LM^6*RR^5*RS + 2*LL^6*LM^6*Omg^6 + 30*LL^2*LM^4*RR^6 + 40*LL^3*LM^3*RR^6 + 30*LL^4*LM^2*RR^6 + 30*LM^6*RR^2*RS^4 + 40*LM^6*RR^3*RS^3 + 30*LM^6*RR^4*RS^2 + 6*LL^2*LM^6*Omg^2*RR^4 + 24*LL^3*LM^5*Omg^2*RR^4 + 36*LL^4*LM^4*Omg^2*RR^4 + 24*LL^5*LM^3*Omg^2*RR^4 + 6*LL^6*LM^2*Omg^2*RR^4 + 6*LL^4*LM^6*Omg^4*RR^2 + 12*LL^5*LM^5*Omg^4*RR^2 + 6*LL^6*LM^4*Omg^4*RR^2 + 6*LL^2*LM^6*Omg^2*RS^4 + 6*LL^4*LM^6*Omg^4*RS^2 + 30*LL^2*LM^4*RR^2*RS^4 + 24*LL^2*LM^4*RR^3*RS^3 - 12*LL^2*LM^4*RR^4*RS^2 - 40*LL^3*LM^3*RR^3*RS^3 + 24*LL^3*LM^3*RR^4*RS^2 + 30*LL^4*LM^2*RR^4*RS^2 + 12*3^(1/2)*LL^6*LM^6*(-(Omg^4*RR^2*RS^2*(LL^4*LM^4*Omg^4 + 2*LL^4*LM^2*Omg^2*RR^2 + LL^4*RR^4 + 4*LL^3*LM^3*Omg^2*RR^2 - 4*LL^3*LM^3*Omg^2*RR*RS + 4*LL^3*LM*RR^4 - 4*LL^3*LM*RR^3*RS + 2*LL^2*LM^4*Omg^2*RR^2 - 12*LL^2*LM^4*Omg^2*RR*RS + 2*LL^2*LM^4*Omg^2*RS^2 + 6*LL^2*LM^2*RR^4 - 4*LL^2*LM^2*RR^3*RS + 6*LL^2*LM^2*RR^2*RS^2 + 4*LL*LM^3*RR^4 + 4*LL*LM^3*RR^3*RS - 4*LL*LM^3*RR^2*RS^2 - 4*LL*LM^3*RR*RS^3 + LM^4*RR^4 + 4*LM^4*RR^3*RS + 6*LM^4*RR^2*RS^2 + 4*LM^4*RR*RS^3 + LM^4*RS^4))/(LL^8*LM^4))^(1/2) - 12*LL*LM^5*RR*RS^5 + 36*LL*LM^5*RR^5*RS - 12*LL^5*LM*RR^5*RS - 36*LL*LM^5*RR^2*RS^4 - 24*LL*LM^5*RR^3*RS^3 + 24*LL*LM^5*RR^4*RS^2 + 24*LL^2*LM^4*RR^5*RS - 24*LL^3*LM^3*RR^5*RS - 36*LL^4*LM^2*RR^5*RS - 24*LL^4*LM^6*Omg^4*RR*RS - 12*LL^5*LM^5*Omg^4*RR*RS - 12*LL^2*LM^6*Omg^2*RR*RS^3 - 12*LL^2*LM^6*Omg^2*RR^3*RS - 24*LL^3*LM^5*Omg^2*RR*RS^3 - 48*LL^3*LM^5*Omg^2*RR^3*RS - 60*LL^4*LM^4*Omg^2*RR^3*RS - 24*LL^5*LM^3*Omg^2*RR^3*RS - 36*LL^2*LM^6*Omg^2*RR^2*RS^2 + 48*LL^3*LM^5*Omg^2*RR^2*RS^2 + 36*LL^4*LM^4*Omg^2*RR^2*RS^2)/(LL^6*LM^6))^(2/3) + (- 2*LL^2*LM^2*Omg^2 + LL^2*RR^2 + 2*LL*LM*RR^2 - 2*LL*LM*RR*RS + LM^2*RR^2 + 2*LM^2*RR*RS + LM^2*RS^2)^2/(2*LL^4*LM^4) - (9*(LL*RR + LM*RR + LM*RS)^4)/(2*LL^4*LM^4) + (24*RS^2*(LM^2*Omg^2 + RR^2))/(LL^2*LM^2) + (6*(LL*RR + LM*RR + LM*RS)^2*(LL^2*LM^2*Omg^2 + LL^2*RR^2 + 2*LL*LM*RR^2 + 4*LL*LM*RR*RS + LM^2*RR^2 + 2*LM^2*RR*RS + LM^2*RS^2))/(LL^4*LM^4) + (2^(2/3)*((2*LL^6*RR^6 + 2*LM^6*RR^6 + 2*LM^6*RS^6 + 12*LL*LM^5*RR^6 + 12*LL^5*LM*RR^6 + 12*LM^6*RR*RS^5 + 12*LM^6*RR^5*RS + 2*LL^6*LM^6*Omg^6 + 30*LL^2*LM^4*RR^6 + 40*LL^3*LM^3*RR^6 + 30*LL^4*LM^2*RR^6 + 30*LM^6*RR^2*RS^4 + 40*LM^6*RR^3*RS^3 + 30*LM^6*RR^4*RS^2 + 6*LL^2*LM^6*Omg^2*RR^4 + 24*LL^3*LM^5*Omg^2*RR^4 + 36*LL^4*LM^4*Omg^2*RR^4 + 24*LL^5*LM^3*Omg^2*RR^4 + 6*LL^6*LM^2*Omg^2*RR^4 + 6*LL^4*LM^6*Omg^4*RR^2 + 12*LL^5*LM^5*Omg^4*RR^2 + 6*LL^6*LM^4*Omg^4*RR^2 + 6*LL^2*LM^6*Omg^2*RS^4 + 6*LL^4*LM^6*Omg^4*RS^2 + 30*LL^2*LM^4*RR^2*RS^4 + 24*LL^2*LM^4*RR^3*RS^3 - 12*LL^2*LM^4*RR^4*RS^2 - 40*LL^3*LM^3*RR^3*RS^3 + 24*LL^3*LM^3*RR^4*RS^2 + 30*LL^4*LM^2*RR^4*RS^2 + 12*3^(1/2)*LL^6*LM^6*(-(Omg^4*RR^2*RS^2*(LL^4*LM^4*Omg^4 + 2*LL^4*LM^2*Omg^2*RR^2 + LL^4*RR^4 + 4*LL^3*LM^3*Omg^2*RR^2 - 4*LL^3*LM^3*Omg^2*RR*RS + 4*LL^3*LM*RR^4 - 4*LL^3*LM*RR^3*RS + 2*LL^2*LM^4*Omg^2*RR^2 - 12*LL^2*LM^4*Omg^2*RR*RS + 2*LL^2*LM^4*Omg^2*RS^2 + 6*LL^2*LM^2*RR^4 - 4*LL^2*LM^2*RR^3*RS + 6*LL^2*LM^2*RR^2*RS^2 + 4*LL*LM^3*RR^4 + 4*LL*LM^3*RR^3*RS - 4*LL*LM^3*RR^2*RS^2 - 4*LL*LM^3*RR*RS^3 + LM^4*RR^4 + 4*LM^4*RR^3*RS + 6*LM^4*RR^2*RS^2 + 4*LM^4*RR*RS^3 + LM^4*RS^4))/(LL^8*LM^4))^(1/2) - 12*LL*LM^5*RR*RS^5 + 36*LL*LM^5*RR^5*RS - 12*LL^5*LM*RR^5*RS - 36*LL*LM^5*RR^2*RS^4 - 24*LL*LM^5*RR^3*RS^3 + 24*LL*LM^5*RR^4*RS^2 + 24*LL^2*LM^4*RR^5*RS - 24*LL^3*LM^3*RR^5*RS - 36*LL^4*LM^2*RR^5*RS - 24*LL^4*LM^6*Omg^4*RR*RS - 12*LL^5*LM^5*Omg^4*RR*RS - 12*LL^2*LM^6*Omg^2*RR*RS^3 - 12*LL^2*LM^6*Omg^2*RR^3*RS - 24*LL^3*LM^5*Omg^2*RR*RS^3 - 48*LL^3*LM^5*Omg^2*RR^3*RS - 60*LL^4*LM^4*Omg^2*RR^3*RS - 24*LL^5*LM^3*Omg^2*RR^3*RS - 36*LL^2*LM^6*Omg^2*RR^2*RS^2 + 48*LL^3*LM^5*Omg^2*RR^2*RS^2 + 36*LL^4*LM^4*Omg^2*RR^2*RS^2)/(LL^6*LM^6))^(1/3)*(- 2*LL^2*LM^2*Omg^2 + LL^2*RR^2 + 2*LL*LM*RR^2 - 2*LL*LM*RR*RS + LM^2*RR^2 + 2*LM^2*RR*RS + LM^2*RS^2))/(LL^2*LM^2) - (24*RS*(LL*RR + LM*RR + LM*RS)*(LL*LM^2*Omg^2 + LM*RR^2 + RS*LM*RR + LL*RR^2))/(LL^3*LM^3))^(1/2))/(12*((2*LL^6*RR^6 + 2*LM^6*RR^6 + 2*LM^6*RS^6 + 12*LL*LM^5*RR^6 + 12*LL^5*LM*RR^6 + 12*LM^6*RR*RS^5 + 12*LM^6*RR^5*RS + 2*LL^6*LM^6*Omg^6 + 30*LL^2*LM^4*RR^6 + 40*LL^3*LM^3*RR^6 + 30*LL^4*LM^2*RR^6 + 30*LM^6*RR^2*RS^4 + 40*LM^6*RR^3*RS^3 + 30*LM^6*RR^4*RS^2 + 6*LL^2*LM^6*Omg^2*RR^4 + 24*LL^3*LM^5*Omg^2*RR^4 + 36*LL^4*LM^4*Omg^2*RR^4 + 24*LL^5*LM^3*Omg^2*RR^4 + 6*LL^6*LM^2*Omg^2*RR^4 + 6*LL^4*LM^6*Omg^4*RR^2 + 12*LL^5*LM^5*Omg^4*RR^2 + 6*LL^6*LM^4*Omg^4*RR^2 + 6*LL^2*LM^6*Omg^2*RS^4 + 6*LL^4*LM^6*Omg^4*RS^2 + 30*LL^2*LM^4*RR^2*RS^4 + 24*LL^2*LM^4*RR^3*RS^3 - 12*LL^2*LM^4*RR^4*RS^2 - 40*LL^3*LM^3*RR^3*RS^3 + 24*LL^3*LM^3*RR^4*RS^2 + 30*LL^4*LM^2*RR^4*RS^2 + 12*3^(1/2)*LL^6*LM^6*(-(Omg^4*RR^2*RS^2*(LL^4*LM^4*Omg^4 + 2*LL^4*LM^2*Omg^2*RR^2 + LL^4*RR^4 + 4*LL^3*LM^3*Omg^2*RR^2 - 4*LL^3*LM^3*Omg^2*RR*RS + 4*LL^3*LM*RR^4 - 4*LL^3*LM*RR^3*RS + 2*LL^2*LM^4*Omg^2*RR^2 - 12*LL^2*LM^4*Omg^2*RR*RS + 2*LL^2*LM^4*Omg^2*RS^2 + 6*LL^2*LM^2*RR^4 - 4*LL^2*LM^2*RR^3*RS + 6*LL^2*LM^2*RR^2*RS^2 + 4*LL*LM^3*RR^4 + 4*LL*LM^3*RR^3*RS - 4*LL*LM^3*RR^2*RS^2 - 4*LL*LM^3*RR*RS^3 + LM^4*RR^4 + 4*LM^4*RR^3*RS + 6*LM^4*RR^2*RS^2 + 4*LM^4*RR*RS^3 + LM^4*RS^4))/(LL^8*LM^4))^(1/2) - 12*LL*LM^5*RR*RS^5 + 36*LL*LM^5*RR^5*RS - 12*LL^5*LM*RR^5*RS - 36*LL*LM^5*RR^2*RS^4 - 24*LL*LM^5*RR^3*RS^3 + 24*LL*LM^5*RR^4*RS^2 + 24*LL^2*LM^4*RR^5*RS - 24*LL^3*LM^3*RR^5*RS - 36*LL^4*LM^2*RR^5*RS - 24*LL^4*LM^6*Omg^4*RR*RS - 12*LL^5*LM^5*Omg^4*RR*RS - 12*LL^2*LM^6*Omg^2*RR*RS^3 - 12*LL^2*LM^6*Omg^2*RR^3*RS - 24*LL^3*LM^5*Omg^2*RR*RS^3 - 48*LL^3*LM^5*Omg^2*RR^3*RS - 60*LL^4*LM^4*Omg^2*RR^3*RS - 24*LL^5*LM^3*Omg^2*RR^3*RS - 36*LL^2*LM^6*Omg^2*RR^2*RS^2 + 48*LL^3*LM^5*Omg^2*RR^2*RS^2 + 36*LL^4*LM^4*Omg^2*RR^2*RS^2)/(LL^6*LM^6))^(1/6)) - RS/(2*LL) - RR/(2*LM) - RR/(2*LL) - (2^(5/12)*3^(1/2)*(24*2^(1/2)*((3*(LL*RR + LM*RR + LM*RS)^4)/(16*LL^4*LM^4) - (RS^2*(LM^2*Omg^2 + RR^2))/(LL^2*LM^2) - ((LL*RR + LM*RR + LM*RS)^2*(LL^2*LM^2*Omg^2 + LL^2*RR^2 + 2*LL*LM*RR^2 + 4*LL*LM*RR*RS + LM^2*RR^2 + 2*LM^2*RR*RS + LM^2*RS^2))/(4*LL^4*LM^4) + (RS*(LL*RR + LM*RR + LM*RS)*(LL*LM^2*Omg^2 + LM*RR^2 + RS*LM*RR + LL*RR^2))/(LL^3*LM^3))*(2^(1/3)*((2*LL^6*RR^6 + 2*LM^6*RR^6 + 2*LM^6*RS^6 + 12*LL*LM^5*RR^6 + 12*LL^5*LM*RR^6 + 12*LM^6*RR*RS^5 + 12*LM^6*RR^5*RS + 2*LL^6*LM^6*Omg^6 + 30*LL^2*LM^4*RR^6 + 40*LL^3*LM^3*RR^6 + 30*LL^4*LM^2*RR^6 + 30*LM^6*RR^2*RS^4 + 40*LM^6*RR^3*RS^3 + 30*LM^6*RR^4*RS^2 + 6*LL^2*LM^6*Omg^2*RR^4 + 24*LL^3*LM^5*Omg^2*RR^4 + 36*LL^4*LM^4*Omg^2*RR^4 + 24*LL^5*LM^3*Omg^2*RR^4 + 6*LL^6*LM^2*Omg^2*RR^4 + 6*LL^4*LM^6*Omg^4*RR^2 + 12*LL^5*LM^5*Omg^4*RR^2 + 6*LL^6*LM^4*Omg^4*RR^2 + 6*LL^2*LM^6*Omg^2*RS^4 + 6*LL^4*LM^6*Omg^4*RS^2 + 30*LL^2*LM^4*RR^2*RS^4 + 24*LL^2*LM^4*RR^3*RS^3 - 12*LL^2*LM^4*RR^4*RS^2 - 40*LL^3*LM^3*RR^3*RS^3 + 24*LL^3*LM^3*RR^4*RS^2 + 30*LL^4*LM^2*RR^4*RS^2 + 12*3^(1/2)*LL^6*LM^6*(-(Omg^4*RR^2*RS^2*(LL^4*LM^4*Omg^4 + 2*LL^4*LM^2*Omg^2*RR^2 + LL^4*RR^4 + 4*LL^3*LM^3*Omg^2*RR^2 - 4*LL^3*LM^3*Omg^2*RR*RS + 4*LL^3*LM*RR^4 - 4*LL^3*LM*RR^3*RS + 2*LL^2*LM^4*Omg^2*RR^2 - 12*LL^2*LM^4*Omg^2*RR*RS + 2*LL^2*LM^4*Omg^2*RS^2 + 6*LL^2*LM^2*RR^4 - 4*LL^2*LM^2*RR^3*RS + 6*LL^2*LM^2*RR^2*RS^2 + 4*LL*LM^3*RR^4 + 4*LL*LM^3*RR^3*RS - 4*LL*LM^3*RR^2*RS^2 - 4*LL*LM^3*RR*RS^3 + LM^4*RR^4 + 4*LM^4*RR^3*RS + 6*LM^4*RR^2*RS^2 + 4*LM^4*RR*RS^3 + LM^4*RS^4))/(LL^8*LM^4))^(1/2) - 12*LL*LM^5*RR*RS^5 + 36*LL*LM^5*RR^5*RS - 12*LL^5*LM*RR^5*RS - 36*LL*LM^5*RR^2*RS^4 - 24*LL*LM^5*RR^3*RS^3 + 24*LL*LM^5*RR^4*RS^2 + 24*LL^2*LM^4*RR^5*RS - 24*LL^3*LM^3*RR^5*RS - 36*LL^4*LM^2*RR^5*RS - 24*LL^4*LM^6*Omg^4*RR*RS - 12*LL^5*LM^5*Omg^4*RR*RS - 12*LL^2*LM^6*Omg^2*RR*RS^3 - 12*LL^2*LM^6*Omg^2*RR^3*RS - 24*LL^3*LM^5*Omg^2*RR*RS^3 - 48*LL^3*LM^5*Omg^2*RR^3*RS - 60*LL^4*LM^4*Omg^2*RR^3*RS - 24*LL^5*LM^3*Omg^2*RR^3*RS - 36*LL^2*LM^6*Omg^2*RR^2*RS^2 + 48*LL^3*LM^5*Omg^2*RR^2*RS^2 + 36*LL^4*LM^4*Omg^2*RR^2*RS^2)/(LL^6*LM^6))^(2/3) + (- 2*LL^2*LM^2*Omg^2 + LL^2*RR^2 + 2*LL*LM*RR^2 - 2*LL*LM*RR*RS + LM^2*RR^2 + 2*LM^2*RR*RS + LM^2*RS^2)^2/(2*LL^4*LM^4) - (9*(LL*RR + LM*RR + LM*RS)^4)/(2*LL^4*LM^4) + (24*RS^2*(LM^2*Omg^2 + RR^2))/(LL^2*LM^2) + (6*(LL*RR + LM*RR + LM*RS)^2*(LL^2*LM^2*Omg^2 + LL^2*RR^2 + 2*LL*LM*RR^2 + 4*LL*LM*RR*RS + LM^2*RR^2 + 2*LM^2*RR*RS + LM^2*RS^2))/(LL^4*LM^4) + (2^(2/3)*((2*LL^6*RR^6 + 2*LM^6*RR^6 + 2*LM^6*RS^6 + 12*LL*LM^5*RR^6 + 12*LL^5*LM*RR^6 + 12*LM^6*RR*RS^5 + 12*LM^6*RR^5*RS + 2*LL^6*LM^6*Omg^6 + 30*LL^2*LM^4*RR^6 + 40*LL^3*LM^3*RR^6 + 30*LL^4*LM^2*RR^6 + 30*LM^6*RR^2*RS^4 + 40*LM^6*RR^3*RS^3 + 30*LM^6*RR^4*RS^2 + 6*LL^2*LM^6*Omg^2*RR^4 + 24*LL^3*LM^5*Omg^2*RR^4 + 36*LL^4*LM^4*Omg^2*RR^4 + 24*LL^5*LM^3*Omg^2*RR^4 + 6*LL^6*LM^2*Omg^2*RR^4 + 6*LL^4*LM^6*Omg^4*RR^2 + 12*LL^5*LM^5*Omg^4*RR^2 + 6*LL^6*LM^4*Omg^4*RR^2 + 6*LL^2*LM^6*Omg^2*RS^4 + 6*LL^4*LM^6*Omg^4*RS^2 + 30*LL^2*LM^4*RR^2*RS^4 + 24*LL^2*LM^4*RR^3*RS^3 - 12*LL^2*LM^4*RR^4*RS^2 - 40*LL^3*LM^3*RR^3*RS^3 + 24*LL^3*LM^3*RR^4*RS^2 + 30*LL^4*LM^2*RR^4*RS^2 + 12*3^(1/2)*LL^6*LM^6*(-(Omg^4*RR^2*RS^2*(LL^4*LM^4*Omg^4 + 2*LL^4*LM^2*Omg^2*RR^2 + LL^4*RR^4 + 4*LL^3*LM^3*Omg^2*RR^2 - 4*LL^3*LM^3*Omg^2*RR*RS + 4*LL^3*LM*RR^4 - 4*LL^3*LM*RR^3*RS + 2*LL^2*LM^4*Omg^2*RR^2 - 12*LL^2*LM^4*Omg^2*RR*RS + 2*LL^2*LM^4*Omg^2*RS^2 + 6*LL^2*LM^2*RR^4 - 4*LL^2*LM^2*RR^3*RS + 6*LL^2*LM^2*RR^2*RS^2 + 4*LL*LM^3*RR^4 + 4*LL*LM^3*RR^3*RS - 4*LL*LM^3*RR^2*RS^2 - 4*LL*LM^3*RR*RS^3 + LM^4*RR^4 + 4*LM^4*RR^3*RS + 6*LM^4*RR^2*RS^2 + 4*LM^4*RR*RS^3 + LM^4*RS^4))/(LL^8*LM^4))^(1/2) - 12*LL*LM^5*RR*RS^5 + 36*LL*LM^5*RR^5*RS - 12*LL^5*LM*RR^5*RS - 36*LL*LM^5*RR^2*RS^4 - 24*LL*LM^5*RR^3*RS^3 + 24*LL*LM^5*RR^4*RS^2 + 24*LL^2*LM^4*RR^5*RS - 24*LL^3*LM^3*RR^5*RS - 36*LL^4*LM^2*RR^5*RS - 24*LL^4*LM^6*Omg^4*RR*RS - 12*LL^5*LM^5*Omg^4*RR*RS - 12*LL^2*LM^6*Omg^2*RR*RS^3 - 12*LL^2*LM^6*Omg^2*RR^3*RS - 24*LL^3*LM^5*Omg^2*RR*RS^3 - 48*LL^3*LM^5*Omg^2*RR^3*RS - 60*LL^4*LM^4*Omg^2*RR^3*RS - 24*LL^5*LM^3*Omg^2*RR^3*RS - 36*LL^2*LM^6*Omg^2*RR^2*RS^2 + 48*LL^3*LM^5*Omg^2*RR^2*RS^2 + 36*LL^4*LM^4*Omg^2*RR^2*RS^2)/(LL^6*LM^6))^(1/3)*(- 2*LL^2*LM^2*Omg^2 + LL^2*RR^2 + 2*LL*LM*RR^2 - 2*LL*LM*RR*RS + LM^2*RR^2 + 2*LM^2*RR*RS + LM^2*RS^2))/(LL^2*LM^2) - (24*RS*(LL*RR + LM*RR + LM*RS)*(LL*LM^2*Omg^2 + LM*RR^2 + RS*LM*RR + LL*RR^2))/(LL^3*LM^3))^(1/2) - 2^(5/6)*((2*LL^6*RR^6 + 2*LM^6*RR^6 + 2*LM^6*RS^6 + 12*LL*LM^5*RR^6 + 12*LL^5*LM*RR^6 + 12*LM^6*RR*RS^5 + 12*LM^6*RR^5*RS + 2*LL^6*LM^6*Omg^6 + 30*LL^2*LM^4*RR^6 + 40*LL^3*LM^3*RR^6 + 30*LL^4*LM^2*RR^6 + 30*LM^6*RR^2*RS^4 + 40*LM^6*RR^3*RS^3 + 30*LM^6*RR^4*RS^2 + 6*LL^2*LM^6*Omg^2*RR^4 + 24*LL^3*LM^5*Omg^2*RR^4 + 36*LL^4*LM^4*Omg^2*RR^4 + 24*LL^5*LM^3*Omg^2*RR^4 + 6*LL^6*LM^2*Omg^2*RR^4 + 6*LL^4*LM^6*Omg^4*RR^2 + 12*LL^5*LM^5*Omg^4*RR^2 + 6*LL^6*LM^4*Omg^4*RR^2 + 6*LL^2*LM^6*Omg^2*RS^4 + 6*LL^4*LM^6*Omg^4*RS^2 + 30*LL^2*LM^4*RR^2*RS^4 + 24*LL^2*LM^4*RR^3*RS^3 - 12*LL^2*LM^4*RR^4*RS^2 - 40*LL^3*LM^3*RR^3*RS^3 + 24*LL^3*LM^3*RR^4*RS^2 + 30*LL^4*LM^2*RR^4*RS^2 + 12*3^(1/2)*LL^6*LM^6*(-(Omg^4*RR^2*RS^2*(LL^4*LM^4*Omg^4 + 2*LL^4*LM^2*Omg^2*RR^2 + LL^4*RR^4 + 4*LL^3*LM^3*Omg^2*RR^2 - 4*LL^3*LM^3*Omg^2*RR*RS + 4*LL^3*LM*RR^4 - 4*LL^3*LM*RR^3*RS + 2*LL^2*LM^4*Omg^2*RR^2 - 12*LL^2*LM^4*Omg^2*RR*RS + 2*LL^2*LM^4*Omg^2*RS^2 + 6*LL^2*LM^2*RR^4 - 4*LL^2*LM^2*RR^3*RS + 6*LL^2*LM^2*RR^2*RS^2 + 4*LL*LM^3*RR^4 + 4*LL*LM^3*RR^3*RS - 4*LL*LM^3*RR^2*RS^2 - 4*LL*LM^3*RR*RS^3 + LM^4*RR^4 + 4*LM^4*RR^3*RS + 6*LM^4*RR^2*RS^2 + 4*LM^4*RR*RS^3 + LM^4*RS^4))/(LL^8*LM^4))^(1/2) - 12*LL*LM^5*RR*RS^5 + 36*LL*LM^5*RR^5*RS - 12*LL^5*LM*RR^5*RS - 36*LL*LM^5*RR^2*RS^4 - 24*LL*LM^5*RR^3*RS^3 + 24*LL*LM^5*RR^4*RS^2 + 24*LL^2*LM^4*RR^5*RS - 24*LL^3*LM^3*RR^5*RS - 36*LL^4*LM^2*RR^5*RS - 24*LL^4*LM^6*Omg^4*RR*RS - 12*LL^5*LM^5*Omg^4*RR*RS - 12*LL^2*LM^6*Omg^2*RR*RS^3 - 12*LL^2*LM^6*Omg^2*RR^3*RS - 24*LL^3*LM^5*Omg^2*RR*RS^3 - 48*LL^3*LM^5*Omg^2*RR^3*RS - 60*LL^4*LM^4*Omg^2*RR^3*RS - 24*LL^5*LM^3*Omg^2*RR^3*RS - 36*LL^2*LM^6*Omg^2*RR^2*RS^2 + 48*LL^3*LM^5*Omg^2*RR^2*RS^2 + 36*LL^4*LM^4*Omg^2*RR^2*RS^2)/(LL^6*LM^6))^(2/3)*(2^(1/3)*((2*LL^6*RR^6 + 2*LM^6*RR^6 + 2*LM^6*RS^6 + 12*LL*LM^5*RR^6 + 12*LL^5*LM*RR^6 + 12*LM^6*RR*RS^5 + 12*LM^6*RR^5*RS + 2*LL^6*LM^6*Omg^6 + 30*LL^2*LM^4*RR^6 + 40*LL^3*LM^3*RR^6 + 30*LL^4*LM^2*RR^6 + 30*LM^6*RR^2*RS^4 + 40*LM^6*RR^3*RS^3 + 30*LM^6*RR^4*RS^2 + 6*LL^2*LM^6*Omg^2*RR^4 + 24*LL^3*LM^5*Omg^2*RR^4 + 36*LL^4*LM^4*Omg^2*RR^4 + 24*LL^5*LM^3*Omg^2*RR^4 + 6*LL^6*LM^2*Omg^2*RR^4 + 6*LL^4*LM^6*Omg^4*RR^2 + 12*LL^5*LM^5*Omg^4*RR^2 + 6*LL^6*LM^4*Omg^4*RR^2 + 6*LL^2*LM^6*Omg^2*RS^4 + 6*LL^4*LM^6*Omg^4*RS^2 + 30*LL^2*LM^4*RR^2*RS^4 + 24*LL^2*LM^4*RR^3*RS^3 - 12*LL^2*LM^4*RR^4*RS^2 - 40*LL^3*LM^3*RR^3*RS^3 + 24*LL^3*LM^3*RR^4*RS^2 + 30*LL^4*LM^2*RR^4*RS^2 + 12*3^(1/2)*LL^6*LM^6*(-(Omg^4*RR^2*RS^2*(LL^4*LM^4*Omg^4 + 2*LL^4*LM^2*Omg^2*RR^2 + LL^4*RR^4 + 4*LL^3*LM^3*Omg^2*RR^2 - 4*LL^3*LM^3*Omg^2*RR*RS + 4*LL^3*LM*RR^4 - 4*LL^3*LM*RR^3*RS + 2*LL^2*LM^4*Omg^2*RR^2 - 12*LL^2*LM^4*Omg^2*RR*RS + 2*LL^2*LM^4*Omg^2*RS^2 + 6*LL^2*LM^2*RR^4 - 4*LL^2*LM^2*RR^3*RS + 6*LL^2*LM^2*RR^2*RS^2 + 4*LL*LM^3*RR^4 + 4*LL*LM^3*RR^3*RS - 