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mmcdara

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@tomleslie Sorry, I have been lured...

mmcdara 7891
August 18 2020
0 0

@tomleslie 

Sorry, I have been lured by the "alpha" of the OP, which is a number, and your "alpha" which is a random variable.
So I take it back, and you did the right thing.

I just maintain that there was no need to define the operator "Expectation" for "Mean" already does the job.

A final remark:

A := RandomVariable(Uniform(c, d)):
f := z -> a*z+b*t/z+exp(z);  # your function
g := f(A);                   # a random variable
diff(g, t);
Mean(%);                     # does exactly the same thing than
                             # diff(Mean(g), t), and is faster

The interest of that? 
Suppose f := z -> b*t/z +H(z) were H is a function whose integral doesn't have a closed form.
Then Mean(f(A)) won't be calculated neither its derivative according to t.
If tou do Mean(diff(f(A), t)) instead you will obtain the result.

@Axel Vogt  I guess you parenthesi...

mmcdara 7891
August 17 2020
0 0

@Axel Vogt 

I guess you parenthesis could refer to my question "Can this be done with IntegrationTools ???" ?
I first defined S as the Fourier transform of the convolution product that is

Sp__fg := xi -> Int((Int(f(x)*g(y-x), x = -infinity .. infinity))*exp(-I*y*xi), y = -infinity .. infinity)

And I did next what we usualy do by hand, that is exchanging the integrals to obtain 
 

S__fg := xi -> Int(f(x)*(Int(g(y-x)*exp(-I*y*xi), y = -infinity .. infinity)), x = -infinity .. infinity)

I wonder if this last expression can be derived from the first one by using some maple's procedure, maybe one from IntegrationTools?
 

@Umang Varshney  @tomleslieHi,...

mmcdara 7891
August 17 2020
0 0

@Umang Varshney  @tomleslie

Hi, 

The simplest and correct way to obtain what you want is given below.
No need to define the operator "E" (it already exists in Maple under a different form). More of this your writting of g is uncorrect (as tomleslie's second part is).
 

restart:
with(Statistics):
A :=RandomVariable(Uniform(c, d))
                               _R
# Remark: Maple unfortunately uses the same name "Mean" for the mean of a sample and for the 
# "mean" of a random variable which, if the things were done correctly, should 
# have been named "(mathematical) expectation" (just as you wrote).
#
# This means you can use the procedure "Mean" to obtain the Expectation of a random variable.

Mean(A);
                           1     1  
                           - c + - d
                           2     2  
# Remark: in the following g is not a random variable


g := a*alpha+b*t/alpha+exp(alpha);

# Here g is a random variable

g := a*A + b*t/A + exp(A);
                             b t              
                  a alpha + ----- + exp(alpha)
                            alpha             
                             b t          
                      a _R + --- + exp(_R)
                             _R           
# and if we are happy enough, and it happens we are, Maple can compute its "Mean" (Mathematical Expectation)

Mean(g)
         /                              
         |                              
piecewise|And(0 < d, c < 0), undefined, 
         \                              

                                   2      2                      \
    2 b t ln(d) - 2 b t ln(c) - a c  + a d  + 2 exp(d) - 2 exp(c)|
  - -------------------------------------------------------------|
                             2 (-d + c)                          /

Corrections.mw

@Test007 @Axel Vogt I guess Axel s...

mmcdara 7891
August 16 2020
0 0

@Test007 @Axel Vogt

I guess Axel suggested to do something like this (?)

I think that it could be possible to define a linear operator called (FT) with ad hoc properties (for instance FT(f * g)=FT(f).FT(g), FT(f(t+h))=exp(-I*h*theta)=FT(f)(theta) and so on... but I do not have enough skill to do this)
 

> 

restart:
> 

with(IntegrationTools):
> 

# Convolution product

C__fg := unapply(Int(f(x)*g(y-x), x=-infinity..+infinity), y)

proc (y) options operator, arrow; Int(f(x)*g(y-x), x = -infinity .. infinity) end proc

 (1)
> 

# Spectral density (possibly up to some constant)

Sp__fg := unapply(Int(C__fg(y)*exp(-I*y*xi), y=-infinity..+infinity), xi);

proc (xi) options operator, arrow; Int((Int(f(x)*g(y-x), x = -infinity .. infinity))*exp(-I*y*xi), y = -infinity .. infinity) end proc

 (2)
> 

# rewrite Sp__fg under this form
# Can this be done with IntegrationTools ???

S__fg := unapply(Int(f(x)*Int(g(y-x)*exp(-I*y*xi), y=-infinity..+infinity), x=-infinity..+infinity), xi);

proc (xi) options operator, arrow; Int(f(x)*(Int(g(y-x)*exp(-I*y*xi), y = -infinity .. infinity)), x = -infinity .. infinity) end proc

 (3)
> 

# Inner Integral

Inner_Integral := op([1, 2], S__fg(xi));

Change(Inner_Integral, y=x+eta, eta);

Expand(%);

Inner_Integral := map(simplify, %);

TF_g := eta -> `#mrow(mover(mo(g),mo("&#94;")))`(eta):

Inner_Integral := eval(Inner_Integral, op(2, Inner_Integral)=TF_g(eta))

Int(g(y-x)*exp(-I*y*xi), y = -infinity .. infinity)

  

Int(g(eta)*exp(-I*(x+eta)*xi), eta = -infinity .. infinity)

  

(Int(g(eta)/exp(I*xi*eta), eta = -infinity .. infinity))/exp(I*xi*x)

  

exp(-I*xi*x)*(Int(g(eta)*exp(-I*xi*eta), eta = -infinity .. infinity))

  

exp(-I*xi*x)*`#mrow(mover(mo(g),mo("^")))`(eta)

 (4)
> 

# Outer integral

eval(S__fg(xi), op([1, 2], S__fg(xi))=Inner_Integral);

Expand(%);

Outer_Integral := simplify(%);

TF_f := eta -> `#mrow(mover(mo(f),mo("&#94;")))`(eta):

Outer_Integral := eval(Outer_Integral, op(2, Outer_Integral)=TF_f(xi))

Int(f(x)*exp(-I*xi*x)*`#mrow(mover(mo(g),mo("^")))`(eta), x = -infinity .. infinity)

  

`#mrow(mover(mo(g),mo("^")))`(eta)*(Int(f(x)/exp(I*xi*x), x = -infinity .. infinity))

  

`#mrow(mover(mo(g),mo("^")))`(eta)*(Int(f(x)*exp(-I*xi*x), x = -infinity .. infinity))

  

`#mrow(mover(mo(g),mo("^")))`(eta)*`#mrow(mover(mo(f),mo("^")))`(xi)

 (5)
> 

# eta being a dummy variable one has

Sp__fg(xi) = eval(Outer_Integral, eta=xi);

Int((Int(f(x)*g(y-x), x = -infinity .. infinity))*exp(-I*y*xi), y = -infinity .. infinity) = `#mrow(mover(mo(g),mo("^")))`(xi)*`#mrow(mover(mo(f),mo("^")))`(xi)

 (6)
> 

 


 

Download TF_of_Convolution.mw

@Preben Alsholm  Thanks Preben,&nb...

mmcdara 7891
August 15 2020
0 0

@Preben Alsholm 

Thanks Preben, it's very interesting.
I have observed myself this kind of instability. No scheme seems able to avoid it and to "folow" the solution I exepcted. More of this the "exponential growth" of the solution doesn't appear at the same place for different schemes (some present an "exponential decay" which ends with an error (X(t)=0 and thus  ( diff(X(t),t)/(1+diff(X(t),t)^2) + F) / X(t) leads to a singularity).



A few words about the orign of this problem.
Let x=x(t), v=v(t) with v=dx.dt, and F=F(t)
The form of the ode is an example of more general odes of the form  f(x)*dv/dt+g(v^2)=F. These odes appear when a solid whose mass depends on its position (f(x)) moves in a fluid, which develops a resistance g(v^2), under an acceleration F(t).

To test different numerical methods I use a method dubbed MMS (Method of Manufactured Solutions) where the principle is the following:

  • Let A some differential operator and S = {A(t, x, v) = F, v=ds/dt, (+ics)} a differential system.
  • To obtain a reference solution of S one choose an arbitrary, but "plausible", function U(t), and plug it in operator A to obtain the expression of F = A(t U(t), diff(U(t),t)) .
  • On then solve numerically the system  {A(t, x, v) = A(t U(t), diff(U(t),t)), v=ds/dt, (+ics)} and compare the solution approximated solution x*(t) to U(t)


MMC is widely used for verification of computational codes (Finite Elements for instance)

@SeyiOshin Do you have any idea how...

mmcdara 7891
August 14 2020
0 0

@SeyiOshin 

Do you have any idea how complex can be the expressions of the eigenvalues of a general square matrix (and thus of the time and the memory required to buid these expressions) ?

The first example concerns the computations of the eigenvalues of a formal squre matrix.
The second the computations of the eigenvalues of a matrix of integers.
The third one the computations of the eigenvalues of a matrix of floats (basically what Matlab does, unless you use its Symbolic Toolbox).

Another point that the second example illustrate: the eigenvalues are the roots of some polynomial and it's known that no general expression does exist to express the roots of any polynomial of order larger than 4.
Of course this problem does not exist for matrix of floats.


Now, if you want to assess the derivative of an eigenvalue V[k] to the element M[i,j] of some matrix M, a simple thing to do (unless your matrix has small size and/or has a "simple" structure)  is to compute
(V'[k]-V[k]) / ((M[i,j]+eps) - M[i,j])  for |eps| "small"
where V'[k] is the corresponding kth eigenvalue of the matrix M' which differs from M only by its [i,j] term now equal to M[i,j]+eps

More cleverly you can also use theorirical results about the eigenvalues of M'=M+eps*E (for |eps| small again) :
V'(k) = V(k) + eps*(vT[k].E.u[k]) where vT[k] and u[k] are the right and left kth eigenvectors of M
'here E could the matrix E[i', j']=1 if i'=' and j=j' and 0 otherwise)


 

> 

restart
> 

with(LinearAlgebra):
> 

for N from 2 to 4 do
  M       := Matrix(N$2, symbol=s):
  eig_||N := CodeTools:-Usage( Eigenvalues(M) ):
  print(length(eig_||N)):  # length(eig_||N) returns the number of words used to represent eig_||N
end do:

memory used=3.70MiB, alloc change=32.00MiB, cpu time=162.00ms, real time=7.10s, gc time=7.54ms

  

302

  

memory used=7.41MiB, alloc change=0 bytes, cpu time=117.00ms, real time=5.73s, gc time=0ns

  

87720

  

memory used=3.16GiB, alloc change=407.25MiB, cpu time=50.70s, real time=36.18s, gc time=23.98s

  

174243014

 (1)
> 

eig_3;  # eigenvalues of a 3 by 3 matrix with generic element s[i,j]