4*LL*LM^3*RR^2*RS^2 - 4*LL*LM^3*RR*RS^3 + LM^4*RR^4 + 4*LM^4*RR^3*RS + 6*LM^4*RR^2*RS^2 + 4*LM^4*RR*RS^3 + LM^4*RS^4))/(LL^8*LM^4))^(1/2) - 12*LL*LM^5*RR*RS^5 + 36*LL*LM^5*RR^5*RS - 12*LL^5*LM*RR^5*RS - 36*LL*LM^5*RR^2*RS^4 - 24*LL*LM^5*RR^3*RS^3 + 24*LL*LM^5*RR^4*RS^2 + 24*LL^2*LM^4*RR^5*RS - 24*LL^3*LM^3*RR^5*RS - 36*LL^4*LM^2*RR^5*RS - 24*LL^4*LM^6*Omg^4*RR*RS - 12*LL^5*LM^5*Omg^4*RR*RS - 12*LL^2*LM^6*Omg^2*RR*RS^3 - 12*LL^2*LM^6*Omg^2*RR^3*RS - 24*LL^3*LM^5*Omg^2*RR*RS^3 - 48*LL^3*LM^5*Omg^2*RR^3*RS - 60*LL^4*LM^4*Omg^2*RR^3*RS - 24*LL^5*LM^3*Omg^2*RR^3*RS - 36*LL^2*LM^6*Omg^2*RR^2*RS^2 + 48*LL^3*LM^5*Omg^2*RR^2*RS^2 + 36*LL^4*LM^4*Omg^2*RR^2*RS^2)/(LL^6*LM^6))^(2/3) + (- 2*LL^2*LM^2*Omg^2 + LL^2*RR^2 + 2*LL*LM*RR^2 - 2*LL*LM*RR*RS + LM^2*RR^2 + 2*LM^2*RR*RS + LM^2*RS^2)^2/(2*LL^4*LM^4) - (9*(LL*RR + LM*RR + LM*RS)^4)/(2*LL^4*LM^4) + (24*RS^2*(LM^2*Omg^2 + RR^2))/(LL^2*LM^2) + (6*(LL*RR + LM*RR + LM*RS)^2*(LL^2*LM^2*Omg^2 + LL^2*RR^2 + 2*LL*LM*RR^2 + 4*LL*LM*RR*RS + LM^2*RR^2 + 2*LM^2*RR*RS + LM^2*RS^2))/(LL^4*LM^4) + (2^(2/3)*((2*LL^6*RR^6 + 2*LM^6*RR^6 + 2*LM^6*RS^6 + 12*LL*LM^5*RR^6 + 12*LL^5*LM*RR^6 + 12*LM^6*RR*RS^5 + 12*LM^6*RR^5*RS + 2*LL^6*LM^6*Omg^6 + 30*LL^2*LM^4*RR^6 + 40*LL^3*LM^3*RR^6 + 30*LL^4*LM^2*RR^6 + 30*LM^6*RR^2*RS^4 + 40*LM^6*RR^3*RS^3 + 30*LM^6*RR^4*RS^2 + 6*LL^2*LM^6*Omg^2*RR^4 + 24*LL^3*LM^5*Omg^2*RR^4 + 36*LL^4*LM^4*Omg^2*RR^4 + 24*LL^5*LM^3*Omg^2*RR^4 + 6*LL^6*LM^2*Omg^2*RR^4 + 6*LL^4*LM^6*Omg^4*RR^2 + 12*LL^5*LM^5*Omg^4*RR^2 + 6*LL^6*LM^4*Omg^4*RR^2 + 6*LL^2*LM^6*Omg^2*RS^4 + 6*LL^4*LM^6*Omg^4*RS^2 + 30*LL^2*LM^4*RR^2*RS^4 + 24*LL^2*LM^4*RR^3*RS^3 - 12*LL^2*LM^4*RR^4*RS^2 - 40*LL^3*LM^3*RR^3*RS^3 + 24*LL^3*LM^3*RR^4*RS^2 + 30*LL^4*LM^2*RR^4*RS^2 + 12*3^(1/2)*LL^6*LM^6*(-(Omg^4*RR^2*RS^2*(LL^4*LM^4*Omg^4 + 2*LL^4*LM^2*Omg^2*RR^2 + LL^4*RR^4 + 4*LL^3*LM^3*Omg^2*RR^2 - 4*LL^3*LM^3*Omg^2*RR*RS + 4*LL^3*LM*RR^4 - 4*LL^3*LM*RR^3*RS + 2*LL^2*LM^4*Omg^2*RR^2 - 12*LL^2*LM^4*Omg^2*RR*RS + 2*LL^2*LM^4*Omg^2*RS^2 + 6*LL^2*LM^2*RR^4 - 4*LL^2*LM^2*RR^3*RS + 6*LL^2*LM^2*RR^2*RS^2 + 4*LL*LM^3*RR^4 + 4*LL*LM^3*RR^3*RS - 4*LL*LM^3*RR^2*RS^2 - 4*LL*LM^3*RR*RS^3 + LM^4*RR^4 + 4*LM^4*RR^3*RS + 6*LM^4*RR^2*RS^2 + 4*LM^4*RR*RS^3 + LM^4*RS^4))/(LL^8*LM^4))^(1/2) - 12*LL*LM^5*RR*RS^5 + 36*LL*LM^5*RR^5*RS - 12*LL^5*LM*RR^5*RS - 36*LL*LM^5*RR^2*RS^4 - 24*LL*LM^5*RR^3*RS^3 + 24*LL*LM^5*RR^4*RS^2 + 24*LL^2*LM^4*RR^5*RS - 24*LL^3*LM^3*RR^5*RS - 36*LL^4*LM^2*RR^5*RS - 24*LL^4*LM^6*Omg^4*RR*RS - 12*LL^5*LM^5*Omg^4*RR*RS - 12*LL^2*LM^6*Omg^2*RR*RS^3 - 12*LL^2*LM^6*Omg^2*RR^3*RS - 24*LL^3*LM^5*Omg^2*RR*RS^3 - 48*LL^3*LM^5*Omg^2*RR^3*RS - 60*LL^4*LM^4*Omg^2*RR^3*RS - 24*LL^5*LM^3*Omg^2*RR^3*RS - 36*LL^2*LM^6*Omg^2*RR^2*RS^2 + 48*LL^3*LM^5*Omg^2*RR^2*RS^2 + 36*LL^4*LM^4*Omg^2*RR^2*RS^2)/(LL^6*LM^6))^(1/3)*(- 2*LL^2*LM^2*Omg^2 + LL^2*RR^2 + 2*LL*LM*RR^2 - 2*LL*LM*RR*RS + LM^2*RR^2 + 2*LM^2*RR*RS + LM^2*RS^2))/(LL^2*LM^2) - (24*RS*(LL*RR + LM*RR + LM*RS)*(LL*LM^2*Omg^2 + LM*RR^2 + RS*LM*RR + LL*RR^2))/(LL^3*LM^3))^(1/2) - (2^(1/2)*(2^(1/3)*((2*LL^6*RR^6 + 2*LM^6*RR^6 + 2*LM^6*RS^6 + 12*LL*LM^5*RR^6 + 12*LL^5*LM*RR^6 + 12*LM^6*RR*RS^5 + 12*LM^6*RR^5*RS + 2*LL^6*LM^6*Omg^6 + 30*LL^2*LM^4*RR^6 + 40*LL^3*LM^3*RR^6 + 30*LL^4*LM^2*RR^6 + 30*LM^6*RR^2*RS^4 + 40*LM^6*RR^3*RS^3 + 30*LM^6*RR^4*RS^2 + 6*LL^2*LM^6*Omg^2*RR^4 + 24*LL^3*LM^5*Omg^2*RR^4 + 36*LL^4*LM^4*Omg^2*RR^4 + 24*LL^5*LM^3*Omg^2*RR^4 + 6*LL^6*LM^2*Omg^2*RR^4 + 6*LL^4*LM^6*Omg^4*RR^2 + 12*LL^5*LM^5*Omg^4*RR^2 + 6*LL^6*LM^4*Omg^4*RR^2 + 6*LL^2*LM^6*Omg^2*RS^4 + 6*LL^4*LM^6*Omg^4*RS^2 + 30*LL^2*LM^4*RR^2*RS^4 + 24*LL^2*LM^4*RR^3*RS^3 - 12*LL^2*LM^4*RR^4*RS^2 - 40*LL^3*LM^3*RR^3*RS^3 + 24*LL^3*LM^3*RR^4*RS^2 + 30*LL^4*LM^2*RR^4*RS^2 + 12*3^(1/2)*LL^6*LM^6*(-(Omg^4*RR^2*RS^2*(LL^4*LM^4*Omg^4 + 2*LL^4*LM^2*Omg^2*RR^2 + LL^4*RR^4 + 4*LL^3*LM^3*Omg^2*RR^2 - 4*LL^3*LM^3*Omg^2*RR*RS + 4*LL^3*LM*RR^4 - 4*LL^3*LM*RR^3*RS + 2*LL^2*LM^4*Omg^2*RR^2 - 12*LL^2*LM^4*Omg^2*RR*RS + 2*LL^2*LM^4*Omg^2*RS^2 + 6*LL^2*LM^2*RR^4 - 4*LL^2*LM^2*RR^3*RS + 6*LL^2*LM^2*RR^2*RS^2 + 4*LL*LM^3*RR^4 + 4*LL*LM^3*RR^3*RS - 4*LL*LM^3*RR^2*RS^2 - 4*LL*LM^3*RR*RS^3 + LM^4*RR^4 + 4*LM^4*RR^3*RS + 6*LM^4*RR^2*RS^2 + 4*LM^4*RR*RS^3 + LM^4*RS^4))/(LL^8*LM^4))^(1/2) - 12*LL*LM^5*RR*RS^5 + 36*LL*LM^5*RR^5*RS - 12*LL^5*LM*RR^5*RS - 36*LL*LM^5*RR^2*RS^4 - 