Vector(3, {(1) = (1/6)*(8*s[1, 1]^3-12*s[1, 1]^2*s[2, 2]-12*s[1, 1]^2*s[3, 3]+36*s[1, 1]*s[1, 2]*s[2, 1]+36*s[1, 1]*s[1, 3]*s[3, 1]-12*s[1, 1]*s[2, 2]^2+48*s[1, 1]*s[2, 2]*s[3, 3]-72*s[1, 1]*s[2, 3]*s[3, 2]-12*s[1, 1]*s[3, 3]^2+36*s[1, 2]*s[2, 1]*s[2, 2]-72*s[1, 2]*s[2, 1]*s[3, 3]+108*s[1, 2]*s[2, 3]*s[3, 1]+108*s[1, 3]*s[2, 1]*s[3, 2]-72*s[1, 3]*s[2, 2]*s[3, 1]+36*s[1, 3]*s[3, 1]*s[3, 3]+8*s[2, 2]^3-12*s[2, 2]^2*s[3, 3]+36*s[2, 2]*s[2, 3]*s[3, 2]-12*s[2, 2]*s[3, 3]^2+36*s[2, 3]*s[3, 2]*s[3, 3]+8*s[3, 3]^3+12*sqrt(-3*s[1, 1]^4*s[2, 2]^2+6*s[1, 1]^4*s[2, 2]*s[3, 3]-12*s[1, 1]^4*s[2, 3]*s[3, 2]-3*s[1, 1]^4*s[3, 3]^2+6*s[1, 1]^3*s[1, 2]*s[2, 1]*s[2, 2]-6*s[1, 1]^3*s[1, 2]*s[2, 1]*s[3, 3]+12*s[1, 1]^3*s[1, 2]*s[2, 3]*s[3, 1]+12*s[1, 1]^3*s[1, 3]*s[2, 1]*s[3, 2]-6*s[1, 1]^3*s[1, 3]*s[2, 2]*s[3, 1]+6*s[1, 1]^3*s[1, 3]*s[3, 1]*s[3, 3]+6*s[1, 1]^3*s[2, 2]^3-6*s[1, 1]^3*s[2, 2]^2*s[3, 3]+24*s[1, 1]^3*s[2, 2]*s[2, 3]*s[3, 2]-6*s[1, 1]^3*s[2, 2]*s[3, 3]^2+24*s[1, 1]^3*s[2, 3]*s[3, 2]*s[3, 3]+6*s[1, 1]^3*s[3, 3]^3-3*s[1, 1]^2*s[1, 2]^2*s[2, 1]^2-6*s[1, 1]^2*s[1, 2]*s[1, 3]*s[2, 1]*s[3, 1]-24*s[1, 1]^2*s[1, 2]*s[2, 1]*s[2, 2]^2+30*s[1, 1]^2*s[1, 2]*s[2, 1]*s[2, 2]*s[3, 3]-60*s[1, 1]^2*s[1, 2]*s[2, 1]*s[2, 3]*s[3, 2]-6*s[1, 1]^2*s[1, 2]*s[2, 1]*s[3, 3]^2-18*s[1, 1]^2*s[1, 2]*s[2, 2]*s[2, 3]*s[3, 1]-18*s[1, 1]^2*s[1, 2]*s[2, 3]*s[3, 1]*s[3, 3]-3*s[1, 1]^2*s[1, 3]^2*s[3, 1]^2-18*s[1, 1]^2*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 2]-18*s[1, 1]^2*s[1, 3]*s[2, 1]*s[3, 2]*s[3, 3]-6*s[1, 1]^2*s[1, 3]*s[2, 2]^2*s[3, 1]+30*s[1, 1]^2*s[1, 3]*s[2, 2]*s[3, 1]*s[3, 3]-60*s[1, 1]^2*s[1, 3]*s[2, 3]*s[3, 1]*s[3, 2]-24*s[1, 1]^2*s[1, 3]*s[3, 1]*s[3, 3]^2-3*s[1, 1]^2*s[2, 2]^4-6*s[1, 1]^2*s[2, 2]^3*s[3, 3]-6*s[1, 1]^2*s[2, 2]^2*s[2, 3]*s[3, 2]+18*s[1, 1]^2*s[2, 2]^2*s[3, 3]^2-60*s[1, 1]^2*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]-6*s[1, 1]^2*s[2, 2]*s[3, 3]^3+24*s[1, 1]^2*s[2, 3]^2*s[3, 2]^2-6*s[1, 1]^2*s[2, 3]*s[3, 2]*s[3, 3]^2-3*s[1, 1]^2*s[3, 3]^4+30*s[1, 1]*s[1, 2]^2*s[2, 1]^2*s[2, 2]-24*s[1, 1]*s[1, 2]^2*s[2, 1]^2*s[3, 3]+54*s[1, 1]*s[1, 2]^2*s[2, 1]*s[2, 3]*s[3, 1]+54*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 1]^2*s[3, 2]+6*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 1]+6*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 1]*s[3, 1]*s[3, 3]+54*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 3]*s[3, 1]^2+6*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]^3+30*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]^2*s[3, 3]+6*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 2]-60*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]*s[3, 3]^2+114*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 3]*s[3, 2]*s[3, 3]+24*s[1, 1]*s[1, 2]*s[2, 1]*s[3, 3]^3-18*s[1, 1]*s[1, 2]*s[2, 2]^2*s[2, 3]*s[3, 1]+72*s[1, 1]*s[1, 2]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 3]-108*s[1, 1]*s[1, 2]*s[2, 3]^2*s[3, 1]*s[3, 2]-18*s[1, 1]*s[1, 2]*s[2, 3]*s[3, 1]*s[3, 3]^2+54*s[1, 1]*s[1, 3]^2*s[2, 1]*s[3, 1]*s[3, 2]-24*s[1, 1]*s[1, 3]^2*s[2, 2]*s[3, 1]^2+30*s[1, 1]*s[1, 3]^2*s[3, 1]^2*s[3, 3]-18*s[1, 1]*s[1, 3]*s[2, 1]*s[2, 2]^2*s[3, 2]+72*s[1, 1]*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 2]*s[3, 3]-108*s[1, 1]*s[1, 3]*s[2, 1]*s[2, 3]*s[3, 2]^2-18*s[1, 1]*s[1, 3]*s[2, 1]*s[3, 2]*s[3, 3]^2+24*s[1, 1]*s[1, 3]*s[2, 2]^3*s[3, 1]-60*s[1, 1]*s[1, 3]*s[2, 2]^2*s[3, 1]*s[3, 3]+114*s[1, 1]*s[1, 3]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 2]+30*s[1, 1]*s[1, 3]*s[2, 2]*s[3, 1]*s[3, 3]^2+6*s[1, 1]*s[1, 3]*s[2, 3]*s[3, 1]*s[3, 2]*s[3, 3]+6*s[1, 1]*s[1, 3]*s[3, 1]*s[3, 3]^3+6*s[1, 1]*s[2, 2]^4*s[3, 3]-6*s[1, 1]*s[2, 2]^3*s[2, 3]*s[3, 2]-6*s[1, 1]*s[2, 2]^3*s[3, 3]^2+30*s[1, 1]*s[2, 2]^2*s[2, 3]*s[3, 2]*s[3, 3]-6*s[1, 1]*s[2, 2]^2*s[3, 3]^3-24*s[1, 1]*s[2, 2]*s[2, 3]^2*s[3, 2]^2+30*s[1, 1]*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]^2+6*s[1, 1]*s[2, 2]*s[3, 3]^4-24*s[1, 1]*s[2, 3]^2*s[3, 2]^2*s[3, 3]-6*s[1, 1]*s[2, 3]*s[3, 2]*s[3, 3]^3-12*s[1, 2]^3*s[2, 1]^3-36*s[1, 2]^2*s[1, 3]*s[2, 1]^2*s[3, 1]-3*s[1, 2]^2*s[2, 1]^2*s[2, 2]^2-24*s[1, 2]^2*s[2, 1]^2*s[2, 2]*s[3, 3]-36*s[1, 2]^2*s[2, 1]^2*s[2, 3]*s[3, 2]+24*s[1, 2]^2*s[2, 1]^2*s[3, 3]^2+54*s[1, 2]^2*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 1]-108*s[1, 2]^2*s[2, 1]*s[2, 3]*s[3, 1]*s[3, 3]+81*s[1, 2]^2*s[2, 3]^2*s[3, 1]^2-36*s[1, 2]*s[1, 3]^2*s[2, 1]*s[3, 1]^2+54*s[1, 2]*s[1, 3]*s[2, 1]^2*s[2, 2]*s[3, 2]-108*s[1, 2]*s[1, 3]*s[2, 1]^2*s[3, 2]*s[3, 3]-60*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 2]^2*s[3, 1]+114*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 1]*s[3, 3]+90*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 3]*s[3, 1]*s[3, 2]-60*s[1, 2]*s[1, 3]*s[2, 1]*s[3, 1]*s[3, 3]^2-108*s[1, 2]*s[1, 3]*s[2, 2]*s[2, 3]*s[3, 1]^2+54*s[1, 2]*s[1, 3]*s[2, 3]*s[3, 1]^2*s[3, 3]-6*s[1, 2]*s[2, 1]*s[2, 2]^3*s[3, 3]-6*s[1, 2]*s[2, 1]*s[2, 2]^2*s[2, 3]*s[3, 2]-6*s[1, 2]*s[2, 1]*s[2, 2]^2*s[3, 3]^2+6*s[1, 2]*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]+24*s[1, 2]*s[2, 1]*s[2, 2]*s[3, 3]^3-36*s[1, 2]*s[2, 1]*s[2, 3]^2*s[3, 2]^2-60*s[1, 2]*s[2, 1]*s[2, 3]*s[3, 2]*s[3, 3]^2-12*s[1, 2]*s[2, 1]*s[3, 3]^4+12*s[1, 2]*s[2, 2]^3*s[2, 3]*s[3, 1]-18*s[1, 2]*s[2, 2]^2*s[2, 3]*s[3, 1]*s[3, 3]+54*s[1, 2]*s[2, 2]*s[2, 3]^2*s[3, 1]*s[3, 2]-18*s[1, 2]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 3]^2+54*s[1, 2]*s[2, 3]^2*s[3, 1]*s[3, 2]*s[3, 3]+12*s[1, 2]*s[2, 3]*s[3, 1]*s[3, 3]^3-12*s[1, 3]^3*s[3, 1]^3+81*s[1, 3]^2*s[2, 1]^2*s[3, 2]^2-108*s[1, 3]^2*s[2, 1]*s[2, 2]*s[3, 1]*s[3, 2]+54*s[1, 3]^2*s[2, 1]*s[3, 1]*s[3, 2]*s[3, 3]+24*s[1, 3]^2*s[2, 2]^2*s[3, 1]^2-24*s[1, 3]^2*s[2, 2]*s[3, 1]^2*s[3, 3]-36*s[1, 3]^2*s[2, 3]*s[3, 1]^2*s[3, 2]-3*s[1, 3]^2*s[3, 1]^2*s[3, 3]^2+12*s[1, 3]*s[2, 1]*s[2, 2]^3*s[3, 2]-18*s[1, 3]*s[2, 1]*s[2, 2]^2*s[3, 2]*s[3, 3]+54*s[1, 3]*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 2]^2-18*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 2]*s[3, 3]^2+54*s[1, 3]*s[2, 1]*s[2, 3]*s[3, 2]^2*s[3, 3]+12*s[1, 3]*s[2, 1]*s[3, 2]*s[3, 3]^3-12*s[1, 3]*s[2, 2]^4*s[3, 1]+24*s[1, 3]*s[2, 2]^3*s[3, 1]*s[3, 3]-60*s[1, 3]*s[2, 2]^2*s[2, 3]*s[3, 1]*s[3, 2]-6*s[1, 3]*s[2, 2]^2*s[3, 1]*s[3, 3]^2+6*s[1, 3]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 2]*s[3, 3]-6*s[1, 3]*s[2, 2]*s[3, 1]*s[3, 3]^3-36*s[1, 3]*s[2, 3]^2*s[3, 1]*s[3, 2]^2-6*s[1, 3]*s[2, 3]*s[3, 1]*s[3, 2]*s[3, 3]^2-3*s[2, 2]^4*s[3, 3]^2+6*s[2, 2]^3*s[2, 3]*s[3, 2]*s[3, 3]+6*s[2, 2]^3*s[3, 3]^3-3*s[2, 2]^2*s[2, 3]^2*s[3, 2]^2-24*s[2, 2]^2*s[2, 3]*s[3, 2]*s[3, 3]^2-3*s[2, 2]^2*s[3, 3]^4+30*s[2, 2]*s[2, 3]^2*s[3, 2]^2*s[3, 3]+6*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]^3-12*s[2, 3]^3*s[3, 2]^3-3*s[2, 3]^2*s[3, 2]^2*s[3, 3]^2))^(1/3)-((2/3)*s[1, 1]*s[2, 2]+(2/3)*s[3, 3]*s[1, 1]-2*s[1, 2]*s[2, 1]-2*s[1, 3]*s[3, 1]+(2/3)*s[3, 3]*s[2, 2]-2*s[2, 3]*s[3, 2]-(2/3)*s[1, 1]^2-(2/3)*s[2, 2]^2-(2/3)*s[3, 3]^2)/(8*s[1, 1]^3-12*s[1, 1]^2*s[2, 2]-12*s[1, 1]^2*s[3, 3]+36*s[1, 1]*s[1, 2]*s[2, 1]+36*s[1, 1]*s[1, 3]*s[3, 1]-12*s[1, 1]*s[2, 2]^2+48*s[1, 1]*s[2, 2]*s[3, 3]-72*s[1, 1]*s[2, 3]*s[3, 2]-12*s[1, 1]*s[3, 3]^2+36*s[1, 2]*s[2, 1]*s[2, 2]-72*s[1, 2]*s[2, 1]*s[3, 3]+108*s[1, 2]*s[2, 3]*s[3, 1]+108*s[1, 3]*s[2, 1]*s[3, 2]-72*s[1, 3]*s[2, 2]*s[3, 1]+36*s[1, 3]*s[3, 1]*s[3, 3]+8*s[2, 2]^3-12*s[2, 2]^2*s[3, 3]+36*s[2, 2]*s[2, 3]*s[3, 2]-12*s[2, 2]*s[3, 3]^2+36*s[2, 3]*s[3, 2]*s[3, 3]+8*s[3, 3]^3+12*sqrt(-3*s[1, 1]^4*s[2, 2]^2+6*s[1, 1]^4*s[2, 2]*s[3, 3]-12*s[1, 1]^4*s[2, 3]*s[3, 2]-3*s[1, 1]^4*s[3, 3]^2+6*s[1, 1]^3*s[1, 2]*s[2, 1]*s[2, 2]-6*s[1, 1]^3*s[1, 2]*s[2, 1]*s[3, 3]+12*s[1, 1]^3*s[1, 2]*s[2, 3]*s[3, 1]+12*s[1, 1]^3*s[1, 3]*s[2, 1]*s[3, 2]-6*s[1, 1]^3*s[1, 3]*s[2, 2]*s[3, 1]+6*s[1, 1]^3*s[1, 3]*s[3, 1]*s[3, 3]+6*s[1, 1]^3*s[2, 2]^3-6*s[1, 1]^3*s[2, 2]^2*s[3, 3]+24*s[1, 1]^3*s[2, 2]*s[2, 3]*s[3, 2]-6*s[1, 1]^3*s[2, 2]*s[3, 3]^2+24*s[1, 1]^3*s[2, 3]*s[3, 2]*s[3, 3]+6*s[1, 1]^3*s[3, 3]^3-3*s[1, 1]^2*s[1, 2]^2*s[2, 1]^2-6*s[1, 1]^2*s[1, 2]*s[1, 3]*s[2, 1]*s[3, 1]-24*s[1, 1]^2*s[1, 2]*s[2, 1]*s[2, 2]^2+30*s[1, 1]^2*s[1, 2]*s[2, 1]*s[2, 2]*s[3, 3]-60*s[1, 1]^2*s[1, 2]*s[2, 1]*s[2, 3]*s[3, 2]-6*s[1, 1]^2*s[1, 2]*s[2, 1]*s[3, 3]^2-18*s[1, 1]^2*s[1, 2]*s[2, 2]*s[2, 3]*s[3, 1]-18*s[1, 1]^2*s[1, 2]*s[2, 3]*s[3, 1]*s[3, 3]-3*s[1, 1]^2*s[1, 3]^2*s[3, 1]^2-18*s[1, 1]^2*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 2]-18*s[1, 1]^2*s[1, 3]*s[2, 1]*s[3, 2]*s[3, 3]-6*s[1, 1]^2*s[1, 3]*s[2, 2]^2*s[3, 1]+30*s[1, 1]^2*s[1, 3]*s[2, 2]*s[3, 1]*s[3, 3]-60*s[1, 1]^2*s[1, 3]*s[2, 3]*s[3, 1]*s[3, 2]-24*s[1, 1]^2*s[1, 3]*s[3, 1]*s[3, 3]^2-3*s[1, 1]^2*s[2, 2]^4-6*s[1, 1]^2*s[2, 2]^3*s[3, 3]-6*s[1, 1]^2*s[2, 2]^2*s[2, 3]*s[3, 2]+18*s[1, 1]^2*s[2, 2]^2*s[3, 3]^2-60*s[1, 1]^2*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]-6*s[1, 1]^2*s[2, 2]*s[3, 3]^3+24*s[1, 1]^2*s[2, 3]^2*s[3, 2]^2-6*s[1, 1]^2*s[2, 3]*s[3, 2]*s[3, 3]^2-3*s[1, 1]^2*s[3, 3]^4+30*s[1, 1]*s[1, 2]^2*s[2, 1]^2*s[2, 2]-24*s[1, 1]*s[1, 2]^2*s[2, 1]^2*s[3, 3]+54*s[1, 1]*s[1, 2]^2*s[2, 1]*s[2, 3]*s[3, 1]+54*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 1]^2*s[3, 2]+6*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 1]+6*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 1]*s[3, 1]*s[3, 3]+54*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 3]*s[3, 1]^2+6*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]^3+30*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]^2*s[3, 3]+6*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 2]-60*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]*s[3, 3]^2+114*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 3]*s[3, 2]*s[3, 3]+24*s[1, 1]*s[1, 2]*s[2, 1]*s[3, 3]^3-18*s[1, 1]*s[1, 2]*s[2, 2]^2*s[2, 3]*s[3, 1]+72*s[1, 1]*s[1, 2]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 3]-108*s[1, 1]*s[1, 2]*s[2, 3]^2*s[3, 1]*s[3, 2]-18*s[1, 1]*s[1, 2]*s[2, 3]*s[3, 1]*s[3, 3]^2+54*s[1, 1]*s[1, 3]^2*s[2, 1]*s[3, 1]*s[3, 2]-24*s[1, 1]*s[1, 3]^2*s[2, 2]*s[3, 1]^2+30*s[1, 1]*s[1, 3]^2*s[3, 1]^2*s[3, 3]-18*s[1, 1]*s[1, 3]*s[2, 1]*s[2, 2]^2*s[3, 2]+72*s[1, 1]*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 2]*s[3, 3]-108*s[1, 1]*s[1, 3]*s[2, 1]*s[2, 3]*s[3, 2]^2-18*s[1, 1]*s[1, 3]*s[2, 1]*s[3, 2]*s[3, 3]^2+24*s[1, 1]*s[1, 3]*s[2, 2]^3*s[3, 1]-60*s[1, 1]*s[1, 3]*s[2, 2]^2*s[3, 1]*s[3, 3]+114*s[1, 1]*s[1, 3]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 2]+30*s[1, 1]*s[1, 3]*s[2, 2]*s[3, 1]*s[3, 3]^2+6*s[1, 1]*s[1, 3]*s[2, 3]*s[3, 1]*s[3, 2]*s[3, 3]+6*s[1, 1]*s[1, 3]*s[3, 1]*s[3, 3]^3+6*s[1, 1]*s[2, 2]^4*s[3, 3]-6*s[1, 1]*s[2, 2]^3*s[2, 3]*s[3, 2]-6*s[1, 1]*s[2, 2]^3*s[3, 3]^2+30*s[1, 1]*s[2, 2]^2*s[2, 3]*s[3, 2]*s[3, 3]-6*s[1, 1]*s[2, 2]^2*s[3, 3]^3-24*s[1, 1]*s[2, 2]*s[2, 3]^2*s[3, 2]^2+30*s[1, 1]*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]^2+6*s[1, 1]*s[2, 2]*s[3, 3]^4-24*s[1, 1]*s[2, 3]^2*s[3, 2]^2*s[3, 3]-6*s[1, 1]*s[2, 3]*s[3, 2]*s[3, 3]^3-12*s[1, 2]^3*s[2, 1]^3-36*s[1, 2]^2*s[1, 3]*s[2, 1]^2*s[3, 1]-3*s[1, 2]^2*s[2, 1]^2*s[2, 2]^2-24*s[1, 2]^2*s[2, 1]^2*s[2, 2]*s[3, 3]-36*s[1, 2]^2*s[2, 1]^2*s[2, 3]*s[3, 2]+24*s[1, 2]^2*s[2, 1]^2*s[3, 3]^2+54*s[1, 2]^2*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 1]-108*s[1, 2]^2*s[2, 1]*s[2, 3]*s[3, 1]*s[3, 3]+81*s[1, 2]^2*s[2, 3]^2*s[3, 1]^2-36*s[1, 2]*s[1, 3]^2*s[2, 1]*s[3, 1]^2+54*s[1, 2]*s[1, 3]*s[2, 1]^2*s[2, 2]*s[3, 2]-108*s[1, 2]*s[1, 3]*s[2, 1]^2*s[3, 2]*s[3, 3]-60*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 2]^2*s[3, 1]+114*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 1]*s[3, 3]+90*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 3]*s[3, 1]*s[3, 2]-60*s[1, 2]*s[1, 3]*s[2, 1]*s[3, 1]*s[3, 3]^2-108*s[1, 2]*s[1, 3]*s[2, 2]*s[2, 3]*s[3, 1]^2+54*s[1, 2]*s[1, 3]*s[2, 3]*s[3, 1]^2*s[3, 3]-6*s[1, 2]*s[2, 1]*s[2, 2]^3*s[3, 3]-6*s[1, 2]*s[2, 1]*s[2, 2]^2*s[2, 3]*s[3, 2]-6*s[1, 2]*s[2, 1]*s[2, 2]^2*s[3, 3]^2+6*s[1, 2]*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]+24*s[1, 2]*s[2, 1]*s[2, 2]*s[3, 3]^3-36*s[1, 2]*s[2, 1]*s[2, 3]^2*s[3, 2]^2-60*s[1, 2]*s[2, 1]*s[2, 3]*s[3, 2]*s[3, 3]^2-12*s[1, 2]*s[2, 1]*s[3, 3]^4+12*s[1, 2]*s[2, 2]^3*s[2, 3]*s[3, 1]-18*s[1, 2]*s[2, 2]^2*s[2, 3]*s[3, 1]*s[3, 3]+54*s[1, 2]*s[2, 2]*s[2, 3]^2*s[3, 1]*s[3, 2]-18*s[1, 2]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 3]^2+54*s[1, 2]*s[2, 3]^2*s[3, 1]*s[3, 2]*s[3, 3]+12*s[1, 2]*s[2, 3]*s[3, 1]*s[3, 3]^3-12*s[1, 3]^3*s[3, 1]^3+81*s[1, 3]^2*s[2, 1]^2*s[3, 2]^2-108*s[1, 3]^2*s[2, 1]*s[2, 2]*s[3, 1]*s[3, 2]+54*s[1, 3]^2*s[2, 1]*s[3, 1]*s[3, 2]*s[3, 3]+24*s[1, 3]^2*s[2, 2]^2*s[3, 1]^2-24*s[1, 3]^2*s[2, 2]*s[3, 1]^2*s[3, 3]-36*s[1, 3]^2*s[2, 3]*s[3, 1]^2*s[3, 2]-3*s[1, 3]^2*s[3, 1]^2*s[3, 3]^2+12*s[1, 3]*s[2, 1]*s[2, 2]^3*s[3, 2]-18*s[1, 3]*s[2, 1]*s[2, 2]^2*s[3, 2]*s[3, 3]+54*s[1, 3]*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 2]^2-18*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 2]*s[3, 3]^2+54*s[1, 3]*s[2, 1]*s[2, 3]*s[3, 2]^2*s[3, 3]+12*s[1, 3]*s[2, 1]*s[3, 2]*s[3, 3]^3-12*s[1, 3]*s[2, 2]^4*s[3, 1]+24*s[1, 3]*s[2, 2]^3*s[3, 1]*s[3, 3]-60*s[1, 3]*s[2, 2]^2*s[2, 3]*s[3, 1]*s[3, 2]-6*s[1, 3]*s[2, 2]^2*s[3, 1]*s[3, 3]^2+6*s[1, 3]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 2]*s[3, 3]-6*s[1, 3]*s[2, 2]*s[3, 1]*s[3, 3]^3-36*s[1, 3]*s[2, 3]^2*s[3, 1]*s[3, 2]^2-6*s[1, 3]*s[2, 3]*s[3, 1]*s[3, 2]*s[3, 3]^2-3*s[2, 2]^4*s[3, 3]^2+6*s[2, 2]^3*s[2, 3]*s[3, 2]*s[3, 3]+6*s[2, 2]^3*s[3, 3]^3-3*s[2, 2]^2*s[2, 3]^2*s[3, 2]^2-24*s[2, 2]^2*s[2, 3]*s[3, 2]*s[3, 3]^2-3*s[2, 2]^2*s[3, 3]^4+30*s[2, 2]*s[2, 3]^2*s[3, 2]^2*s[3, 3]+6*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]^3-12*s[2, 3]^3*s[3, 2]^3-3*s[2, 3]^2*s[3, 2]^2*s[3, 3]^2))^(1/3)+(1/3)*s[3, 3]+(1/3)*s[2, 2]+(1/3)*s[1, 1], (2) = -(1/12)*(8*s[1, 1]^3-12*s[1, 1]^2*s[2, 2]-12*s[1, 1]^2*s[3, 3]+36*s[1, 1]*s[1, 2]*s[2, 1]+36*s[1, 1]*s[1, 3]*s[3, 1]-12*s[1, 1]*s[2, 2]^2+48*s[1, 1]*s[2, 2]*s[3, 3]-72*s[1, 1]*s[2, 3]*s[3, 2]-12*s[1, 1]*s[3, 3]^2+36*s[1, 2]*s[2, 1]*s[2, 2]-72*s[1, 2]*s[2, 1]*s[3, 3]+108*s[1, 2]*s[2, 3]*s[3, 1]+108*s[1, 3]*s[2, 1]*s[3, 2]-72*s[1, 3]*s[2, 2]*s[3, 1]+36*s[1, 3]*s[3, 1]*s[3, 3]+8*s[2, 2]^3-12*s[2, 2]^2*s[3, 3]+36*s[2, 2]*s[2, 3]*s[3, 2]-12*s[2, 2]*s[3, 3]^2+36*s[2, 3]*s[3, 2]*s[3, 3]+8*s[3, 3]^3+12*sqrt(-3*s[1, 1]^4*s[2, 2]^2+6*s[1, 1]^4*s[2, 2]*s[3, 3]-12*s[1, 1]^4*s[2, 3]*s[3, 2]-3*s[1, 1]^4*s[3, 3]^2+6*s[1, 1]^3*s[1, 2]*s[2, 1]*s[2, 2]-6*s[1, 1]^3*s[1, 2]*s[2, 1]*s[3, 3]+12*s[1, 1]^3*s[1, 2]*s[2, 3]*s[3, 1]+12*s[1, 1]^3*s[1, 3]*s[2, 1]*s[3, 2]-6*s[1, 1]^3*s[1, 3]*s[2, 2]*s[3, 1]+6*s[1, 1]^3*s[1, 3]*s[3, 1]*s[3, 3]+6*s[1, 1]^3*s[2, 2]^3-6*s[1, 1]^3*s[2, 2]^2*s[3, 3]+24*s[1, 1]^3*s[2, 2]*s[2, 3]*s[3, 2]-6*s[1, 1]^3*s[2, 2]*s[3, 3]^2+24*s[1, 1]^3*s[2, 3]*s[3, 2]*s[3, 3]+6*s[1, 1]^3*s[3, 3]^3-3*s[1, 1]^2*s[1, 2]^2*s[2, 1]^2-6*s[1, 1]^2*s[1, 2]*s[1, 3]*s[2, 1]*s[3, 1]-24*s[1, 1]^2*s[1, 2]*s[2, 1]*s[2, 2]^2+30*s[1, 1]^2*s[1, 2]*s[2, 1]*s[2, 2]*s[3, 3]-60*s[1, 1]^2*s[1, 2]*s[2, 1]*s[2, 3]*s[3, 2]-6*s[1, 1]^2*s[1, 2]*s[2, 1]*s[3, 3]^2-18*s[1, 1]^2*s[1, 2]*s[2, 2]*s[2, 3]*s[3, 1]-18*s[1, 1]^2*s[1, 2]*s[2, 3]*s[3, 1]*s[3, 3]-3*s[1, 1]^2*s[1, 3]^2*s[3, 1]^2-18*s[1, 1]^2*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 2]-18*s[1, 1]^2*s[1, 3]*s[2, 1]*s[3, 2]*s[3, 3]-6*s[1, 1]^2*s[1, 3]*s[2, 2]^2*s[3, 1]+30*s[1, 1]^2*s[1, 3]*s[2, 2]*s[3, 1]*s[3, 3]-60*s[1, 1]^2*s[1, 3]*s[2, 3]*s[3, 1]*s[3, 2]-24*s[1, 1]^2*s[1, 3]*s[3, 1]*s[3, 3]^2-3*s[1, 1]^2*s[2, 2]^4-6*s[1, 1]^2*s[2, 2]^3*s[3, 3]-6*s[1, 1]^2*s[2, 2]^2*s[2, 3]*s[3, 2]+18*s[1, 1]^2*s[2, 2]^2*s[3, 3]^2-60*s[1, 1]^2*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]-6*s[1, 1]^2*s[2, 2]*s[3, 3]^3+24*s[1, 1]^2*s[2, 3]^2*s[3, 2]^2-6*s[1, 1]^2*s[2, 3]*s[3, 2]*s[3, 3]^2-3*s[1, 1]^2*s[3, 3]^4+30*s[1, 1]*s[1, 2]^2*s[2, 1]^2*s[2, 2]-24*s[1, 1]*s[1, 2]^2*s[2, 1]^2*s[3, 3]+54*s[1, 1]*s[1, 2]^2*s[2, 1]*s[2, 3]*s[3, 1]+54*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 1]^2*s[3, 2]+6*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 1]+6*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 1]*s[3, 1]*s[3, 3]+54*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 3]*s[3, 1]^2+6*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]^3+30*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]^2*s[3, 3]+6*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 2]-60*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]*s[3, 3]^2+114*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 3]*s[3, 2]*s[3, 3]+24*s[1, 1]*s[1, 2]*s[2, 1]*s[3, 3]^3-18*s[1, 1]*s[1, 2]*s[2, 2]^2*s[2, 3]*s[3, 1]+72*s[1, 1]*s[1, 2]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 3]-108*s[1, 1]*s[1, 2]*s[2, 3]^2*s[3, 1]*s[3, 2]-18*s[1, 1]*s[1, 2]*s[2, 3]*s[3, 1]*s[3, 3]^2+54*s[1, 1]*s[1, 3]^2*s[2, 1]*s[3, 1]*s[3, 2]-24*s[1, 1]*s[1, 3]^2*s[2, 2]*s[3, 1]^2+30*s[1, 1]*s[1, 3]^2*s[3, 1]^2*s[3, 3]-18*s[1, 1]*s[1, 3]*s[2, 1]*s[2, 2]^2*s[3, 2]+72*s[1, 1]*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 2]*s[3, 3]-108*s[1, 1]*s[1, 3]*s[2, 1]*s[2, 3]*s[3, 2]^2-18*s[1, 1]*s[1, 3]*s[2, 1]*s[3, 2]*s[3, 3]^2+24*s[1, 1]*s[1, 3]*s[2, 2]^3*s[3, 1]-60*s[1, 1]*s[1, 3]*s[2, 2]^2*s[3, 1]*s[3, 3]+114*s[1, 1]*s[1, 3]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 2]+30*s[1, 1]*s[1, 3]*s[2, 2]*s[3, 1]*s[3, 3]^2+6*s[1, 1]*s[1, 3]*s[2, 3]*s[3, 1]*s[3, 2]*s[3, 3]+6*s[1, 1]*s[1, 3]*s[3, 1]*s[3, 3]^3+6*s[1, 1]*s[2, 2]^4*s[3, 3]-6*s[1, 1]*s[2, 2]^3*s[2, 3]*s[3, 2]-6*s[1, 1]*s[2, 2]^3*s[3, 3]^2+30*s[1, 1]*s[2, 2]^2*s[2, 3]*s[3, 2]*s[3, 3]-6*s[1, 1]*s[2, 2]^2*s[3, 3]^3-24*s[1, 1]*s[2, 2]*s[2, 3]^2*s[3, 2]^2+30*s[1, 1]*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]^2+6*s[1, 1]*s[2, 2]*s[3, 3]^4-24*s[1, 1]*s[2, 3]^2*s[3, 2]^2*s[3, 3]-6*s[1, 1]*s[2, 3]*s[3, 2]*s[3, 3]^3-12*s[1, 2]^3*s[2, 1]^3-36*s[1, 2]^2*s[1, 3]*s[2, 1]^2*s[3, 1]-3*s[1, 2]^2*s[2, 1]^2*s[2, 2]^2-24*s[1, 2]^2*s[2, 1]^2*s[2, 2]*s[3, 3]-36*s[1, 2]^2*s[2, 1]^2*s[2, 3]*s[3, 2]+24*s[1, 2]^2*s[2, 1]^2*s[3, 3]^2+54*s[1, 2]^2*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 1]-108*s[1, 2]^2*s[2, 1]*s[2, 3]*s[3, 1]*s[3, 3]+81*s[1, 2]^2*s[2, 3]^2*s[3, 1]^2-36*s[1, 2]*s[1, 3]^2*s[2, 1]*s[3, 1]^2+54*s[1, 2]*s[1, 3]*s[2, 1]^2*s[2, 2]*s[3, 2]-108*s[1, 2]*s[1, 3]*s[2, 1]^2*s[3, 2]*s[3, 3]-60*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 2]^2*s[3, 1]+114*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 1]*s[3, 3]+90*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 