24*LL*LM^5*RR^3*RS^3 + 24*LL*LM^5*RR^4*RS^2 + 24*LL^2*LM^4*RR^5*RS - 24*LL^3*LM^3*RR^5*RS - 36*LL^4*LM^2*RR^5*RS - 24*LL^4*LM^6*Omg^4*RR*RS - 12*LL^5*LM^5*Omg^4*RR*RS - 12*LL^2*LM^6*Omg^2*RR*RS^3 - 12*LL^2*LM^6*Omg^2*RR^3*RS - 24*LL^3*LM^5*Omg^2*RR*RS^3 - 48*LL^3*LM^5*Omg^2*RR^3*RS - 60*LL^4*LM^4*Omg^2*RR^3*RS - 24*LL^5*LM^3*Omg^2*RR^3*RS - 36*LL^2*LM^6*Omg^2*RR^2*RS^2 + 48*LL^3*LM^5*Omg^2*RR^2*RS^2 + 36*LL^4*LM^4*Omg^2*RR^2*RS^2)/(LL^6*LM^6))^(2/3) + (- 2*LL^2*LM^2*Omg^2 + LL^2*RR^2 + 2*LL*LM*RR^2 - 2*LL*LM*RR*RS + LM^2*RR^2 + 2*LM^2*RR*RS + LM^2*RS^2)^2/(2*LL^4*LM^4) - (9*(LL*RR + LM*RR + LM*RS)^4)/(2*LL^4*LM^4) + (24*RS^2*(LM^2*Omg^2 + RR^2))/(LL^2*LM^2) + (6*(LL*RR + LM*RR + LM*RS)^2*(LL^2*LM^2*Omg^2 + LL^2*RR^2 + 2*LL*LM*RR^2 + 4*LL*LM*RR*RS + LM^2*RR^2 + 2*LM^2*RR*RS + LM^2*RS^2))/(LL^4*LM^4) + (2^(2/3)*((2*LL^6*RR^6 + 2*LM^6*RR^6 + 2*LM^6*RS^6 + 12*LL*LM^5*RR^6 + 12*LL^5*LM*RR^6 + 12*LM^6*RR*RS^5 + 12*LM^6*RR^5*RS + 2*LL^6*LM^6*Omg^6 + 30*LL^2*LM^4*RR^6 + 40*LL^3*LM^3*RR^6 + 30*LL^4*LM^2*RR^6 + 30*LM^6*RR^2*RS^4 + 40*LM^6*RR^3*RS^3 + 30*LM^6*RR^4*RS^2 + 6*LL^2*LM^6*Omg^2*RR^4 + 24*LL^3*LM^5*Omg^2*RR^4 + 36*LL^4*LM^4*Omg^2*RR^4 + 24*LL^5*LM^3*Omg^2*RR^4 + 6*LL^6*LM^2*Omg^2*RR^4 + 6*LL^4*LM^6*Omg^4*RR^2 + 12*LL^5*LM^5*Omg^4*RR^2 + 6*LL^6*LM^4*Omg^4*RR^2 + 6*LL^2*LM^6*Omg^2*RS^4 + 6*LL^4*LM^6*Omg^4*RS^2 + 30*LL^2*LM^4*RR^2*RS^4 + 24*LL^2*LM^4*RR^3*RS^3 - 12*LL^2*LM^4*RR^4*RS^2 - 40*LL^3*LM^3*RR^3*RS^3 + 24*LL^3*LM^3*RR^4*RS^2 + 30*LL^4*LM^2*RR^4*RS^2 + 12*3^(1/2)*LL^6*LM^6*(-(Omg^4*RR^2*RS^2*(LL^4*LM^4*Omg^4 + 2*LL^4*LM^2*Omg^2*RR^2 + LL^4*RR^4 + 4*LL^3*LM^3*Omg^2*RR^2 - 4*LL^3*LM^3*Omg^2*RR*RS + 4*LL^3*LM*RR^4 - 4*LL^3*LM*RR^3*RS + 2*LL^2*LM^4*Omg^2*RR^2 - 12*LL^2*LM^4*Omg^2*RR*RS + 2*LL^2*LM^4*Omg^2*RS^2 + 6*LL^2*LM^2*RR^4 - 4*LL^2*LM^2*RR^3*RS + 6*LL^2*LM^2*RR^2*RS^2 + 4*LL*LM^3*RR^4 + 4*LL*LM^3*RR^3*RS - 4*LL*LM^3*RR^2*RS^2 - 4*LL*LM^3*RR*RS^3 + LM^4*RR^4 + 4*LM^4*RR^3*RS + 6*LM^4*RR^2*RS^2 + 4*LM^4*RR*RS^3 + LM^4*RS^4))/(LL^8*LM^4))^(1/2) - 12*LL*LM^5*RR*RS^5 + 36*LL*LM^5*RR^5*RS - 12*LL^5*LM*RR^5*RS - 36*LL*LM^5*RR^2*RS^4 - 24*LL*LM^5*RR^3*RS^3 + 24*LL*LM^5*RR^4*RS^2 + 24*LL^2*LM^4*RR^5*RS - 24*LL^3*LM^3*RR^5*RS - 36*LL^4*LM^2*RR^5*RS - 24*LL^4*LM^6*Omg^4*RR*RS - 12*LL^5*LM^5*Omg^4*RR*RS - 12*LL^2*LM^6*Omg^2*RR*RS^3 - 12*LL^2*LM^6*Omg^2*RR^3*RS - 24*LL^3*LM^5*Omg^2*RR*RS^3 - 48*LL^3*LM^5*Omg^2*RR^3*RS - 60*LL^4*LM^4*Omg^2*RR^3*RS - 24*LL^5*LM^3*Omg^2*RR^3*RS - 36*LL^2*LM^6*Omg^2*RR^2*RS^2 + 48*LL^3*LM^5*Omg^2*RR^2*RS^2 + 36*LL^4*LM^4*Omg^2*RR^2*RS^2)/(LL^6*LM^6))^(1/3)*(- 2*LL^2*LM^2*Omg^2 + LL^2*RR^2 + 2*LL*LM*RR^2 - 2*LL*LM*RR*RS + LM^2*RR^2 + 2*LM^2*RR*RS + LM^2*RS^2))/(LL^2*LM^2) - (24*RS*(LL*RR + LM*RR + LM*RS)*(LL*LM^2*Omg^2 + LM*RR^2 + RS*LM*RR + LL*RR^2))/(LL^3*LM^3))^(1/2)*(- 2*LL^2*LM^2*Omg^2 + LL^2*RR^2 + 2*LL*LM*RR^2 - 2*LL*LM*RR*RS + LM^2*RR^2 + 2*LM^2*RR*RS + LM^2*RS^2)^2)/(2*LL^4*LM^4) + (4*2^(1/6)*((2*LL^6*RR^6 + 2*LM^6*RR^6 + 2*LM^6*RS^6 + 12*LL*LM^5*RR^6 + 12*LL^5*LM*RR^6 + 12*LM^6*RR*RS^5 + 12*LM^6*RR^5*RS + 2*LL^6*LM^6*Omg^6 + 30*LL^2*LM^4*RR^6 + 40*LL^3*LM^3*RR^6 + 30*LL^4*LM^2*RR^6 + 30*LM^6*RR^2*RS^4 + 40*LM^6*RR^3*RS^3 + 30*LM^6*RR^4*RS^2 + 6*LL^2*LM^6*Omg^2*RR^4 + 24*LL^3*LM^5*Omg^2*RR^4 + 36*LL^4*LM^4*Omg^2*RR^4 + 24*LL^5*LM^3*Omg^2*RR^4 + 6*LL^6*LM^2*Omg^2*RR^4 + 6*LL^4*LM^6*Omg^4*RR^2 + 12*LL^5*LM^5*Omg^4*RR^2 + 6*LL^6*LM^4*Omg^4*RR^2 + 6*LL^2*LM^6*Omg^2*RS^4 + 6*LL^4*LM^6*Omg^4*RS^2 + 30*LL^2*LM^4*RR^2*RS^4 + 24*LL^2*LM^4*RR^3*RS^3 - 12*LL^2*LM^4*RR^4*RS^2 - 40*LL^3*LM^3*RR^3*RS^3 + 24*LL^3*LM^3*RR^4*RS^2 + 30*LL^4*LM^2*RR^4*RS^2 + 12*3^(1/2)*LL^6*LM^6*(-(Omg^4*RR^2*RS^2*(LL^4*LM^4*Omg^4 + 2*LL^4*LM^2*Omg^2*RR^2 + LL^4*RR^4 + 4*LL^3*LM^3*Omg^2*RR^2 - 4*LL^3*LM^3*Omg^2*RR*RS + 4*LL^3*LM*RR^4 - 4*LL^3*LM*RR^3*RS + 2*LL^2*LM^4*Omg^2*RR^2 - 12*LL^2*LM^4*Omg^2*RR*RS + 2*LL^2*LM^4*Omg^2*RS^2 + 6*LL^2*LM^2*RR^4 - 4*LL^2*LM^2*RR^3*RS + 6*LL^2*LM^2*RR^2*RS^2 + 4*LL*LM^3*RR^4 + 4*LL*LM^3*RR^3*RS - 4*LL*LM^3*RR^2*RS^2 - 4*LL*LM^3*RR*RS^3 + LM^4*RR^4 + 4*LM^4*RR^3*RS + 6*LM^4*RR^2*RS^2 + 4*LM^4*RR*RS^3 + LM^4*RS^4))/(LL^8*LM^4))^(1/2) - 12*LL*LM^5*RR*RS^5 + 36*LL*LM^5*RR^5*RS - 12*LL^5*LM*RR^5*RS - 36*LL*LM^5*RR^2*RS^4 - 24*LL*LM^5*RR^3*RS^3 + 24*LL*LM^5*RR^4*RS^2 + 24*LL^2*LM^4*RR^5*RS - 24*LL^3*LM^3*RR^5*RS - 36*LL^4*LM^2*RR^5*RS - 24*LL^4*LM^6*Omg^4*RR*RS - 12*LL^5*LM^5*Omg^4*RR*RS - 12*LL^2*LM^6*Omg^2*RR*RS^3 - 12*LL^2*LM^6*Omg^2*RR^3*RS - 24*LL^3*LM^5*Omg^2*RR*RS^3 - 48*LL^3*LM^5*Omg^2*RR^3*RS - 60*LL^4*LM^4*Omg^2*RR^3*RS - 24*LL^5*LM^3*Omg^2*RR^3*RS - 36*LL^2*LM^6*Omg^2*RR^2*RS^2 + 48*LL^3*LM^5*Omg^2*RR^2*RS^2 + 36*LL^4*LM^4*Omg^2*RR^2*RS^2)/(LL^6*LM^6))^(1/3)*(2^(1/3)*((2*LL^6*RR^6 + 2*LM^6*RR^6 + 2*LM^6*RS^6 + 12*LL*LM^5*RR^6 + 12*LL^5*LM*RR^6 + 12*LM^6*RR*RS^5 + 12*LM^6*RR^5*RS + 2*LL^6*LM^6*Omg^6 + 30*LL^2*LM^4*RR^6 + 40*LL^3*LM^3*RR^6 + 30*LL^4*LM^2*RR^6 + 30*LM^6*RR^2*RS^4 + 40*LM^6*RR^3*RS^3 + 30*LM^6*RR^4*RS^2 + 6*LL^2*LM^6*Omg^2*RR^4 + 24*LL^3*LM^5*Omg^2*RR^4 + 36*LL^4*LM^4*Omg^2*RR^4 + 24*LL^5*LM^3*Omg^2*RR^4 + 6*LL^6*LM^2*Omg^2*RR^4 + 6*LL^4*LM^6*Omg^4*RR^2 + 12*LL^5*LM^5*Omg^4*RR^2 + 6*LL^6*LM^4*Omg^4*RR^2 + 6*LL^2*LM^6*Omg^2*RS^4 + 6*LL^4*LM^6*Omg^4*RS^2 + 30*LL^2*LM^4*RR^2*RS^4 + 24*LL^2*LM^4*RR^3*RS^3 - 12*LL^2*LM^4*RR^4*RS^2 - 40*LL^3*LM^3*RR^3*RS^3 + 24*LL^3*LM^3*RR^4*RS^2 + 30*LL^4*LM^2*RR^4*RS^2 + 12*3^(1/2)*LL^6*LM^6*(-(Omg^4*RR^2*RS^2*(LL^4*LM^4*Omg^4 + 2*LL^4*LM^2*Omg^2*RR^2 + LL^4*RR^4 + 4*LL^3*LM^3*Omg^2*RR^2 - 4*LL^3*LM^3*Omg^2*RR*RS + 4*LL^3*LM*RR^4 - 4*LL^3*LM*RR^3*RS + 2*LL^2*LM^4*Omg^2*RR^2 - 12*LL^2*LM^4*Omg^2*RR*RS + 2*LL^2*LM^4*Omg^2*RS^2 + 6*LL^2*LM^2*RR^4 - 4*LL^2*LM^2*RR^3*RS + 6*LL^2*LM^2*RR^2*RS^2 + 4*LL*LM^3*RR^4 + 4*LL*LM^3*RR^3*RS - 4*LL*LM^3*RR^2*RS^2 - 4*LL*LM^3*RR*RS^3 + LM^4*RR^4 + 4*LM^4*RR^3*RS + 6*LM^4*RR^2*RS^2 + 4*LM^4*RR*RS^3 + LM^4*RS^4))/(LL^8*LM^4))^(1/2) - 12*LL*LM^5*RR*RS^5 + 36*LL*LM^5*RR^5*RS - 12*LL^5*LM*RR^5*RS - 36*LL*LM^5*RR^2*RS^4 - 24*LL*LM^5*RR^3*RS^3 + 24*LL*LM^5*RR^4*RS^2 + 24*LL^2*LM^4*RR^5*RS - 24*LL^3*LM^3*RR^5*RS - 36*LL^4*LM^2*RR^5*RS - 24*LL^4*LM^6*Omg^4*RR*RS - 12*LL^5*LM^5*Omg^4*RR*RS - 12*LL^2*LM^6*Omg^2*RR*RS^3 - 12*LL^2*LM^6*Omg^2*RR^3*RS - 24*LL^3*LM^5*Omg^2*RR*RS^3 - 48*LL^3*LM^5*Omg^2*RR^3*RS - 60*LL^4*LM^4*Omg^2*RR^3*RS - 24*LL^5*LM^3*Omg^2*RR^3*RS - 36*LL^2*LM^6*Omg^2*RR^2*RS^2 + 48*LL^3*LM^5*Omg^2*RR^2*RS^2 + 36*LL^4*LM^4*Omg^2*RR^2*RS^2)/(LL^6*LM^6))^(2/3) + (- 2*LL^2*LM^2*Omg^2 + LL^2*RR^2 + 2*LL*LM*RR^2 - 2*LL*LM*RR*RS + LM^2*RR^2 + 2*LM^2*RR*RS + LM^2*RS^2)^2/(2*LL^4*LM^4) - (9*(LL*RR + LM*RR + LM*RS)^4)/(2*LL^4*LM^4) + (24*RS^2*(LM^2*Omg^2 + RR^2))/(LL^2*LM^2) + (6*(LL*RR + LM*RR + LM*RS)^2*(LL^2*LM^2*Omg^2 + LL^2*RR^2 + 2*LL*LM*RR^2 + 4*LL*LM*RR*RS + LM^2*RR^2 + 2*LM^2*RR*RS + LM^2*RS^2))/(LL^4*LM^4) + (2^(2/3)*((2*LL^6*RR^6 + 2*LM^6*RR^6 + 2*LM^6*RS^6 + 12*LL*LM^5*RR^6 + 12*LL^5*LM*RR^6 + 12*LM^6*RR*RS^5 + 12*LM^6*RR^5*RS + 2*LL^6*LM^6*Omg^6 + 30*LL^2*LM^4*RR^6 + 40*LL^3*LM^3*RR^6 + 30*LL^4*LM^2*RR^6 + 30*LM^6*RR^2*RS^4 + 40*LM^6*RR^3*RS^3 + 30*LM^6*RR^4*RS^2 + 6*LL^2*LM^6*Omg^2*RR^4 + 24*LL^3*LM^5*Omg^2*RR^4 + 36*LL^4*LM^4*Omg^2*RR^4 + 24*LL^5*LM^3*Omg^2*RR^4 + 6*LL^6*LM^2*Omg^2*RR^4 + 6*LL^4*LM^6*Omg^4*RR^2 + 12*LL^5*LM^5*Omg^4*RR^2 + 6*LL^6*LM^4*Omg^4*RR^2 + 6*LL^2*LM^6*Omg^2*RS^4 + 6*LL^4*LM^6*Omg^4*RS^2 + 30*LL^2*LM^4*RR^2*RS^4 + 24*LL^2*LM^4*RR^3*RS^3 - 12*LL^2*LM^4*RR^4*RS^2 - 40*LL^3*LM^3*RR^3*RS^3 + 24*LL^3*LM^3*RR^4*RS^2 + 30*LL^4*LM^2*RR^4*RS^2 + 12*3^(1/2)*LL^6*LM^6*(-(Omg^4*RR^2*RS^2*(LL^4*LM^4*Omg^4 + 2*LL^4*LM^2*Omg^2*RR^2 + LL^4*RR^4 + 4*LL^3*LM^3*Omg^2*RR^2 - 4*LL^3*LM^3*Omg^2*RR*RS + 4*LL^3*LM*RR^4 - 4*LL^3*LM*RR^3*RS + 2*LL^2*LM^4*Omg^2*RR^2 - 12*LL^2*LM^4*Omg^2*RR*RS + 2*LL^2*LM^4*Omg^2*RS^2 + 6*LL^2*LM^2*RR^4 - 4*LL^2*LM^2*RR^3*RS + 6*LL^2*LM^2*RR^2*RS^2 + 4*LL*LM^3*RR^4 + 4*LL*LM^3*RR^3*RS - 4*LL*LM^3*RR^2*RS^2 - 4*LL*LM^3*RR*RS^3 + LM^4*RR^4 + 4*LM^4*RR^3*RS + 6*LM^4*RR^2*RS^2 + 4*LM^4*RR*RS^3 + LM^4*RS^4))/(LL^8*LM^4))^(1/2) - 12*LL*LM^5*RR*RS^5 + 36*LL*LM^5*RR^5*RS - 12*LL^5*LM*RR^5*RS - 36*LL*LM^5*RR^2*RS^4 - 24*LL*LM^5*RR^3*RS^3 + 24*LL*LM^5*RR^4*RS^2 + 24*LL^2*LM^4*RR^5*RS - 24*LL^3*LM^3*RR^5*RS - 36*LL^4*LM^2*RR^5*RS - 24*LL^4*LM^6*Omg^4*RR*RS - 12*LL^5*LM^5*Omg^4*RR*RS - 12*LL^2*LM^6*Omg^2*RR*RS^3 - 12*LL^2*LM^6*Omg^2*RR^3*RS - 24*LL^3*LM^5*Omg^2*RR*RS^3 - 48*LL^3*LM^5*Omg^2*RR^3*RS - 60*LL^4*LM^4*Omg^2*RR^3*RS - 24*LL^5*LM^3*Omg^2*RR^3*RS - 36*LL^2*LM^6*Omg^2*RR^2*RS^2 + 48*LL^3*LM^5*Omg^2*RR^2*RS^2 + 36*LL^4*LM^4*Omg^2*RR^2*RS^2)/(LL^6*LM^6))^(1/3)*(- 2*LL^2*LM^2*Omg^2 + LL^2*RR^2 + 2*LL*LM*RR^2 - 2*LL*LM*RR*RS + LM^2*RR^2 + 2*LM^2*RR*RS + LM^2*RS^2))/(LL^2*LM^2) - (24*RS*(LL*RR + LM*RR + LM*RS)*(LL*LM^2*Omg^2 + LM*RR^2 + RS*LM*RR + LL*RR^2))/(LL^3*LM^3))^(1/2)*(- 2*LL^2*LM^2*Omg^2 + LL^2*RR^2 + 2*LL*LM*RR^2 - 2*LL*LM*RR*RS + LM^2*RR^2 + 2*LM^2*RR*RS + LM^2*RS^2))/(LL^2*LM^2) + (12*6^(1/2)*Omg^2*(LL*RR + LM*RR - LM*RS)*(12*3^(1/2)*(-(Omg^4*RR^2*RS^2*(LL^4*LM^4*Omg^4 + 2*LL^4*LM^2*Omg^2*RR^2 + LL^4*RR^4 + 4*LL^3*LM^3*Omg^2*RR^2 - 4*LL^3*LM^3*Omg^2*RR*RS + 4*LL^3*LM*RR^4 - 4*LL^3*LM*RR^3*RS + 2*LL^2*LM^4*Omg^2*RR^2 - 12*LL^2*LM^4*Omg^2*RR*RS + 2*LL^2*LM^4*Omg^2*RS^2 + 6*LL^2*LM^2*RR^4 - 4*LL^2*LM^2*RR^3*RS + 6*LL^2*LM^2*RR^2*RS^2 + 4*LL*LM^3*RR^4 + 4*LL*LM^3*RR^3*RS - 4*LL*LM^3*RR^2*RS^2 - 4*LL*LM^3*RR*RS^3 + LM^4*RR^4 + 4*LM^4*RR^3*RS + 6*LM^4*RR^2*RS^2 + 4*LM^4*RR*RS^3 + LM^4*RS^4))/(LL^8*LM^4))^(1/2) - (- 2*LL^2*LM^2*Omg^2 + LL^2*RR^2 + 2*LL*LM*RR^2 - 2*LL*LM*RR*RS + LM^2*RR^2 + 2*LM^2*RR*RS + LM^2*RS^2)^3/(4*LL^6*LM^6) + (27*Omg^4*(LL*RR + LM*RR - LM*RS)^2)/(LL^2*LM^2) - (36*((3*(LL*RR + LM*RR + LM*RS)^4)/(16*LL^4*LM^4) - (RS^2*(LM^2*Omg^2 + RR^2))/(LL^2*LM^2) - ((LL*RR + LM*RR + LM*RS)^2*(LL^2*LM^2*Omg^2 + LL^2*RR^2 + 2*LL*LM*RR^2 + 4*LL*LM*RR*RS + LM^2*RR^2 + 2*LM^2*RR*RS + LM^2*RS^2))/(4*LL^4*LM^4) + (RS*(LL*RR + LM*RR + LM*RS)*(LL*LM^2*Omg^2 + LM*RR^2 + RS*LM*RR + LL*RR^2))/(LL^3*LM^3))*(- 2*LL^2*LM^2*Omg^2 + LL^2*RR^2 + 2*LL*LM*RR^2 - 2*LL*LM*RR*RS + LM^2*RR^2 + 2*LM^2*RR*RS + LM^2*RS^2))/(LL^2*LM^2))^(1/2))/(LL*LM))^(1/2))/(12*((2*LL^6*RR^6 + 2*LM^6*RR^6 + 2*LM^6*RS^6 + 12*LL*LM^5*RR^6 + 12*LL^5*LM*RR^6 + 12*LM^6*RR*RS^5 + 12*LM^6*RR^5*RS + 2*LL^6*LM^6*Omg^6 + 30*LL^2*LM^4*RR^6 + 40*LL^3*LM^3*RR^6 + 30*LL^4*LM^2*RR^6 + 30*LM^6*RR^2*RS^4 + 40*LM^6*RR^3*RS^3 + 30*LM^6*RR^4*RS^2 + 6*LL^2*LM^6*Omg^2*RR^4 + 24*LL^3*LM^5*Omg^2*RR^4 + 36*LL^4*LM^4*Omg^2*RR^4 + 24*LL^5*LM^3*Omg^2*RR^4 + 6*LL^6*LM^2*Omg^2*RR^4 + 6*LL^4*LM^6*Omg^4*RR^2 + 12*LL^5*LM^5*Omg^4*RR^2 + 6*LL^6*LM^4*Omg^4*RR^2 + 6*LL^2*LM^6*Omg^2*RS^4 + 6*LL^4*LM^6*Omg^4*RS^2 + 30*LL^2*LM^4*RR^2*RS^4 + 24*LL^2*LM^4*RR^3*RS^3 - 12*LL^2*LM^4*RR^4*RS^2 - 40*LL^3*LM^3*RR^3*RS^3 + 24*LL^3*LM^3*RR^4*RS^2 + 30*LL^4*LM^2*RR^4*RS^2 + 12*3^(1/2)*LL^6*LM^6*(-(Omg^4*RR^2*RS^2*(LL^4*LM^4*Omg^4 + 2*LL^4*LM^2*Omg^2*RR^2 + LL^4*RR^4 + 4*LL^3*LM^3*Omg^2*RR^2 - 4*LL^3*LM^3*Omg^2*RR*RS + 4*LL^3*LM*RR^4 - 4*LL^3*LM*RR^3*RS + 2*LL^2*LM^4*Omg^2*RR^2 - 12*LL^2*LM^4*Omg^2*RR*RS + 2*LL^2*LM^4*Omg^2*RS^2 + 6*LL^2*LM^2*RR^4 - 4*LL^2*LM^2*RR^3*RS + 6*LL^2*LM^2*RR^2*RS^2 + 4*LL*LM^3*RR^4 + 4*LL*LM^3*RR^3*RS - 4*LL*LM^3*RR^2*RS^2 - 4*LL*LM^3*RR*RS^3 + LM^4*RR^4 + 4*LM^4*RR^3*RS + 6*LM^4*RR^2*RS^2 + 4*LM^4*RR*RS^3 + LM^4*RS^4))/(LL^8*LM^4))^(1/2) - 12*LL*LM^5*RR*RS^5 + 36*LL*LM^5*RR^5*RS - 12*LL^5*LM*RR^5*RS - 36*LL*LM^5*RR^2*RS^4 - 24*LL*LM^5*RR^3*RS^3 + 24*LL*LM^5*RR^4*RS^2 + 24*LL^2*LM^4*RR^5*RS - 24*LL^3*LM^3*RR^5*RS - 36*LL^4*LM^2*RR^5*RS - 24*LL^4*LM^6*Omg^4*RR*RS - 12*LL^5*LM^5*Omg^4*RR*RS - 12*LL^2*LM^6*Omg^2*RR*RS^3 - 12*LL^2*LM^6*Omg^2*RR^3*RS - 24*LL^3*LM^5*Omg^2*RR*RS^3 - 48*LL^3*LM^5*Omg^2*RR^3*RS - 60*LL^4*LM^4*Omg^2*RR^3*RS - 24*LL^5*LM^3*Omg^2*RR^3*RS - 36*LL^2*LM^6*Omg^2*RR^2*RS^2 + 48*LL^3*LM^5*Omg^2*RR^2*RS^2 + 36*LL^4*LM^4*Omg^2*RR^2*RS^2)/(LL^6*LM^6))^(1/6)*(2^(1/3)*((2*LL^6*RR^6 + 2*LM^6*RR^6 + 2*LM^6*RS^6 + 12*LL*LM^5*RR^6 + 12*LL^5*LM*RR^6 + 12*LM^6*RR*RS^5 + 12*LM^6*RR^5*RS + 2*LL^6*LM^6*Omg^6 + 30*LL^2*LM^4*RR^6 + 40*LL^3*LM^3*RR^6 + 30*LL^4*LM^2*RR^6 + 30*LM^6*RR^2*RS^4 + 40*LM^6*RR^3*RS^3 + 30*LM^6*RR^4*RS^2 + 6*LL^2*LM^6*Omg^2*RR^4 + 24*LL^3*LM^5*Omg^2*RR^4 + 36*LL^4*LM^4*Omg^2*RR^4 + 24*LL^5*LM^3*Omg^2*RR^4 + 6*LL^6*LM^2*Omg^2*RR^4 + 6*LL^4*LM^6*Omg^4*RR^2 + 12*LL^5*LM^5*Omg^4*RR^2 + 6*LL^6*LM^4*Omg^4*RR^2 + 6*LL^2*LM^6*Omg^2*RS^4 + 6*LL^4*LM^6*Omg^4*RS^2 + 30*LL^2*LM^4*RR^2*RS^4 + 24*LL^2*LM^4*RR^3*RS^3 - 12*LL^2*LM^4*RR^4*RS^2 - 40*LL^3*LM^3*RR^3*RS^3 + 24*LL^3*LM^3*RR^4*RS^2 + 30*LL^4*LM^2*RR^4*RS^2 + 12*3^(1/2)*LL^6*LM^6*(-(Omg^4*RR^2*RS^2*(LL^4*LM^4*Omg^4 + 2*LL^4*LM^2*Omg^2*RR^2 + LL^4*RR^4 + 4*LL^3*LM^3*Omg^2*RR^2 - 4*LL^3*LM^3*Omg^2*RR*RS + 4*LL^3*LM*RR^4 - 4*LL^3*LM*RR^3*RS + 2*LL^2*LM^4*Omg^2*RR^2 - 12*LL^2*LM^4*Omg^2*RR*RS + 2*LL^2*LM^4*Omg^2*RS^2 + 6*LL^2*LM^2*RR^4 - 4*LL^2*LM^2*RR^3*RS + 6*LL^2*LM^2*RR^2*RS^2 + 4*LL*LM^3*RR^4 + 4*LL*LM^3*RR^3*RS - 4*LL*LM^3*RR^2*RS^2 - 4*LL*LM^3*RR*RS^3 + LM^4*RR^4 + 4*LM^4*RR^3*RS + 6*LM^4*RR^2*RS^2 + 4*LM^4*RR*RS^3 + LM^4*RS^4))/(LL^8*LM^4))^(1/2) - 