3]*s[3, 1]*s[3, 2]-60*s[1, 2]*s[1, 3]*s[2, 1]*s[3, 1]*s[3, 3]^2-108*s[1, 2]*s[1, 3]*s[2, 2]*s[2, 3]*s[3, 1]^2+54*s[1, 2]*s[1, 3]*s[2, 3]*s[3, 1]^2*s[3, 3]-6*s[1, 2]*s[2, 1]*s[2, 2]^3*s[3, 3]-6*s[1, 2]*s[2, 1]*s[2, 2]^2*s[2, 3]*s[3, 2]-6*s[1, 2]*s[2, 1]*s[2, 2]^2*s[3, 3]^2+6*s[1, 2]*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]+24*s[1, 2]*s[2, 1]*s[2, 2]*s[3, 3]^3-36*s[1, 2]*s[2, 1]*s[2, 3]^2*s[3, 2]^2-60*s[1, 2]*s[2, 1]*s[2, 3]*s[3, 2]*s[3, 3]^2-12*s[1, 2]*s[2, 1]*s[3, 3]^4+12*s[1, 2]*s[2, 2]^3*s[2, 3]*s[3, 1]-18*s[1, 2]*s[2, 2]^2*s[2, 3]*s[3, 1]*s[3, 3]+54*s[1, 2]*s[2, 2]*s[2, 3]^2*s[3, 1]*s[3, 2]-18*s[1, 2]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 3]^2+54*s[1, 2]*s[2, 3]^2*s[3, 1]*s[3, 2]*s[3, 3]+12*s[1, 2]*s[2, 3]*s[3, 1]*s[3, 3]^3-12*s[1, 3]^3*s[3, 1]^3+81*s[1, 3]^2*s[2, 1]^2*s[3, 2]^2-108*s[1, 3]^2*s[2, 1]*s[2, 2]*s[3, 1]*s[3, 2]+54*s[1, 3]^2*s[2, 1]*s[3, 1]*s[3, 2]*s[3, 3]+24*s[1, 3]^2*s[2, 2]^2*s[3, 1]^2-24*s[1, 3]^2*s[2, 2]*s[3, 1]^2*s[3, 3]-36*s[1, 3]^2*s[2, 3]*s[3, 1]^2*s[3, 2]-3*s[1, 3]^2*s[3, 1]^2*s[3, 3]^2+12*s[1, 3]*s[2, 1]*s[2, 2]^3*s[3, 2]-18*s[1, 3]*s[2, 1]*s[2, 2]^2*s[3, 2]*s[3, 3]+54*s[1, 3]*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 2]^2-18*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 2]*s[3, 3]^2+54*s[1, 3]*s[2, 1]*s[2, 3]*s[3, 2]^2*s[3, 3]+12*s[1, 3]*s[2, 1]*s[3, 2]*s[3, 3]^3-12*s[1, 3]*s[2, 2]^4*s[3, 1]+24*s[1, 3]*s[2, 2]^3*s[3, 1]*s[3, 3]-60*s[1, 3]*s[2, 2]^2*s[2, 3]*s[3, 1]*s[3, 2]-6*s[1, 3]*s[2, 2]^2*s[3, 1]*s[3, 3]^2+6*s[1, 3]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 2]*s[3, 3]-6*s[1, 3]*s[2, 2]*s[3, 1]*s[3, 3]^3-36*s[1, 3]*s[2, 3]^2*s[3, 1]*s[3, 2]^2-6*s[1, 3]*s[2, 3]*s[3, 1]*s[3, 2]*s[3, 3]^2-3*s[2, 2]^4*s[3, 3]^2+6*s[2, 2]^3*s[2, 3]*s[3, 2]*s[3, 3]+6*s[2, 2]^3*s[3, 3]^3-3*s[2, 2]^2*s[2, 3]^2*s[3, 2]^2-24*s[2, 2]^2*s[2, 3]*s[3, 2]*s[3, 3]^2-3*s[2, 2]^2*s[3, 3]^4+30*s[2, 2]*s[2, 3]^2*s[3, 2]^2*s[3, 3]+6*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]^3-12*s[2, 3]^3*s[3, 2]^3-3*s[2, 3]^2*s[3, 2]^2*s[3, 3]^2))^(1/3)+((1/3)*s[1, 1]*s[2, 2]+(1/3)*s[3, 3]*s[1, 1]-s[1, 2]*s[2, 1]-s[1, 3]*s[3, 1]+(1/3)*s[3, 3]*s[2, 2]-s[2, 3]*s[3, 2]-(1/3)*s[1, 1]^2-(1/3)*s[2, 2]^2-(1/3)*s[3, 3]^2)/(8*s[1, 1]^3-12*s[1, 1]^2*s[2, 2]-12*s[1, 1]^2*s[3, 3]+36*s[1, 1]*s[1, 2]*s[2, 1]+36*s[1, 1]*s[1, 3]*s[3, 1]-12*s[1, 1]*s[2, 2]^2+48*s[1, 1]*s[2, 2]*s[3, 3]-72*s[1, 1]*s[2, 3]*s[3, 2]-12*s[1, 1]*s[3, 3]^2+36*s[1, 2]*s[2, 1]*s[2, 2]-72*s[1, 2]*s[2, 1]*s[3, 3]+108*s[1, 2]*s[2, 3]*s[3, 1]+108*s[1, 3]*s[2, 1]*s[3, 2]-72*s[1, 3]*s[2, 2]*s[3, 1]+36*s[1, 3]*s[3, 1]*s[3, 3]+8*s[2, 2]^3-12*s[2, 2]^2*s[3, 3]+36*s[2, 2]*s[2, 3]*s[3, 2]-12*s[2, 2]*s[3, 3]^2+36*s[2, 3]*s[3, 2]*s[3, 3]+8*s[3, 3]^3+12*sqrt(-3*s[1, 1]^4*s[2, 2]^2+6*s[1, 1]^4*s[2, 2]*s[3, 3]-12*s[1, 1]^4*s[2, 3]*s[3, 2]-3*s[1, 1]^4*s[3, 3]^2+6*s[1, 1]^3*s[1, 2]*s[2, 1]*s[2, 2]-6*s[1, 1]^3*s[1, 2]*s[2, 1]*s[3, 3]+12*s[1, 1]^3*s[1, 2]*s[2, 3]*s[3, 1]+12*s[1, 1]^3*s[1, 3]*s[2, 1]*s[3, 2]-6*s[1, 1]^3*s[1, 3]*s[2, 2]*s[3, 1]+6*s[1, 1]^3*s[1, 3]*s[3, 1]*s[3, 3]+6*s[1, 1]^3*s[2, 2]^3-6*s[1, 1]^3*s[2, 2]^2*s[3, 3]+24*s[1, 1]^3*s[2, 2]*s[2, 3]*s[3, 2]-6*s[1, 1]^3*s[2, 2]*s[3, 3]^2+24*s[1, 1]^3*s[2, 3]*s[3, 2]*s[3, 3]+6*s[1, 1]^3*s[3, 3]^3-3*s[1, 1]^2*s[1, 2]^2*s[2, 1]^2-6*s[1, 1]^2*s[1, 2]*s[1, 3]*s[2, 1]*s[3, 1]-24*s[1, 1]^2*s[1, 2]*s[2, 1]*s[2, 2]^2+30*s[1, 1]^2*s[1, 2]*s[2, 1]*s[2, 2]*s[3, 3]-60*s[1, 1]^2*s[1, 2]*s[2, 1]*s[2, 3]*s[3, 2]-6*s[1, 1]^2*s[1, 2]*s[2, 1]*s[3, 3]^2-18*s[1, 1]^2*s[1, 2]*s[2, 2]*s[2, 3]*s[3, 1]-18*s[1, 1]^2*s[1, 2]*s[2, 3]*s[3, 1]*s[3, 3]-3*s[1, 1]^2*s[1, 3]^2*s[3, 1]^2-18*s[1, 1]^2*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 2]-18*s[1, 1]^2*s[1, 3]*s[2, 1]*s[3, 2]*s[3, 3]-6*s[1, 1]^2*s[1, 3]*s[2, 2]^2*s[3, 1]+30*s[1, 1]^2*s[1, 3]*s[2, 2]*s[3, 1]*s[3, 3]-60*s[1, 1]^2*s[1, 3]*s[2, 3]*s[3, 1]*s[3, 2]-24*s[1, 1]^2*s[1, 3]*s[3, 1]*s[3, 3]^2-3*s[1, 1]^2*s[2, 2]^4-6*s[1, 1]^2*s[2, 2]^3*s[3, 3]-6*s[1, 1]^2*s[2, 2]^2*s[2, 3]*s[3, 2]+18*s[1, 1]^2*s[2, 2]^2*s[3, 3]^2-60*s[1, 1]^2*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]-6*s[1, 1]^2*s[2, 2]*s[3, 3]^3+24*s[1, 1]^2*s[2, 3]^2*s[3, 2]^2-6*s[1, 1]^2*s[2, 3]*s[3, 2]*s[3, 3]^2-3*s[1, 1]^2*s[3, 3]^4+30*s[1, 1]*s[1, 2]^2*s[2, 1]^2*s[2, 2]-24*s[1, 1]*s[1, 2]^2*s[2, 1]^2*s[3, 3]+54*s[1, 1]*s[1, 2]^2*s[2, 1]*s[2, 3]*s[3, 1]+54*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 1]^2*s[3, 2]+6*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 1]+6*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 1]*s[3, 1]*s[3, 3]+54*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 3]*s[3, 1]^2+6*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]^3+30*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]^2*s[3, 3]+6*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 2]-60*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]*s[3, 3]^2+114*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 3]*s[3, 2]*s[3, 3]+24*s[1, 1]*s[1, 2]*s[2, 1]*s[3, 3]^3-18*s[1, 1]*s[1, 2]*s[2, 2]^2*s[2, 3]*s[3, 1]+72*s[1, 1]*s[1, 2]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 3]-108*s[1, 1]*s[1, 2]*s[2, 3]^2*s[3, 1]*s[3, 2]-18*s[1, 1]*s[1, 2]*s[2, 3]*s[3, 1]*s[3, 3]^2+54*s[1, 1]*s[1, 3]^2*s[2, 1]*s[3, 1]*s[3, 2]-24*s[1, 1]*s[1, 3]^2*s[2, 2]*s[3, 1]^2+30*s[1, 1]*s[1, 3]^2*s[3, 1]^2*s[3, 3]-18*s[1, 1]*s[1, 3]*s[2, 1]*s[2, 2]^2*s[3, 2]+72*s[1, 1]*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 2]*s[3, 3]-108*s[1, 1]*s[1, 3]*s[2, 1]*s[2, 3]*s[3, 2]^2-18*s[1, 1]*s[1, 3]*s[2, 1]*s[3, 2]*s[3, 3]^2+24*s[1, 1]*s[1, 3]*s[2, 2]^3*s[3, 1]-60*s[1, 1]*s[1, 3]*s[2, 2]^2*s[3, 1]*s[3, 3]+114*s[1, 1]*s[1, 3]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 2]+30*s[1, 1]*s[1, 3]*s[2, 2]*s[3, 1]*s[3, 3]^2+6*s[1, 1]*s[1, 3]*s[2, 3]*s[3, 1]*s[3, 2]*s[3, 3]+6*s[1, 1]*s[1, 3]*s[3, 1]*s[3, 3]^3+6*s[1, 1]*s[2, 2]^4*s[3, 3]-6*s[1, 1]*s[2, 2]^3*s[2, 3]*s[3, 2]-6*s[1, 1]*s[2, 2]^3*s[3, 3]^2+30*s[1, 1]*s[2, 2]^2*s[2, 3]*s[3, 2]*s[3, 3]-6*s[1, 1]*s[2, 2]^2*s[3, 3]^3-24*s[1, 1]*s[2, 2]*s[2, 3]^2*s[3, 2]^2+30*s[1, 1]*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]^2+6*s[1, 1]*s[2, 2]*s[3, 3]^4-24*s[1, 1]*s[2, 3]^2*s[3, 2]^2*s[3, 3]-6*s[1, 1]*s[2, 3]*s[3, 2]*s[3, 3]^3-12*s[1, 2]^3*s[2, 1]^3-36*s[1, 2]^2*s[1, 3]*s[2, 1]^2*s[3, 1]-3*s[1, 2]^2*s[2, 1]^2*s[2, 2]^2-24*s[1, 2]^2*s[2, 1]^2*s[2, 2]*s[3, 3]-36*s[1, 2]^2*s[2, 1]^2*s[2, 3]*s[3, 2]+24*s[1, 2]^2*s[2, 1]^2*s[3, 3]^2+54*s[1, 2]^2*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 1]-108*s[1, 2]^2*s[2, 1]*s[2, 3]*s[3, 1]*s[3, 3]+81*s[1, 2]^2*s[2, 3]^2*s[3, 1]^2-36*s[1, 2]*s[1, 3]^2*s[2, 1]*s[3, 1]^2+54*s[1, 2]*s[1, 3]*s[2, 1]^2*s[2, 2]*s[3, 2]-108*s[1, 2]*s[1, 3]*s[2, 1]^2*s[3, 2]*s[3, 3]-60*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 2]^2*s[3, 1]+114*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 1]*s[3, 3]+90*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 3]*s[3, 1]*s[3, 2]-60*s[1, 2]*s[1, 3]*s[2, 1]*s[3, 1]*s[3, 3]^2-108*s[1, 2]*s[1, 3]*s[2, 2]*s[2, 3]*s[3, 1]^2+54*s[1, 2]*s[1, 3]*s[2, 3]*s[3, 1]^2*s[3, 3]-6*s[1, 2]*s[2, 1]*s[2, 2]^3*s[3, 3]-6*s[1, 2]*s[2, 1]*s[2, 2]^2*s[2, 3]*s[3, 2]-6*s[1, 2]*s[2, 1]*s[2, 2]^2*s[3, 3]^2+6*s[1, 2]*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]+24*s[1, 2]*s[2, 1]*s[2, 2]*s[3, 3]^3-36*s[1, 2]*s[2, 1]*s[2, 3]^2*s[3, 2]^2-60*s[1, 2]*s[2, 1]*s[2, 3]*s[3, 2]*s[3, 3]^2-12*s[1, 2]*s[2, 1]*s[3, 3]^4+12*s[1, 2]*s[2, 2]^3*s[2, 3]*s[3, 1]-18*s[1, 2]*s[2, 2]^2*s[2, 3]*s[3, 1]*s[3, 3]+54*s[1, 2]*s[2, 2]*s[2, 3]^2*s[3, 1]*s[3, 2]-18*s[1, 2]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 3]^2+54*s[1, 2]*s[2, 3]^2*s[3, 1]*s[3, 2]*s[3, 3]+12*s[1, 2]*s[2, 3]*s[3, 1]*s[3, 3]^3-12*s[1, 3]^3*s[3, 1]^3+81*s[1, 3]^2*s[2, 1]^2*s[3, 2]^2-108*s[1, 3]^2*s[2, 1]*s[2, 2]*s[3, 1]*s[3, 2]+54*s[1, 3]^2*s[2, 1]*s[3, 1]*s[3, 2]*s[3, 3]+24*s[1, 3]^2*s[2, 2]^2*s[3, 1]^2-24*s[1, 3]^2*s[2, 2]*s[3, 1]^2*s[3, 3]-36*s[1, 3]^2*s[2, 3]*s[3, 1]^2*s[3, 2]-3*s[1, 3]^2*s[3, 1]^2*s[3, 3]^2+12*s[1, 3]*s[2, 1]*s[2, 2]^3*s[3, 2]-18*s[1, 3]*s[2, 1]*s[2, 2]^2*s[3, 2]*s[3, 3]+54*s[1, 3]*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 2]^2-18*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 2]*s[3, 3]^2+54*s[1, 3]*s[2, 1]*s[2, 3]*s[3, 2]^2*s[3, 3]+12*s[1, 3]*s[2, 1]*s[3, 2]*s[3, 3]^3-12*s[1, 3]*s[2, 2]^4*s[3, 1]+24*s[1, 3]*s[2, 2]^3*s[3, 1]*s[3, 3]-60*s[1, 3]*s[2, 2]^2*s[2, 3]*s[3, 1]*s[3, 2]-6*s[1, 3]*s[2, 2]^2*s[3, 1]*s[3, 3]^2+6*s[1, 3]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 2]*s[3, 3]-6*s[1, 3]*s[2, 2]*s[3, 1]*s[3, 3]^3-36*s[1, 3]*s[2, 3]^2*s[3, 1]*s[3, 2]^2-6*s[1, 3]*s[2, 3]*s[3, 1]*s[3, 2]*s[3, 3]^2-3*s[2, 2]^4*s[3, 3]^2+6*s[2, 2]^3*s[2, 3]*s[3, 2]*s[3, 3]+6*s[2, 2]^3*s[3, 3]^3-3*s[2, 2]^2*s[2, 3]^2*s[3, 2]^2-24*s[2, 2]^2*s[2, 3]*s[3, 2]*s[3, 3]^2-3*s[2, 2]^2*s[3, 3]^4+30*s[2, 2]*s[2, 3]^2*s[3, 2]^2*s[3, 3]+6*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]^3-12*s[2, 3]^3*s[3, 2]^3-3*s[2, 3]^2*s[3, 2]^2*s[3, 3]^2))^(1/3)+(1/3)*s[3, 3]+(1/3)*s[2, 2]+(1/3)*s[1, 1]+((1/2)*I)*sqrt(3)*((1/6)*(8*s[1, 1]^3-12*s[1, 1]^2*s[2, 2]-12*s[1, 1]^2*s[3, 3]+36*s[1, 1]*s[1, 2]*s[2, 1]+36*s[1, 1]*s[1, 3]*s[3, 1]-12*s[1, 1]*s[2, 2]^2+48*s[1, 1]*s[2, 2]*s[3, 3]-72*s[1, 1]*s[2, 3]*s[3, 2]-12*s[1, 1]*s[3, 3]^2+36*s[1, 2]*s[2, 1]*s[2, 2]-72*s[1, 2]*s[2, 1]*s[3, 3]+108*s[1, 2]*s[2, 3]*s[3, 1]+108*s[1, 3]*s[2, 1]*s[3, 2]-72*s[1, 3]*s[2, 2]*s[3, 1]+36*s[1, 3]*s[3, 1]*s[3, 3]+8*s[2, 2]^3-12*s[2, 2]^2*s[3, 3]+36*s[2, 2]*s[2, 3]*s[3, 2]-12*s[2, 2]*s[3, 3]^2+36*s[2, 3]*s[3, 2]*s[3, 3]+8*s[3, 3]^3+12*sqrt(-3*s[1, 1]^4*s[2, 2]^2+6*s[1, 1]^4*s[2, 2]*s[3, 3]-12*s[1, 1]^4*s[2, 3]*s[3, 2]-3*s[1, 1]^4*s[3, 3]^2+6*s[1, 1]^3*s[1, 2]*s[2, 1]*s[2, 2]-6*s[1, 1]^3*s[1, 2]*s[2, 1]*s[3, 3]+12*s[1, 1]^3*s[1, 2]*s[2, 3]*s[3, 1]+12*s[1, 1]^3*s[1, 3]*s[2, 1]*s[3, 2]-6*s[1, 1]^3*s[1, 3]*s[2, 2]*s[3, 1]+6*s[1, 1]^3*s[1, 3]*s[3, 1]*s[3, 3]+6*s[1, 1]^3*s[2, 2]^3-6*s[1, 1]^3*s[2, 2]^2*s[3, 3]+24*s[1, 1]^3*s[2, 2]*s[2, 3]*s[3, 2]-6*s[1, 1]^3*s[2, 2]*s[3, 3]^2+24*s[1, 1]^3*s[2, 3]*s[3, 2]*s[3, 3]+6*s[1, 1]^3*s[3, 3]^3-3*s[1, 1]^2*s[1, 2]^2*s[2, 1]^2-6*s[1, 1]^2*s[1, 2]*s[1, 3]*s[2, 1]*s[3, 1]-24*s[1, 1]^2*s[1, 2]*s[2, 1]*s[2, 2]^2+30*s[1, 1]^2*s[1, 2]*s[2, 1]*s[2, 2]*s[3, 3]-60*s[1, 1]^2*s[1, 2]*s[2, 1]*s[2, 3]*s[3, 2]-6*s[1, 1]^2*s[1, 2]*s[2, 1]*s[3, 3]^2-18*s[1, 1]^2*s[1, 2]*s[2, 2]*s[2, 3]*s[3, 1]-18*s[1, 1]^2*s[1, 2]*s[2, 3]*s[3, 1]*s[3, 3]-3*s[1, 1]^2*s[1, 3]^2*s[3, 1]^2-18*s[1, 1]^2*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 2]-18*s[1, 1]^2*s[1, 3]*s[2, 1]*s[3, 2]*s[3, 3]-6*s[1, 1]^2*s[1, 3]*s[2, 2]^2*s[3, 1]+30*s[1, 1]^2*s[1, 3]*s[2, 2]*s[3, 1]*s[3, 3]-60*s[1, 1]^2*s[1, 3]*s[2, 3]*s[3, 1]*s[3, 2]-24*s[1, 1]^2*s[1, 3]*s[3, 1]*s[3, 3]^2-3*s[1, 1]^2*s[2, 2]^4-6*s[1, 1]^2*s[2, 2]^3*s[3, 3]-6*s[1, 1]^2*s[2, 2]^2*s[2, 3]*s[3, 2]+18*s[1, 1]^2*s[2, 2]^2*s[3, 3]^2-60*s[1, 1]^2*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]-6*s[1, 1]^2*s[2, 2]*s[3, 3]^3+24*s[1, 1]^2*s[2, 3]^2*s[3, 2]^2-6*s[1, 1]^2*s[2, 3]*s[3, 2]*s[3, 3]^2-3*s[1, 1]^2*s[3, 3]^4+30*s[1, 1]*s[1, 2]^2*s[2, 1]^2*s[2, 2]-24*s[1, 1]*s[1, 2]^2*s[2, 1]^2*s[3, 3]+54*s[1, 1]*s[1, 2]^2*s[2, 1]*s[2, 3]*s[3, 1]+54*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 1]^2*s[3, 2]+6*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 1]+6*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 1]*s[3, 1]*s[3, 3]+54*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 3]*s[3, 1]^2+6*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]^3+30*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]^2*s[3, 3]+6*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 2]-60*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]*s[3, 3]^2+114*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 3]*s[3, 2]*s[3, 3]+24*s[1, 1]*s[1, 2]*s[2, 1]*s[3, 3]^3-18*s[1, 1]*s[1, 2]*s[2, 2]^2*s[2, 3]*s[3, 1]+72*s[1, 1]*s[1, 2]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 3]-108*s[1, 1]*s[1, 2]*s[2, 3]^2*s[3, 1]*s[3, 2]-18*s[1, 1]*s[1, 2]*s[2, 3]*s[3, 1]*s[3, 3]^2+54*s[1, 1]*s[1, 3]^2*s[2, 1]*s[3, 1]*s[3, 2]-24*s[1, 1]*s[1, 3]^2*s[2, 2]*s[3, 1]^2+30*s[1, 1]*s[1, 3]^2*s[3, 1]^2*s[3, 3]-18*s[1, 1]*s[1, 3]*s[2, 1]*s[2, 2]^2*s[3, 2]+72*s[1, 1]*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 2]*s[3, 3]-108*s[1, 1]*s[1, 3]*s[2, 1]*s[2, 3]*s[3, 2]^2-18*s[1, 1]*s[1, 3]*s[2, 1]*s[3, 2]*s[3, 3]^2+24*s[1, 1]*s[1, 3]*s[2, 2]^3*s[3, 1]-60*s[1, 1]*s[1, 3]*s[2, 2]^2*s[3, 1]*s[3, 3]+114*s[1, 1]*s[1, 3]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 2]+30*s[1, 1]*s[1, 3]*s[2, 2]*s[3, 1]*s[3, 3]^2+6*s[1, 1]*s[1, 3]*s[2, 3]*s[3, 1]*s[3, 2]*s[3, 3]+6*s[1, 1]*s[1, 3]*s[3, 1]*s[3, 3]^3+6*s[1, 1]*s[2, 2]^4*s[3, 3]-6*s[1, 1]*s[2, 2]^3*s[2, 3]*s[3, 2]-6*s[1, 1]*s[2, 2]^3*s[3, 3]^2+30*s[1, 1]*s[2, 2]^2*s[2, 3]*s[3, 2]*s[3, 3]-6*s[1, 1]*s[2, 2]^2*s[3, 3]^3-24*s[1, 1]*s[2, 2]*s[2, 3]^2*s[3, 2]^2+30*s[1, 1]*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]^2+6*s[1, 1]*s[2, 2]*s[3, 3]^4-24*s[1, 1]*s[2, 3]^2*s[3, 2]^2*s[3, 3]-6*s[1, 1]*s[2, 3]*s[3, 2]*s[3, 3]^3-12*s[1, 2]^3*s[2, 1]^3-36*s[1, 2]^2*s[1, 3]*s[2, 1]^2*s[3, 1]-3*s[1, 2]^2*s[2, 1]^2*s[2, 2]^2-24*s[1, 2]^2*s[2, 1]^2*s[2, 2]*s[3, 3]-36*s[1, 2]^2*s[2, 1]^2*s[2, 3]*s[3, 2]+24*s[1, 2]^2*s[2, 1]^2*s[3, 3]^2+54*s[1, 2]^2*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 1]-108*s[1, 2]^2*s[2, 1]*s[2, 3]*s[3, 1]*s[3, 3]+81*s[1, 2]^2*s[2, 3]^2*s[3, 1]^2-36*s[1, 2]*s[1, 3]^2*s[2, 1]*s[3, 1]^2+54*s[1, 2]*s[1, 3]*s[2, 1]^2*s[2, 2]*s[3, 2]-108*s[1, 2]*s[1, 3]*s[2, 1]^2*s[3, 2]*s[3, 3]-60*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 2]^2*s[3, 1]+114*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 1]*s[3, 3]+90*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 3]*s[3, 1]*s[3, 2]-60*s[1, 2]*s[1, 3]*s[2, 1]*s[3, 1]*s[3, 3]^2-108*s[1, 2]*s[1, 3]*s[2, 2]*s[2, 3]*s[3, 1]^2+54*s[1, 2]*s[1, 3]*s[2, 3]*s[3, 1]^2*s[3, 3]-6*s[1, 2]*s[2, 1]*s[2, 2]^3*s[3, 3]-6*s[1, 2]*s[2, 1]*s[2, 2]^2*s[2, 3]*s[3, 2]-6*s[1, 2]*s[2, 1]*s[2, 2]^2*s[3, 3]^2+6*s[1, 2]*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]+24*s[1, 2]*s[2, 1]*s[2, 2]*s[3, 3]^3-36*s[1, 2]*s[2, 1]*s[2, 3]^2*s[3, 2]^2-60*s[1, 2]*s[2, 1]*s[2, 3]*s[3, 2]*s[3, 3]^2-12*s[1, 2]*s[2, 1]*s[3, 3]^4+12*s[1, 2]*s[2, 2]^3*s[2, 3]*s[3, 1]-18*s[1, 2]*s[2, 2]^2*s[2, 3]*s[3, 1]*s[3, 3]+54*s[1, 2]*s[2, 2]*s[2, 3]^2*s[3, 1]*s[3, 2]-18*s[1, 2]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 3]^2+54*s[1, 2]*s[2, 3]^2*s[3, 1]*s[3, 2]*s[3, 3]+12*s[1, 2]*s[2, 3]*s[3, 1]*s[3, 3]^3-12*s[1, 3]^3*s[3, 1]^3+81*s[1, 3]^2*s[2, 1]^2*s[3, 2]^2-108*s[1, 3]^2*s[2, 1]*s[2, 2]*s[3, 1]*s[3, 2]+54*s[1, 3]^2*s[2, 1]*s[3, 1]*s[3, 2]*s[3, 3]+24*s[1, 3]^2*s[2, 2]^2*s[3, 1]^2-24*s[1, 3]^2*s[2, 2]*s[3, 1]^2*s[3, 3]-36*s[1, 3]^2*s[2, 3]*s[3, 1]^2*s[3, 2]-3*s[1, 3]^2*s[3, 1]^2*s[3, 3]^2+12*s[1, 3]*s[2, 1]*s[2, 2]^3*s[3, 2]-18*s[1, 3]*s[2, 1]*s[2, 2]^2*s[3, 2]*s[3, 3]+54*s[1, 3]*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 2]^2-18*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 2]*s[3, 3]^2+54*s[1, 3]*s[2, 1]*s[2, 3]*s[3, 2]^2*s[3, 3]+12*s[1, 3]*s[2, 1]*s[3, 2]*s[3, 3]^3-12*s[1, 3]*s[2, 2]^4*s[3, 1]+24*s[1, 3]*s[2, 2]^3*s[3, 1]*s[3, 3]-60*s[1, 3]*s[2, 2]^2*s[2, 3]*s[3, 1]*s[3, 2]-6*s[1, 3]*s[2, 2]^2*s[3, 1]*s[3, 3]^2+6*s[1, 3]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 2]*s[3, 3]-6*s[1, 3]*s[2, 2]*s[3, 1]*s[3, 3]^3-36*s[1, 3]*s[2, 3]^2*s[3, 1]*s[3, 2]^2-6*s[1, 3]*s[2, 3]*s[3, 1]*s[3, 2]*s[3, 3]^2-3*s[2, 2]^4*s[3, 3]^2+6*s[2, 2]^3*s[2, 3]*s[3, 2]*s[3, 3]+6*s[2, 2]^3*s[3, 3]^3-3*s[2, 2]^2*s[2, 3]^2*s[3, 2]^2-24*s[2, 2]^2*s[2, 3]*s[3, 2]*s[3, 3]^2-3*s[2, 2]^2*s[3, 3]^4+30*s[2, 2]*s[2, 3]^2*s[3, 2]^2*s[3, 3]+6*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]^3-12*s[2, 3]^3*s[3, 2]^3-3*s[2, 3]^2*s[3, 2]^2*s[3, 3]^2))^(1/3)+((2/3)*s[1, 1]*s[2, 2]+(2/3)*s[3, 3]*s[1, 1]-2*s[1, 2]*s[2, 1]-2*s[1, 3]*s[3, 1]+(2/3)*s[3, 3]*s[2, 2]-2*s[2, 3]*s[3, 2]-(2/3)*s[1, 1]^2-(2/3)*s[2, 2]^2-(2/3)*s[3, 3]^2)/(8*s[1, 1]^3-12*s[1, 1]^2*s[2, 2]-12*s[1, 1]^2*s[3, 3]+36*s[1, 1]*s[1, 2]*s[2, 1]+36*s[1, 1]*s[1, 3]*s[3, 1]-12*s[1, 1]*s[2, 2]^2+48*s[1, 1]*s[2, 2]*s[3, 3]-72*s[1, 1]*s[2, 3]*s[3, 2]-12*s[1, 1]*s[3, 3]^2+36*s[1, 2]*s[2, 1]*s[2, 2]-72*s[1, 2]*s[2, 1]*s[3, 3]+108*s[1, 2]*s[2, 3]*s[3, 1]+108*s[1, 3]*s[2, 1]*s[3, 2]-72*s[1, 3]*s[2, 2]*s[3, 1]+36*s[1, 3]*s[3, 1]*s[3, 3]+8*s[2, 2]^3-12*s[2, 2]^2*s[3, 3]+36*s[2, 2]*s[2, 3]*s[3, 2]-12*s[2, 2]*s[3, 3]^2+36*s[2, 3]*s[3, 2]*s[3, 3]+8*s[3, 3]^3+12*sqrt(-3*s[1, 1]^4*s[2, 2]^2+6*s[1, 1]^4*s[2, 2]*s[3, 3]-12*s[1, 1]^4*s[2, 3]*s[3, 2]-3*s[1, 1]^4*s[3, 3]^2+6*s[1, 1]^3*s[1, 2]*s[2, 1]*s[2, 2]-6*s[1, 1]^3*s[1, 2]*s[2, 1]*s[3, 3]+12*s[1, 1]^3*s[1, 2]*s[2, 3]*s[3, 1]+12*s[1, 1]^3*s[1, 3]*s[2, 1]*s[3, 2]-6*s[1, 1]^3*s[1, 3]*s[2, 2]*s[3, 1]+6*s[1, 1]^3*s[1, 3]*s[3, 1]*s[3, 3]+6*s[1, 1]^3*s[2, 2]^3-6*s[1, 1]^3*s[2, 2]^2*s[3, 3]+24*s[1, 1]^3*s[2, 2]*s[2, 3]*s[3, 2]-6*s[1, 1]^3*s[2, 2]*s[3, 3]^2+24*s[1, 1]^3*s[2, 3]*s[3, 2]*s[3, 3]+6*s[1, 1]^3*s[3, 3]^3-3*s[1, 1]^2*s[1, 2]^2*s[2, 1]^2-6*s[1, 1]^2*s[1, 2]*s[1, 3]*s[2, 1]*s[3, 1]-24*s[1, 1]^2*s[1, 2]*s[2, 1]*s[2, 2]^2+30*s[1, 1]^2*s[1, 2]*s[2, 1]*s[2, 2]*s[3, 3]-60*s[1, 1]^2*s[1, 2]*s[2, 1]*s[2, 3]*s[3, 2]-6*s[1, 1]^2*s[1, 2]*s[2, 1]*s[3, 3]^2-18*s[1, 1]^2*s[1, 2]*s[2, 2]*s[2, 3]*s[3, 1]-18*s[1, 1]^2*s[1, 2]*s[2, 3]*s[3, 1]*s[3, 3]-3*s[1, 1]^2*s[1, 3]^2*s[3, 1]^2-18*s[1, 1]^2*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 2]-18*s[1, 1]^2*s[1, 3]*s[2, 1]*s[3, 2]*s[3, 3]-6*s[1, 1]^2*s[1, 3]*s[2, 2]^2*s[3, 1]+30*s[1, 1]^2*s[1, 3]*s[2, 2]*s[3, 1]*s[3, 3]-60*s[1, 1]^2*s[1, 3]*s[2, 3]*s[3, 1]*s[3, 2]-24*s[1, 1]^2*s[1, 3]*s[3, 1]*s[3, 3]^2-3*s[1, 1]^2*s[2, 2]^4-6*s[1, 1]^2*s[2, 2]^3*s[3, 3]-6*s[1, 1]^2*s[2, 2]^2*s[2, 3]*s[3, 2]+18*s[1, 1]^2*s[2, 2]^2*s[3, 3]^2-60*s[1, 1]^2*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]-6*s[1, 1]^2*s[2, 2]*s[3, 3]^3+24*s[1, 1]^2*s[2, 3]^2*s[3, 2]^2-6*s[1, 1]^2*s[2, 3]*s[3, 2]*s[3, 3]^2-3*s[1, 1]^2*s[3, 3]^4+30*s[1, 1]*s[1, 2]^2*s[2, 1]^2*s[2, 2]-24*s[1, 1]*s[1, 2]^2*s[2, 1]^2*s[3, 3]+54*s[1, 1]*s[1, 2]^2*s[2, 1]*s[2, 3]*s[3, 1]+54*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 1]^2*s[3, 2]+6*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 1]+6*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 1]*s[3, 1]*s[3, 3]+54*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 3]*s[3, 