12*LL*LM^5*RR*RS^5 + 36*LL*LM^5*RR^5*RS - 12*LL^5*LM*RR^5*RS - 36*LL*LM^5*RR^2*RS^4 - 24*LL*LM^5*RR^3*RS^3 + 24*LL*LM^5*RR^4*RS^2 + 24*LL^2*LM^4*RR^5*RS - 24*LL^3*LM^3*RR^5*RS - 36*LL^4*LM^2*RR^5*RS - 24*LL^4*LM^6*Omg^4*RR*RS - 12*LL^5*LM^5*Omg^4*RR*RS - 12*LL^2*LM^6*Omg^2*RR*RS^3 - 12*LL^2*LM^6*Omg^2*RR^3*RS - 24*LL^3*LM^5*Omg^2*RR*RS^3 - 48*LL^3*LM^5*Omg^2*RR^3*RS - 60*LL^4*LM^4*Omg^2*RR^3*RS - 24*LL^5*LM^3*Omg^2*RR^3*RS - 36*LL^2*LM^6*Omg^2*RR^2*RS^2 + 48*LL^3*LM^5*Omg^2*RR^2*RS^2 + 36*LL^4*LM^4*Omg^2*RR^2*RS^2)/(LL^6*LM^6))^(2/3) + (- 2*LL^2*LM^2*Omg^2 + LL^2*RR^2 + 2*LL*LM*RR^2 - 2*LL*LM*RR*RS + LM^2*RR^2 + 2*LM^2*RR*RS + LM^2*RS^2)^2/(2*LL^4*LM^4) - (9*(LL*RR + LM*RR + LM*RS)^4)/(2*LL^4*LM^4) + (24*RS^2*(LM^2*Omg^2 + RR^2))/(LL^2*LM^2) + (6*(LL*RR + LM*RR + LM*RS)^2*(LL^2*LM^2*Omg^2 + LL^2*RR^2 + 2*LL*LM*RR^2 + 4*LL*LM*RR*RS + LM^2*RR^2 + 2*LM^2*RR*RS + LM^2*RS^2))/(LL^4*LM^4) + (2^(2/3)*((2*LL^6*RR^6 + 2*LM^6*RR^6 + 2*LM^6*RS^6 + 12*LL*LM^5*RR^6 + 12*LL^5*LM*RR^6 + 12*LM^6*RR*RS^5 + 12*LM^6*RR^5*RS + 2*LL^6*LM^6*Omg^6 + 30*LL^2*LM^4*RR^6 + 40*LL^3*LM^3*RR^6 + 30*LL^4*LM^2*RR^6 + 30*LM^6*RR^2*RS^4 + 40*LM^6*RR^3*RS^3 + 30*LM^6*RR^4*RS^2 + 6*LL^2*LM^6*Omg^2*RR^4 + 24*LL^3*LM^5*Omg^2*RR^4 + 36*LL^4*LM^4*Omg^2*RR^4 + 24*LL^5*LM^3*Omg^2*RR^4 + 6*LL^6*LM^2*Omg^2*RR^4 + 6*LL^4*LM^6*Omg^4*RR^2 + 12*LL^5*LM^5*Omg^4*RR^2 + 6*LL^6*LM^4*Omg^4*RR^2 + 6*LL^2*LM^6*Omg^2*RS^4 + 6*LL^4*LM^6*Omg^4*RS^2 + 30*LL^2*LM^4*RR^2*RS^4 + 24*LL^2*LM^4*RR^3*RS^3 - 12*LL^2*LM^4*RR^4*RS^2 - 40*LL^3*LM^3*RR^3*RS^3 + 24*LL^3*LM^3*RR^4*RS^2 + 30*LL^4*LM^2*RR^4*RS^2 + 12*3^(1/2)*LL^6*LM^6*(-(Omg^4*RR^2*RS^2*(LL^4*LM^4*Omg^4 + 2*LL^4*LM^2*Omg^2*RR^2 + LL^4*RR^4 + 4*LL^3*LM^3*Omg^2*RR^2 - 4*LL^3*LM^3*Omg^2*RR*RS + 4*LL^3*LM*RR^4 - 4*LL^3*LM*RR^3*RS + 2*LL^2*LM^4*Omg^2*RR^2 - 12*LL^2*LM^4*Omg^2*RR*RS + 2*LL^2*LM^4*Omg^2*RS^2 + 6*LL^2*LM^2*RR^4 - 4*LL^2*LM^2*RR^3*RS + 6*LL^2*LM^2*RR^2*RS^2 + 4*LL*LM^3*RR^4 + 4*LL*LM^3*RR^3*RS - 4*LL*LM^3*RR^2*RS^2 - 4*LL*LM^3*RR*RS^3 + LM^4*RR^4 + 4*LM^4*RR^3*RS + 6*LM^4*RR^2*RS^2 + 4*LM^4*RR*RS^3 + LM^4*RS^4))/(LL^8*LM^4))^(1/2) - 12*LL*LM^5*RR*RS^5 + 36*LL*LM^5*RR^5*RS - 12*LL^5*LM*RR^5*RS - 36*LL*LM^5*RR^2*RS^4 - 24*LL*LM^5*RR^3*RS^3 + 24*LL*LM^5*RR^4*RS^2 + 24*LL^2*LM^4*RR^5*RS - 24*LL^3*LM^3*RR^5*RS - 36*LL^4*LM^2*RR^5*RS - 24*LL^4*LM^6*Omg^4*RR*RS - 12*LL^5*LM^5*Omg^4*RR*RS - 12*LL^2*LM^6*Omg^2*RR*RS^3 - 12*LL^2*LM^6*Omg^2*RR^3*RS - 24*LL^3*LM^5*Omg^2*RR*RS^3 - 48*LL^3*LM^5*Omg^2*RR^3*RS - 60*LL^4*LM^4*Omg^2*RR^3*RS - 24*LL^5*LM^3*Omg^2*RR^3*RS - 36*LL^2*LM^6*Omg^2*RR^2*RS^2 + 48*LL^3*LM^5*Omg^2*RR^2*RS^2 + 36*LL^4*LM^4*Omg^2*RR^2*RS^2)/(LL^6*LM^6))^(1/3)*(- 2*LL^2*LM^2*Omg^2 + LL^2*RR^2 + 2*LL*LM*RR^2 - 2*LL*LM*RR*RS + LM^2*RR^2 + 2*LM^2*RR*RS + LM^2*RS^2))/(LL^2*LM^2) - (24*RS*(LL*RR + LM*RR + LM*RS)*(LL*LM^2*Omg^2 + LM*RR^2 + RS*LM*RR + LL*RR^2))/(LL^3*LM^3))^(1/4))
These are mathematically correct. They are also, for most human purposes, useless; I would submit to you that it is past the capacities of most people to look at solutions such as that and just by reading them, say "Ah yes, that's the closed form solution for the roots of (particular degree 4 polynomial), or to look at a solution such as that and say, "Wow, that is almost the root of a degree 4 polynomial, if only the 28 were a 30 it would be perfect!"
Eigenvalues. Of. A. 4x4. Symbolic. Matrix. Are. Seldom. Going. To. Be. Understandable.
You shouldn't be asking for "usable" solutions for such a situation. There aren't any "usable" solutions.
... This is why approximation and linearization and similar techniques are used.
  3 件のコメント
Stepan Podhorsky
Stepan Podhorsky 2020 年 8 月 14 日
Thank you for your reaction. Do you think that the solution could be simplified by the fact that the original matrix contains symmetries and can be reduced to the 2x2 matrix in case matrix notation of complex numbers is used or these facts are unusable?
Walter Roberson
Walter Roberson 2020 年 8 月 14 日
No, When you have matrices of symbolic entries and you have multiplication, then it automatically expands, so I don't think you can do any better.

サインインしてコメントする。

カテゴリ

Help Center および File ExchangeOperators and Elementary Operations についてさらに検索

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!

Translated by