1]^2+6*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]^3+30*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]^2*s[3, 3]+6*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 2]-60*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]*s[3, 3]^2+114*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 3]*s[3, 2]*s[3, 3]+24*s[1, 1]*s[1, 2]*s[2, 1]*s[3, 3]^3-18*s[1, 1]*s[1, 2]*s[2, 2]^2*s[2, 3]*s[3, 1]+72*s[1, 1]*s[1, 2]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 3]-108*s[1, 1]*s[1, 2]*s[2, 3]^2*s[3, 1]*s[3, 2]-18*s[1, 1]*s[1, 2]*s[2, 3]*s[3, 1]*s[3, 3]^2+54*s[1, 1]*s[1, 3]^2*s[2, 1]*s[3, 1]*s[3, 2]-24*s[1, 1]*s[1, 3]^2*s[2, 2]*s[3, 1]^2+30*s[1, 1]*s[1, 3]^2*s[3, 1]^2*s[3, 3]-18*s[1, 1]*s[1, 3]*s[2, 1]*s[2, 2]^2*s[3, 2]+72*s[1, 1]*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 2]*s[3, 3]-108*s[1, 1]*s[1, 3]*s[2, 1]*s[2, 3]*s[3, 2]^2-18*s[1, 1]*s[1, 3]*s[2, 1]*s[3, 2]*s[3, 3]^2+24*s[1, 1]*s[1, 3]*s[2, 2]^3*s[3, 1]-60*s[1, 1]*s[1, 3]*s[2, 2]^2*s[3, 1]*s[3, 3]+114*s[1, 1]*s[1, 3]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 2]+30*s[1, 1]*s[1, 3]*s[2, 2]*s[3, 1]*s[3, 3]^2+6*s[1, 1]*s[1, 3]*s[2, 3]*s[3, 1]*s[3, 2]*s[3, 3]+6*s[1, 1]*s[1, 3]*s[3, 1]*s[3, 3]^3+6*s[1, 1]*s[2, 2]^4*s[3, 3]-6*s[1, 1]*s[2, 2]^3*s[2, 3]*s[3, 2]-6*s[1, 1]*s[2, 2]^3*s[3, 3]^2+30*s[1, 1]*s[2, 2]^2*s[2, 3]*s[3, 2]*s[3, 3]-6*s[1, 1]*s[2, 2]^2*s[3, 3]^3-24*s[1, 1]*s[2, 2]*s[2, 3]^2*s[3, 2]^2+30*s[1, 1]*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]^2+6*s[1, 1]*s[2, 2]*s[3, 3]^4-24*s[1, 1]*s[2, 3]^2*s[3, 2]^2*s[3, 3]-6*s[1, 1]*s[2, 3]*s[3, 2]*s[3, 3]^3-12*s[1, 2]^3*s[2, 1]^3-36*s[1, 2]^2*s[1, 3]*s[2, 1]^2*s[3, 1]-3*s[1, 2]^2*s[2, 1]^2*s[2, 2]^2-24*s[1, 2]^2*s[2, 1]^2*s[2, 2]*s[3, 3]-36*s[1, 2]^2*s[2, 1]^2*s[2, 3]*s[3, 2]+24*s[1, 2]^2*s[2, 1]^2*s[3, 3]^2+54*s[1, 2]^2*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 1]-108*s[1, 2]^2*s[2, 1]*s[2, 3]*s[3, 1]*s[3, 3]+81*s[1, 2]^2*s[2, 3]^2*s[3, 1]^2-36*s[1, 2]*s[1, 3]^2*s[2, 1]*s[3, 1]^2+54*s[1, 2]*s[1, 3]*s[2, 1]^2*s[2, 2]*s[3, 2]-108*s[1, 2]*s[1, 3]*s[2, 1]^2*s[3, 2]*s[3, 3]-60*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 2]^2*s[3, 1]+114*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 1]*s[3, 3]+90*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 3]*s[3, 1]*s[3, 2]-60*s[1, 2]*s[1, 3]*s[2, 1]*s[3, 1]*s[3, 3]^2-108*s[1, 2]*s[1, 3]*s[2, 2]*s[2, 3]*s[3, 1]^2+54*s[1, 2]*s[1, 3]*s[2, 3]*s[3, 1]^2*s[3, 3]-6*s[1, 2]*s[2, 1]*s[2, 2]^3*s[3, 3]-6*s[1, 2]*s[2, 1]*s[2, 2]^2*s[2, 3]*s[3, 2]-6*s[1, 2]*s[2, 1]*s[2, 2]^2*s[3, 3]^2+6*s[1, 2]*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]+24*s[1, 2]*s[2, 1]*s[2, 2]*s[3, 3]^3-36*s[1, 2]*s[2, 1]*s[2, 3]^2*s[3, 2]^2-60*s[1, 2]*s[2, 1]*s[2, 3]*s[3, 2]*s[3, 3]^2-12*s[1, 2]*s[2, 1]*s[3, 3]^4+12*s[1, 2]*s[2, 2]^3*s[2, 3]*s[3, 1]-18*s[1, 2]*s[2, 2]^2*s[2, 3]*s[3, 1]*s[3, 3]+54*s[1, 2]*s[2, 2]*s[2, 3]^2*s[3, 1]*s[3, 2]-18*s[1, 2]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 3]^2+54*s[1, 2]*s[2, 3]^2*s[3, 1]*s[3, 2]*s[3, 3]+12*s[1, 2]*s[2, 3]*s[3, 1]*s[3, 3]^3-12*s[1, 3]^3*s[3, 1]^3+81*s[1, 3]^2*s[2, 1]^2*s[3, 2]^2-108*s[1, 3]^2*s[2, 1]*s[2, 2]*s[3, 1]*s[3, 2]+54*s[1, 3]^2*s[2, 1]*s[3, 1]*s[3, 2]*s[3, 3]+24*s[1, 3]^2*s[2, 2]^2*s[3, 1]^2-24*s[1, 3]^2*s[2, 2]*s[3, 1]^2*s[3, 3]-36*s[1, 3]^2*s[2, 3]*s[3, 1]^2*s[3, 2]-3*s[1, 3]^2*s[3, 1]^2*s[3, 3]^2+12*s[1, 3]*s[2, 1]*s[2, 2]^3*s[3, 2]-18*s[1, 3]*s[2, 1]*s[2, 2]^2*s[3, 2]*s[3, 3]+54*s[1, 3]*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 2]^2-18*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 2]*s[3, 3]^2+54*s[1, 3]*s[2, 1]*s[2, 3]*s[3, 2]^2*s[3, 3]+12*s[1, 3]*s[2, 1]*s[3, 2]*s[3, 3]^3-12*s[1, 3]*s[2, 2]^4*s[3, 1]+24*s[1, 3]*s[2, 2]^3*s[3, 1]*s[3, 3]-60*s[1, 3]*s[2, 2]^2*s[2, 3]*s[3, 1]*s[3, 2]-6*s[1, 3]*s[2, 2]^2*s[3, 1]*s[3, 3]^2+6*s[1, 3]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 2]*s[3, 3]-6*s[1, 3]*s[2, 2]*s[3, 1]*s[3, 3]^3-36*s[1, 3]*s[2, 3]^2*s[3, 1]*s[3, 2]^2-6*s[1, 3]*s[2, 3]*s[3, 1]*s[3, 2]*s[3, 3]^2-3*s[2, 2]^4*s[3, 3]^2+6*s[2, 2]^3*s[2, 3]*s[3, 2]*s[3, 3]+6*s[2, 2]^3*s[3, 3]^3-3*s[2, 2]^2*s[2, 3]^2*s[3, 2]^2-24*s[2, 2]^2*s[2, 3]*s[3, 2]*s[3, 3]^2-3*s[2, 2]^2*s[3, 3]^4+30*s[2, 2]*s[2, 3]^2*s[3, 2]^2*s[3, 3]+6*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]^3-12*s[2, 3]^3*s[3, 2]^3-3*s[2, 3]^2*s[3, 2]^2*s[3, 3]^2))^(1/3)), (3) = -(1/12)*(8*s[1, 1]^3-12*s[1, 1]^2*s[2, 2]-12*s[1, 1]^2*s[3, 3]+36*s[1, 1]*s[1, 2]*s[2, 1]+36*s[1, 1]*s[1, 3]*s[3, 1]-12*s[1, 1]*s[2, 2]^2+48*s[1, 1]*s[2, 2]*s[3, 3]-72*s[1, 1]*s[2, 3]*s[3, 2]-12*s[1, 1]*s[3, 3]^2+36*s[1, 2]*s[2, 1]*s[2, 2]-72*s[1, 2]*s[2, 1]*s[3, 3]+108*s[1, 2]*s[2, 3]*s[3, 1]+108*s[1, 3]*s[2, 1]*s[3, 2]-72*s[1, 3]*s[2, 2]*s[3, 1]+36*s[1, 3]*s[3, 1]*s[3, 3]+8*s[2, 2]^3-12*s[2, 2]^2*s[3, 3]+36*s[2, 2]*s[2, 3]*s[3, 2]-12*s[2, 2]*s[3, 3]^2+36*s[2, 3]*s[3, 2]*s[3, 3]+8*s[3, 3]^3+12*sqrt(-3*s[1, 1]^4*s[2, 2]^2+6*s[1, 1]^4*s[2, 2]*s[3, 3]-12*s[1, 1]^4*s[2, 3]*s[3, 2]-3*s[1, 1]^4*s[3, 3]^2+6*s[1, 1]^3*s[1, 2]*s[2, 1]*s[2, 2]-6*s[1, 1]^3*s[1, 2]*s[2, 1]*s[3, 3]+12*s[1, 1]^3*s[1, 2]*s[2, 3]*s[3, 1]+12*s[1, 1]^3*s[1, 3]*s[2, 1]*s[3, 2]-6*s[1, 1]^3*s[1, 3]*s[2, 2]*s[3, 1]+6*s[1, 1]^3*s[1, 3]*s[3, 1]*s[3, 3]+6*s[1, 1]^3*s[2, 2]^3-6*s[1, 1]^3*s[2, 2]^2*s[3, 3]+24*s[1, 1]^3*s[2, 2]*s[2, 3]*s[3, 2]-6*s[1, 1]^3*s[2, 2]*s[3, 3]^2+24*s[1, 1]^3*s[2, 3]*s[3, 2]*s[3, 3]+6*s[1, 1]^3*s[3, 3]^3-3*s[1, 1]^2*s[1, 2]^2*s[2, 1]^2-6*s[1, 1]^2*s[1, 2]*s[1, 3]*s[2, 1]*s[3, 1]-24*s[1, 1]^2*s[1, 2]*s[2, 1]*s[2, 2]^2+30*s[1, 1]^2*s[1, 2]*s[2, 1]*s[2, 2]*s[3, 3]-60*s[1, 1]^2*s[1, 2]*s[2, 1]*s[2, 3]*s[3, 2]-6*s[1, 1]^2*s[1, 2]*s[2, 1]*s[3, 3]^2-18*s[1, 1]^2*s[1, 2]*s[2, 2]*s[2, 3]*s[3, 1]-18*s[1, 1]^2*s[1, 2]*s[2, 3]*s[3, 1]*s[3, 3]-3*s[1, 1]^2*s[1, 3]^2*s[3, 1]^2-18*s[1, 1]^2*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 2]-18*s[1, 1]^2*s[1, 3]*s[2, 1]*s[3, 2]*s[3, 3]-6*s[1, 1]^2*s[1, 3]*s[2, 2]^2*s[3, 1]+30*s[1, 1]^2*s[1, 3]*s[2, 2]*s[3, 1]*s[3, 3]-60*s[1, 1]^2*s[1, 3]*s[2, 3]*s[3, 1]*s[3, 2]-24*s[1, 1]^2*s[1, 3]*s[3, 1]*s[3, 3]^2-3*s[1, 1]^2*s[2, 2]^4-6*s[1, 1]^2*s[2, 2]^3*s[3, 3]-6*s[1, 1]^2*s[2, 2]^2*s[2, 3]*s[3, 2]+18*s[1, 1]^2*s[2, 2]^2*s[3, 3]^2-60*s[1, 1]^2*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]-6*s[1, 1]^2*s[2, 2]*s[3, 3]^3+24*s[1, 1]^2*s[2, 3]^2*s[3, 2]^2-6*s[1, 1]^2*s[2, 3]*s[3, 2]*s[3, 3]^2-3*s[1, 1]^2*s[3, 3]^4+30*s[1, 1]*s[1, 2]^2*s[2, 1]^2*s[2, 2]-24*s[1, 1]*s[1, 2]^2*s[2, 1]^2*s[3, 3]+54*s[1, 1]*s[1, 2]^2*s[2, 1]*s[2, 3]*s[3, 1]+54*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 1]^2*s[3, 2]+6*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 1]+6*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 1]*s[3, 1]*s[3, 3]+54*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 3]*s[3, 1]^2+6*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]^3+30*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]^2*s[3, 3]+6*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 2]-60*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]*s[3, 3]^2+114*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 3]*s[3, 2]*s[3, 3]+24*s[1, 1]*s[1, 2]*s[2, 1]*s[3, 3]^3-18*s[1, 1]*s[1, 2]*s[2, 2]^2*s[2, 3]*s[3, 1]+72*s[1, 1]*s[1, 2]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 3]-108*s[1, 1]*s[1, 2]*s[2, 3]^2*s[3, 1]*s[3, 2]-18*s[1, 1]*s[1, 2]*s[2, 3]*s[3, 1]*s[3, 3]^2+54*s[1, 1]*s[1, 3]^2*s[2, 1]*s[3, 1]*s[3, 2]-24*s[1, 1]*s[1, 3]^2*s[2, 2]*s[3, 1]^2+30*s[1, 1]*s[1, 3]^2*s[3, 1]^2*s[3, 3]-18*s[1, 1]*s[1, 3]*s[2, 1]*s[2, 2]^2*s[3, 2]+72*s[1, 1]*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 2]*s[3, 3]-108*s[1, 1]*s[1, 3]*s[2, 1]*s[2, 3]*s[3, 2]^2-18*s[1, 1]*s[1, 3]*s[2, 1]*s[3, 2]*s[3, 3]^2+24*s[1, 1]*s[1, 3]*s[2, 2]^3*s[3, 1]-60*s[1, 1]*s[1, 3]*s[2, 2]^2*s[3, 1]*s[3, 3]+114*s[1, 1]*s[1, 3]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 2]+30*s[1, 1]*s[1, 3]*s[2, 2]*s[3, 1]*s[3, 3]^2+6*s[1, 1]*s[1, 3]*s[2, 3]*s[3, 1]*s[3, 2]*s[3, 3]+6*s[1, 1]*s[1, 3]*s[3, 1]*s[3, 3]^3+6*s[1, 1]*s[2, 2]^4*s[3, 3]-6*s[1, 1]*s[2, 2]^3*s[2, 3]*s[3, 2]-6*s[1, 1]*s[2, 2]^3*s[3, 3]^2+30*s[1, 1]*s[2, 2]^2*s[2, 3]*s[3, 2]*s[3, 3]-6*s[1, 1]*s[2, 2]^2*s[3, 3]^3-24*s[1, 1]*s[2, 2]*s[2, 3]^2*s[3, 2]^2+30*s[1, 1]*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]^2+6*s[1, 1]*s[2, 2]*s[3, 3]^4-24*s[1, 1]*s[2, 3]^2*s[3, 2]^2*s[3, 3]-6*s[1, 1]*s[2, 3]*s[3, 2]*s[3, 3]^3-12*s[1, 2]^3*s[2, 1]^3-36*s[1, 2]^2*s[1, 3]*s[2, 1]^2*s[3, 1]-3*s[1, 2]^2*s[2, 1]^2*s[2, 2]^2-24*s[1, 2]^2*s[2, 1]^2*s[2, 2]*s[3, 3]-36*s[1, 2]^2*s[2, 1]^2*s[2, 3]*s[3, 2]+24*s[1, 2]^2*s[2, 1]^2*s[3, 3]^2+54*s[1, 2]^2*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 1]-108*s[1, 2]^2*s[2, 1]*s[2, 3]*s[3, 1]*s[3, 3]+81*s[1, 2]^2*s[2, 3]^2*s[3, 1]^2-36*s[1, 2]*s[1, 3]^2*s[2, 1]*s[3, 1]^2+54*s[1, 2]*s[1, 3]*s[2, 1]^2*s[2, 2]*s[3, 2]-108*s[1, 2]*s[1, 3]*s[2, 1]^2*s[3, 2]*s[3, 3]-60*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 2]^2*s[3, 1]+114*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 1]*s[3, 3]+90*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 3]*s[3, 1]*s[3, 2]-60*s[1, 2]*s[1, 3]*s[2, 1]*s[3, 1]*s[3, 3]^2-108*s[1, 2]*s[1, 3]*s[2, 2]*s[2, 3]*s[3, 1]^2+54*s[1, 2]*s[1, 3]*s[2, 3]*s[3, 1]^2*s[3, 3]-6*s[1, 2]*s[2, 1]*s[2, 2]^3*s[3, 3]-6*s[1, 2]*s[2, 1]*s[2, 2]^2*s[2, 3]*s[3, 2]-6*s[1, 2]*s[2, 1]*s[2, 2]^2*s[3, 3]^2+6*s[1, 2]*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]+24*s[1, 2]*s[2, 1]*s[2, 2]*s[3, 3]^3-36*s[1, 2]*s[2, 1]*s[2, 3]^2*s[3, 2]^2-60*s[1, 2]*s[2, 1]*s[2, 3]*s[3, 2]*s[3, 3]^2-12*s[1, 2]*s[2, 1]*s[3, 3]^4+12*s[1, 2]*s[2, 2]^3*s[2, 3]*s[3, 1]-18*s[1, 2]*s[2, 2]^2*s[2, 3]*s[3, 1]*s[3, 3]+54*s[1, 2]*s[2, 2]*s[2, 3]^2*s[3, 1]*s[3, 2]-18*s[1, 2]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 3]^2+54*s[1, 2]*s[2, 3]^2*s[3, 1]*s[3, 2]*s[3, 3]+12*s[1, 2]*s[2, 3]*s[3, 1]*s[3, 3]^3-12*s[1, 3]^3*s[3, 1]^3+81*s[1, 3]^2*s[2, 1]^2*s[3, 2]^2-108*s[1, 3]^2*s[2, 1]*s[2, 2]*s[3, 1]*s[3, 2]+54*s[1, 3]^2*s[2, 1]*s[3, 1]*s[3, 2]*s[3, 3]+24*s[1, 3]^2*s[2, 2]^2*s[3, 1]^2-24*s[1, 3]^2*s[2, 2]*s[3, 1]^2*s[3, 3]-36*s[1, 3]^2*s[2, 3]*s[3, 1]^2*s[3, 2]-3*s[1, 3]^2*s[3, 1]^2*s[3, 3]^2+12*s[1, 3]*s[2, 1]*s[2, 2]^3*s[3, 2]-18*s[1, 3]*s[2, 1]*s[2, 2]^2*s[3, 2]*s[3, 3]+54*s[1, 3]*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 2]^2-18*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 2]*s[3, 3]^2+54*s[1, 3]*s[2, 1]*s[2, 3]*s[3, 2]^2*s[3, 3]+12*s[1, 3]*s[2, 1]*s[3, 2]*s[3, 3]^3-12*s[1, 3]*s[2, 2]^4*s[3, 1]+24*s[1, 3]*s[2, 2]^3*s[3, 1]*s[3, 3]-60*s[1, 3]*s[2, 2]^2*s[2, 3]*s[3, 1]*s[3, 2]-6*s[1, 3]*s[2, 2]^2*s[3, 1]*s[3, 3]^2+6*s[1, 3]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 2]*s[3, 3]-6*s[1, 3]*s[2, 2]*s[3, 1]*s[3, 3]^3-36*s[1, 3]*s[2, 3]^2*s[3, 1]*s[3, 2]^2-6*s[1, 3]*s[2, 3]*s[3, 1]*s[3, 2]*s[3, 3]^2-3*s[2, 2]^4*s[3, 3]^2+6*s[2, 2]^3*s[2, 3]*s[3, 2]*s[3, 3]+6*s[2, 2]^3*s[3, 3]^3-3*s[2, 2]^2*s[2, 3]^2*s[3, 2]^2-24*s[2, 2]^2*s[2, 3]*s[3, 2]*s[3, 3]^2-3*s[2, 2]^2*s[3, 3]^4+30*s[2, 2]*s[2, 3]^2*s[3, 2]^2*s[3, 3]+6*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]^3-12*s[2, 3]^3*s[3, 2]^3-3*s[2, 3]^2*s[3, 2]^2*s[3, 3]^2))^(1/3)+((1/3)*s[1, 1]*s[2, 2]+(1/3)*s[3, 3]*s[1, 1]-s[1, 2]*s[2, 1]-s[1, 3]*s[3, 1]+(1/3)*s[3, 3]*s[2, 2]-s[2, 3]*s[3, 2]-(1/3)*s[1, 1]^2-(1/3)*s[2, 2]^2-(1/3)*s[3, 3]^2)/(8*s[1, 1]^3-12*s[1, 1]^2*s[2, 2]-12*s[1, 1]^2*s[3, 3]+36*s[1, 1]*s[1, 2]*s[2, 1]+36*s[1, 1]*s[1, 3]*s[3, 1]-12*s[1, 1]*s[2, 2]^2+48*s[1, 1]*s[2, 2]*s[3, 3]-72*s[1, 1]*s[2, 3]*s[3, 2]-12*s[1, 1]*s[3, 3]^2+36*s[1, 2]*s[2, 1]*s[2, 2]-72*s[1, 2]*s[2, 1]*s[3, 3]+108*s[1, 2]*s[2, 3]*s[3, 1]+108*s[1, 3]*s[2, 1]*s[3, 2]-72*s[1, 3]*s[2, 2]*s[3, 1]+36*s[1, 3]*s[3, 1]*s[3, 3]+8*s[2, 2]^3-12*s[2, 2]^2*s[3, 3]+36*s[2, 2]*s[2, 3]*s[3, 2]-12*s[2, 2]*s[3, 3]^2+36*s[2, 3]*s[3, 2]*s[3, 3]+8*s[3, 3]^3+12*sqrt(-3*s[1, 1]^4*s[2, 2]^2+6*s[1, 1]^4*s[2, 2]*s[3, 3]-12*s[1, 1]^4*s[2, 3]*s[3, 2]-3*s[1, 1]^4*s[3, 3]^2+6*s[1, 1]^3*s[1, 2]*s[2, 1]*s[2, 2]-6*s[1, 1]^3*s[1, 2]*s[2, 1]*s[3, 3]+12*s[1, 1]^3*s[1, 2]*s[2, 3]*s[3, 1]+12*s[1, 1]^3*s[1, 3]*s[2, 1]*s[3, 2]-6*s[1, 1]^3*s[1, 3]*s[2, 2]*s[3, 1]+6*s[1, 1]^3*s[1, 3]*s[3, 1]*s[3, 3]+6*s[1, 1]^3*s[2, 2]^3-6*s[1, 1]^3*s[2, 2]^2*s[3, 3]+24*s[1, 1]^3*s[2, 2]*s[2, 3]*s[3, 2]-6*s[1, 1]^3*s[2, 2]*s[3, 3]^2+24*s[1, 1]^3*s[2, 3]*s[3, 2]*s[3, 3]+6*s[1, 1]^3*s[3, 3]^3-3*s[1, 1]^2*s[1, 2]^2*s[2, 1]^2-6*s[1, 1]^2*s[1, 2]*s[1, 3]*s[2, 1]*s[3, 1]-24*s[1, 1]^2*s[1, 2]*s[2, 1]*s[2, 2]^2+30*s[1, 1]^2*s[1, 2]*s[2, 1]*s[2, 2]*s[3, 3]-60*s[1, 1]^2*s[1, 2]*s[2, 1]*s[2, 3]*s[3, 2]-6*s[1, 1]^2*s[1, 2]*s[2, 1]*s[3, 3]^2-18*s[1, 1]^2*s[1, 2]*s[2, 2]*s[2, 3]*s[3, 1]-18*s[1, 1]^2*s[1, 2]*s[2, 3]*s[3, 1]*s[3, 3]-3*s[1, 1]^2*s[1, 3]^2*s[3, 1]^2-18*s[1, 1]^2*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 2]-18*s[1, 1]^2*s[1, 3]*s[2, 1]*s[3, 2]*s[3, 3]-6*s[1, 1]^2*s[1, 3]*s[2, 2]^2*s[3, 1]+30*s[1, 1]^2*s[1, 3]*s[2, 2]*s[3, 1]*s[3, 3]-60*s[1, 1]^2*s[1, 3]*s[2, 3]*s[3, 1]*s[3, 2]-24*s[1, 1]^2*s[1, 3]*s[3, 1]*s[3, 3]^2-3*s[1, 1]^2*s[2, 2]^4-6*s[1, 1]^2*s[2, 2]^3*s[3, 3]-6*s[1, 1]^2*s[2, 2]^2*s[2, 3]*s[3, 2]+18*s[1, 1]^2*s[2, 2]^2*s[3, 3]^2-60*s[1, 1]^2*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]-6*s[1, 1]^2*s[2, 2]*s[3, 3]^3+24*s[1, 1]^2*s[2, 3]^2*s[3, 2]^2-6*s[1, 1]^2*s[2, 3]*s[3, 2]*s[3, 3]^2-3*s[1, 1]^2*s[3, 3]^4+30*s[1, 1]*s[1, 2]^2*s[2, 1]^2*s[2, 2]-24*s[1, 1]*s[1, 2]^2*s[2, 1]^2*s[3, 3]+54*s[1, 1]*s[1, 2]^2*s[2, 1]*s[2, 3]*s[3, 1]+54*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 1]^2*s[3, 2]+6*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 1]+6*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 1]*s[3, 1]*s[3, 3]+54*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 3]*s[3, 1]^2+6*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]^3+30*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]^2*s[3, 3]+6*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 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3]*s[2, 1]*s[2, 2]*s[3, 2]*s[3, 3]^2+54*s[1, 3]*s[2, 1]*s[2, 3]*s[3, 2]^2*s[3, 3]+12*s[1, 3]*s[2, 1]*s[3, 2]*s[3, 3]^3-12*s[1, 3]*s[2, 2]^4*s[3, 1]+24*s[1, 3]*s[2, 2]^3*s[3, 1]*s[3, 3]-60*s[1, 3]*s[2, 2]^2*s[2, 3]*s[3, 1]*s[3, 2]-6*s[1, 3]*s[2, 2]^2*s[3, 1]*s[3, 3]^2+6*s[1, 3]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 2]*s[3, 3]-6*s[1, 3]*s[2, 2]*s[3, 1]*s[3, 3]^3-36*s[1, 3]*s[2, 3]^2*s[3, 1]*s[3, 2]^2-6*s[1, 3]*s[2, 3]*s[3, 1]*s[3, 2]*s[3, 3]^2-3*s[2, 2]^4*s[3, 3]^2+6*s[2, 2]^3*s[2, 3]*s[3, 2]*s[3, 3]+6*s[2, 2]^3*s[3, 3]^3-3*s[2, 2]^2*s[2, 3]^2*s[3, 2]^2-24*s[2, 2]^2*s[2, 3]*s[3, 2]*s[3, 3]^2-3*s[2, 2]^2*s[3, 3]^4+30*s[2, 2]*s[2, 3]^2*s[3, 2]^2*s[3, 3]+6*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]^3-12*s[2, 3]^3*s[3, 2]^3-3*s[2, 3]^2*s[3, 2]^2*s[3, 3]^2))^(1/3)+(1/3)*s[3, 3]+(1/3)*s[2, 2]+(1/3)*s[1, 1]-((1/2)*I)*sqrt(3)*((1/6)*(8*s[1, 1]^3-12*s[1, 1]^2*s[2, 2]-12*s[1, 1]^2*s[3, 3]+36*s[1, 1]*s[1, 2]*s[2, 1]+36*s[1, 1]*s[1, 3]*s[3, 1]-12*s[1, 1]*s[2, 2]^2+48*s[1, 1]*s[2, 2]*s[3, 3]-72*s[1, 1]*s[2, 3]*s[3, 2]-12*s[1, 1]*s[3, 3]^2+36*s[1, 2]*s[2, 1]*s[2, 2]-72*s[1, 2]*s[2, 1]*s[3, 3]+108*s[1, 2]*s[2, 3]*s[3, 1]+108*s[1, 3]*s[2, 1]*s[3, 2]-72*s[1, 3]*s[2, 2]*s[3, 1]+36*s[1, 3]*s[3, 1]*s[3, 3]+8*s[2, 2]^3-12*s[2, 2]^2*s[3, 3]+36*s[2, 2]*s[2, 3]*s[3, 2]-12*s[2, 2]*s[3, 3]^2+36*s[2, 3]*s[3, 2]*s[3, 3]+8*s[3, 3]^3+12*sqrt(-3*s[1, 1]^4*s[2, 2]^2+6*s[1, 1]^4*s[2, 2]*s[3, 3]-12*s[1, 1]^4*s[2, 3]*s[3, 2]-3*s[1, 1]^4*s[3, 3]^2+6*s[1, 1]^3*s[1, 2]*s[2, 1]*s[2, 2]-6*s[1, 1]^3*s[1, 2]*s[2, 1]*s[3, 3]+12*s[1, 1]^3*s[1, 2]*s[2, 3]*s[3, 1]+12*s[1, 1]^3*s[1, 3]*s[2, 1]*s[3, 2]-6*s[1, 1]^3*s[1, 3]*s[2, 2]*s[3, 1]+6*s[1, 1]^3*s[1, 3]*s[3, 1]*s[3, 3]+6*s[1, 1]^3*s[2, 2]^3-6*s[1, 1]^3*s[2, 2]^2*s[3, 3]+24*s[1, 1]^3*s[2, 2]*s[2, 3]*s[3, 2]-6*s[1, 1]^3*s[2, 2]*s[3, 3]^2+24*s[1, 1]^3*s[2, 3]*s[3, 2]*s[3, 3]+6*s[1, 1]^3*s[3, 3]^3-3*s[1, 1]^2*s[1, 2]^2*s[2, 1]^2-6*s[1, 1]^2*s[1, 2]*s[1, 3]*s[2, 1]*s[3, 1]-24*s[1, 1]^2*s[1, 2]*s[2, 1]*s[2, 2]^2+30*s[1, 1]^2*s[1, 2]*s[2, 1]*s[2, 2]*s[3, 3]-60*s[1, 1]^2*s[1, 2]*s[2, 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3]+54*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 3]*s[3, 1]^2+6*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]^3+30*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]^2*s[3, 3]+6*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 2]-60*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]*s[3, 3]^2+114*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 3]*s[3, 2]*s[3, 3]+24*s[1, 1]*s[1, 2]*s[2, 1]*s[3, 3]^3-18*s[1, 1]*s[1, 2]*s[2, 2]^2*s[2, 3]*s[3, 1]+72*s[1, 1]*s[1, 2]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 3]-108*s[1, 1]*s[1, 2]*s[2, 3]^2*s[3, 1]*s[3, 2]-18*s[1, 1]*s[1, 2]*s[2, 3]*s[3, 1]*s[3, 3]^2+54*s[1, 1]*s[1, 3]^2*s[2, 1]*s[3, 1]*s[3, 2]-24*s[1, 1]*s[1, 3]^2*s[2, 2]*s[3, 1]^2+30*s[1, 1]*s[1, 3]^2*s[3, 1]^2*s[3, 3]-18*s[1, 1]*s[1, 3]*s[2, 1]*s[2, 2]^2*s[3, 2]+72*s[1, 1]*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 2]*s[3, 3]-108*s[1, 1]*s[1, 3]*s[2, 1]*s[2, 3]*s[3, 2]^2-18*s[1, 1]*s[1, 3]*s[2, 1]*s[3, 2]*s[3, 3]^2+24*s[1, 1]*s[1, 3]*s[2, 2]^3*s[3, 1]-60*s[1, 1]*s[1, 3]*s[2, 2]^2*s[3, 1]*s[3, 3]+114*s[1, 1]*s[1, 3]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 2]+30*s[1, 1]*s[1, 3]*s[2, 2]*s[3, 1]*s[3, 3]^2+6*s[1, 1]*s[1, 3]*s[2, 3]*s[3, 1]*s[3, 2]*s[3, 3]+6*s[1, 1]*s[1, 3]*s[3, 1]*s[3, 3]^3+6*s[1, 1]*s[2, 2]^4*s[3, 3]-6*s[1, 1]*s[2, 2]^3*s[2, 3]*s[3, 2]-6*s[1, 1]*s[2, 2]^3*s[3, 3]^2+30*s[1, 1]*s[2, 2]^2*s[2, 3]*s[3, 2]*s[3, 3]-6*s[1, 1]*s[2, 2]^2*s[3, 3]^3-24*s[1, 1]*s[2, 2]*s[2, 3]^2*s[3, 2]^2+30*s[1, 1]*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]^2+6*s[1, 1]*s[2, 2]*s[3, 3]^4-24*s[1, 1]*s[2, 3]^2*s[3, 2]^2*s[3, 3]-6*s[1, 1]*s[2, 3]*s[3, 2]*s[3, 3]^3-12*s[1, 2]^3*s[2, 1]^3-36*s[1, 2]^2*s[1, 3]*s[2, 1]^2*s[3, 1]-3*s[1, 2]^2*s[2, 1]^2*s[2, 2]^2-24*s[1, 2]^2*s[2, 1]^2*s[2, 2]*s[3, 3]-36*s[1, 2]^2*s[2, 1]^2*s[2, 3]*s[3, 2]+24*s[1, 2]^2*s[2, 1]^2*s[3, 3]^2+54*s[1, 2]^2*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 1]-108*s[1, 2]^2*s[2, 1]*s[2, 3]*s[3, 1]*s[3, 3]+81*s[1, 2]^2*s[2, 3]^2*s[3, 1]^2-36*s[1, 2]*s[1, 3]^2*s[2, 1]*s[3, 1]^2+54*s[1, 2]*s[1, 3]*s[2, 1]^2*s[2, 2]*s[3, 2]-108*s[1, 2]*s[1, 3]*s[2, 1]^2*s[3, 2]*s[3, 3]-60*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 2]^2*s[3, 1]+114*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 1]*s[3, 3]+90*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 3]*s[3, 1]*s[3, 2]-60*s[1, 2]*s[1, 3]*s[2, 1]*s[3, 1]*s[3, 3]^2-108*s[1, 2]*s[1, 3]*s[2, 2]*s[2, 3]*s[3, 1]^2+54*s[1, 2]*s[1, 3]*s[2, 3]*s[3, 1]^2*s[3, 3]-6*s[1, 2]*s[2, 1]*s[2, 2]^3*s[3, 3]-6*s[1, 2]*s[2, 1]*s[2, 2]^2*s[2, 3]*s[3, 2]-6*s[1, 2]*s[2, 1]*s[2, 2]^2*s[3, 3]^2+6*s[1, 2]*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]+24*s[1, 2]*s[2, 1]*s[2, 2]*s[3, 3]^3-36*s[1, 2]*s[2, 1]*s[2, 3]^2*s[3, 2]^2-60*s[1, 2]*s[2, 1]*s[2, 3]*s[3, 2]*s[3, 3]^2-12*s[1, 2]*s[2, 1]*s[3, 3]^4+12*s[1, 2]*s[2, 2]^3*s[2, 3]*s[3, 1]-18*s[1, 2]*s[2, 2]^2*s[2, 3]*s[3, 1]*s[3, 3]+54*s[1, 2]*s[2, 2]*s[2, 3]^2*s[3, 1]*s[3, 2]-18*s[1, 2]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 3]^2+54*s[1, 2]*s[2, 3]^2*s[3, 1]*s[3, 2]*s[3, 3]+12*s[1, 2]*s[2, 3]*s[3, 1]*s[3, 3]^3-12*s[1, 3]^3*s[3, 1]^3+81*s[1, 3]^2*s[2, 1]^2*s[3, 2]^2-108*s[1, 3]^2*s[2, 1]*s[2, 2]*s[3, 1]*s[3, 2]+54*s[1, 3]^2*s[2, 1]*s[3, 1]*s[3, 2]*s[3, 3]+24*s[1, 3]^2*s[2, 2]^2*s[3, 1]^2-24*s[1, 3]^2*s[2, 2]*s[3, 1]^2*s[3, 3]-36*s[1, 3]^2*s[2, 3]*s[3, 1]^2*s[3, 2]-3*s[1, 3]^2*s[3, 1]^2*s[3, 3]^2+12*s[1, 3]*s[2, 1]*s[2, 2]^3*s[3, 2]-18*s[1, 3]*s[2, 1]*s[2, 2]^2*s[3, 2]*s[3, 3]+54*s[1, 3]*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 2]^2-18*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 2]*s[3, 3]^2+54*s[1, 3]*s[2, 1]*s[2, 3]*s[3, 2]^2*s[3, 3]+12*s[1, 3]*s[2, 1]*s[3, 2]*s[3, 3]^3-12*s[1, 3]*s[2, 2]^4*s[3, 1]+24*s[1, 3]*s[2, 2]^3*s[3, 1]*s[3, 3]-60*s[1, 3]*s[2, 2]^2*s[2, 3]*s[3, 1]*s[3, 2]-6*s[1, 3]*s[2, 2]^2*s[3, 1]*s[3, 3]^2+6*s[1, 3]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 2]*s[3, 3]-6*s[1, 3]*s[2, 2]*s[3, 1]*s[3, 3]^3-36*s[1, 3]*s[2, 3]^2*s[3, 1]*s[3, 2]^2-6*s[1, 3]*s[2, 3]*s[3, 1]*s[3, 2]*s[3, 3]^2-3*s[2, 2]^4*s[3, 3]^2+6*s[2, 2]^3*s[2, 3]*s[3, 2]*s[3, 3]+6*s[2, 2]^3*s[3, 3]^3-3*s[2, 2]^2*s[2, 3]^2*s[3, 2]^2-24*s[2, 2]^2*s[2, 3]*s[3, 2]*s[3, 3]^2-3*s[2, 2]^2*s[3, 3]^4+30*s[2, 2]*s[2, 3]^2*s[3, 2]^2*s[3, 3]+6*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]^3-12*s[2, 3]^3*s[3, 2]^3-3*s[2, 3]^2*s[3, 2]^2*s[3, 3]^2))^(1/3)+((2/3)*s[1, 1]*s[2, 2]+(2/3)*s[3, 3]*s[1, 1]-2*s[1, 2]*s[2, 1]-2*s[1, 3]*s[3, 1]+(2/3)*s[3, 3]*s[2, 2]-2*s[2, 3]*s[3, 2]-(2/3)*s[1, 1]^2-(2/3)*s[2, 2]^2-(2/3)*s[3, 3]^2)/(8*s[1, 1]^3-12*s[1, 1]^2*s[2, 2]-12*s[1, 1]^2*s[3, 3]+36*s[1, 1]*s[1, 2]*s[2, 1]+36*s[1, 1]*s[1, 3]*s[3, 1]-12*s[1, 1]*s[2, 2]^2+48*s[1, 1]*s[2, 2]*s[3, 3]-72*s[1, 1]*s[2, 3]*s[3, 2]-12*s[1, 1]*s[3, 3]^2+36*s[1, 2]*s[2, 1]*s[2, 2]-72*s[1, 2]*s[2, 1]*s[3, 3]+108*s[1, 2]*s[2, 3]*s[3, 1]+108*s[1, 3]*s[2, 1]*s[3, 2]-72*s[1, 3]*s[2, 2]*s[3, 1]+36*s[1, 3]*s[3, 1]*s[3, 3]+8*s[2, 2]^3-12*s[2, 2]^2*s[3, 3]+36*s[2, 2]*s[2, 3]*s[3, 2]-12*s[2, 2]*s[3, 3]^2+36*s[2, 3]*s[3, 2]*s[3, 3]+8*s[3, 3]^3+12*sqrt(-3*s[1, 1]^4*s[2, 2]^2+6*s[1, 1]^4*s[2, 2]*s[3, 3]-12*s[1, 1]^4*s[2, 3]*s[3, 2]-3*s[1, 1]^4*s[3, 3]^2+6*s[1, 1]^3*s[1, 2]*s[2, 1]*s[2, 2]-6*s[1, 1]^3*s[1, 2]*s[2, 1]*s[3, 3]+12*s[1, 1]^3*s[1, 2]*s[2, 3]*s[3, 1]+12*s[1, 1]^3*s[1, 3]*s[2, 1]*s[3, 2]-6*s[1, 1]^3*s[1, 3]*s[2, 2]*s[3, 1]+6*s[1, 1]^3*s[1, 3]*s[3, 1]*s[3, 3]+6*s[1, 1]^3*s[2, 2]^3-6*s[1, 1]^3*s[2, 2]^2*s[3, 3]+24*s[1, 1]^3*s[2, 2]*s[2, 3]*s[3, 2]-6*s[1, 1]^3*s[2, 2]*s[3, 3]^2+24*s[1, 1]^3*s[2, 3]*s[3, 2]*s[3, 3]+6*s[1, 1]^3*s[3, 3]^3-3*s[1, 1]^2*s[1, 2]^2*s[2, 1]^2-6*s[1, 1]^2*s[1, 2]*s[1, 3]*s[2, 1]*s[3, 1]-24*s[1, 1]^2*s[1, 2]*s[2, 1]*s[2, 2]^2+30*s[1, 1]^2*s[1, 2]*s[2, 1]*s[2, 2]*s[3, 3]-60*s[1, 1]^2*s[1, 2]*s[2, 1]*s[2, 3]*s[3, 2]-6*s[1, 1]^2*s[1, 2]*s[2, 1]*s[3, 3]^2-18*s[1, 1]^2*s[1, 2]*s[2, 2]*s[2, 3]*s[3, 1]-18*s[1, 1]^2*s[1, 2]*s[2, 3]*s[3, 1]*s[3, 3]-3*s[1, 1]^2*s[1, 3]^2*s[3, 1]^2-18*s[1, 1]^2*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 2]-18*s[1, 1]^2*s[1, 3]*s[2, 1]*s[3, 2]*s[3, 3]-6*s[1, 1]^2*s[1, 3]*s[2, 2]^2*s[3, 1]+30*s[1, 1]^2*s[1, 3]*s[2, 2]*s[3, 1]*s[3, 3]-60*s[1, 1]^2*s[1, 3]*s[2, 3]*s[3, 1]*s[3, 2]-24*s[1, 1]^2*s[1, 3]*s[3, 1]*s[3, 3]^2-3*s[1, 1]^2*s[2, 2]^4-6*s[1, 1]^2*s[2, 2]^3*s[3, 3]-6*s[1, 1]^2*s[2, 2]^2*s[2, 3]*s[3, 2]+18*s[1, 1]^2*s[2, 2]^2*s[3, 3]^2-60*s[1, 1]^2*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]-6*s[1, 1]^2*s[2, 2]*s[3, 3]^3+24*s[1, 1]^2*s[2, 3]^2*s[3, 2]^2-6*s[1, 1]^2*s[2, 3]*s[3, 2]*s[3, 3]^2-3*s[1, 1]^2*s[3, 3]^4+30*s[1, 1]*s[1, 2]^2*s[2, 1]^2*s[2, 2]-24*s[1, 1]*s[1, 2]^2*s[2, 1]^2*s[3, 3]+54*s[1, 1]*s[1, 2]^2*s[2, 1]*s[2, 3]*s[3, 1]+54*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 1]^2*s[3, 2]+6*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 1]+6*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 1]*s[3, 1]*s[3, 3]+54*s[1, 1]*s[1, 2]*s[1, 3]*s[2, 3]*s[3, 1]^2+6*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]^3+30*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]^2*s[3, 3]+6*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 2]-60*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 2]*s[3, 3]^2+114*s[1, 1]*s[1, 2]*s[2, 1]*s[2, 3]*s[3, 2]*s[3, 3]+24*s[1, 1]*s[1, 2]*s[2, 1]*s[3, 3]^3-18*s[1, 1]*s[1, 2]*s[2, 2]^2*s[2, 3]*s[3, 1]+72*s[1, 1]*s[1, 2]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 3]-108*s[1, 1]*s[1, 2]*s[2, 3]^2*s[3, 1]*s[3, 2]-18*s[1, 1]*s[1, 2]*s[2, 3]*s[3, 1]*s[3, 3]^2+54*s[1, 1]*s[1, 3]^2*s[2, 1]*s[3, 1]*s[3, 2]-24*s[1, 1]*s[1, 3]^2*s[2, 2]*s[3, 1]^2+30*s[1, 1]*s[1, 3]^2*s[3, 1]^2*s[3, 3]-18*s[1, 1]*s[1, 3]*s[2, 1]*s[2, 2]^2*s[3, 2]+72*s[1, 1]*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 2]*s[3, 3]-108*s[1, 1]*s[1, 3]*s[2, 1]*s[2, 3]*s[3, 2]^2-18*s[1, 1]*s[1, 3]*s[2, 1]*s[3, 2]*s[3, 3]^2+24*s[1, 1]*s[1, 3]*s[2, 2]^3*s[3, 1]-60*s[1, 1]*s[1, 3]*s[2, 2]^2*s[3, 1]*s[3, 3]+114*s[1, 1]*s[1, 3]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 2]+30*s[1, 1]*s[1, 3]*s[2, 2]*s[3, 1]*s[3, 3]^2+6*s[1, 1]*s[1, 3]*s[2, 3]*s[3, 1]*s[3, 2]*s[3, 3]+6*s[1, 1]*s[1, 3]*s[3, 1]*s[3, 3]^3+6*s[1, 1]*s[2, 2]^4*s[3, 3]-6*s[1, 1]*s[2, 2]^3*s[2, 3]*s[3, 2]-6*s[1, 1]*s[2, 2]^3*s[3, 3]^2+30*s[1, 1]*s[2, 2]^2*s[2, 3]*s[3, 2]*s[3, 3]-6*s[1, 1]*s[2, 2]^2*s[3, 3]^3-24*s[1, 1]*s[2, 2]*s[2, 3]^2*s[3, 2]^2+30*s[1, 1]*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]^2+6*s[1, 1]*s[2, 2]*s[3, 3]^4-24*s[1, 1]*s[2, 3]^2*s[3, 2]^2*s[3, 3]-6*s[1, 1]*s[2, 3]*s[3, 2]*s[3, 3]^3-12*s[1, 2]^3*s[2, 1]^3-36*s[1, 2]^2*s[1, 3]*s[2, 1]^2*s[3, 1]-3*s[1, 2]^2*s[2, 1]^2*s[2, 2]^2-24*s[1, 2]^2*s[2, 1]^2*s[2, 2]*s[3, 3]-36*s[1, 2]^2*s[2, 1]^2*s[2, 3]*s[3, 2]+24*s[1, 2]^2*s[2, 1]^2*s[3, 3]^2+54*s[1, 2]^2*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 1]-108*s[1, 2]^2*s[2, 1]*s[2, 3]*s[3, 1]*s[3, 3]+81*s[1, 2]^2*s[2, 3]^2*s[3, 1]^2-36*s[1, 2]*s[1, 3]^2*s[2, 1]*s[3, 1]^2+54*s[1, 2]*s[1, 3]*s[2, 1]^2*s[2, 2]*s[3, 2]-108*s[1, 2]*s[1, 3]*s[2, 1]^2*s[3, 2]*s[3, 3]-60*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 2]^2*s[3, 1]+114*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 1]*s[3, 3]+90*s[1, 2]*s[1, 3]*s[2, 1]*s[2, 3]*s[3, 1]*s[3, 2]-60*s[1, 2]*s[1, 3]*s[2, 1]*s[3, 1]*s[3, 3]^2-108*s[1, 2]*s[1, 3]*s[2, 2]*s[2, 3]*s[3, 1]^2+54*s[1, 2]*s[1, 3]*s[2, 3]*s[3, 1]^2*s[3, 3]-6*s[1, 2]*s[2, 1]*s[2, 2]^3*s[3, 3]-6*s[1, 2]*s[2, 1]*s[2, 2]^2*s[2, 3]*s[3, 2]-6*s[1, 2]*s[2, 1]*s[2, 2]^2*s[3, 3]^2+6*s[1, 2]*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]+24*s[1, 2]*s[2, 1]*s[2, 2]*s[3, 3]^3-36*s[1, 2]*s[2, 1]*s[2, 3]^2*s[3, 2]^2-60*s[1, 2]*s[2, 1]*s[2, 3]*s[3, 2]*s[3, 3]^2-12*s[1, 2]*s[2, 1]*s[3, 3]^4+12*s[1, 2]*s[2, 2]^3*s[2, 3]*s[3, 1]-18*s[1, 2]*s[2, 2]^2*s[2, 3]*s[3, 1]*s[3, 3]+54*s[1, 2]*s[2, 2]*s[2, 3]^2*s[3, 1]*s[3, 2]-18*s[1, 2]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 3]^2+54*s[1, 2]*s[2, 3]^2*s[3, 1]*s[3, 2]*s[3, 3]+12*s[1, 2]*s[2, 3]*s[3, 1]*s[3, 3]^3-12*s[1, 3]^3*s[3, 1]^3+81*s[1, 3]^2*s[2, 1]^2*s[3, 2]^2-108*s[1, 3]^2*s[2, 1]*s[2, 2]*s[3, 1]*s[3, 2]+54*s[1, 3]^2*s[2, 1]*s[3, 1]*s[3, 2]*s[3, 3]+24*s[1, 3]^2*s[2, 2]^2*s[3, 1]^2-24*s[1, 3]^2*s[2, 2]*s[3, 1]^2*s[3, 3]-36*s[1, 3]^2*s[2, 3]*s[3, 1]^2*s[3, 2]-3*s[1, 3]^2*s[3, 1]^2*s[3, 3]^2+12*s[1, 3]*s[2, 1]*s[2, 2]^3*s[3, 2]-18*s[1, 3]*s[2, 1]*s[2, 2]^2*s[3, 2]*s[3, 3]+54*s[1, 3]*s[2, 1]*s[2, 2]*s[2, 3]*s[3, 2]^2-18*s[1, 3]*s[2, 1]*s[2, 2]*s[3, 2]*s[3, 3]^2+54*s[1, 3]*s[2, 1]*s[2, 3]*s[3, 2]^2*s[3, 3]+12*s[1, 3]*s[2, 1]*s[3, 2]*s[3, 3]^3-12*s[1, 3]*s[2, 2]^4*s[3, 1]+24*s[1, 3]*s[2, 2]^3*s[3, 1]*s[3, 3]-60*s[1, 3]*s[2, 2]^2*s[2, 3]*s[3, 1]*s[3, 2]-6*s[1, 3]*s[2, 2]^2*s[3, 1]*s[3, 3]^2+6*s[1, 3]*s[2, 2]*s[2, 3]*s[3, 1]*s[3, 2]*s[3, 3]-6*s[1, 3]*s[2, 2]*s[3, 1]*s[3, 3]^3-36*s[1, 3]*s[2, 3]^2*s[3, 1]*s[3, 2]^2-6*s[1, 3]*s[2, 3]*s[3, 1]*s[3, 2]*s[3, 3]^2-3*s[2, 2]^4*s[3, 3]^2+6*s[2, 2]^3*s[2, 3]*s[3, 2]*s[3, 3]+6*s[2, 2]^3*s[3, 3]^3-3*s[2, 2]^2*s[2, 3]^2*s[3, 2]^2-24*s[2, 2]^2*s[2, 3]*s[3, 2]*s[3, 3]^2-3*s[2, 2]^2*s[3, 3]^4+30*s[2, 2]*s[2, 3]^2*s[3, 2]^2*s[3, 3]+6*s[2, 2]*s[2, 3]*s[3, 2]*s[3, 3]^3-12*s[2, 3]^3*s[3, 2]^3-3*s[2, 3]^2*s[3, 2]^2*s[3, 3]^2))^(1/3))})

 (2)
> 

restart
> 

with(LinearAlgebra):
> 

for N from 2 to 5 do
  M       := RandomMatrix(N, generator=-2^31..2^31);
  eig_||N := CodeTools:-Usage( Eigenvalues(M) ):
  print(length(eig_||N)):
end do:

memory used=2.99MiB, alloc change=32.00MiB, cpu time=77.00ms, real time=520.00ms, gc time=8.46ms

  

126

  

memory used=2.84MiB, alloc change=0 bytes, cpu time=84.00ms, real time=2.71s, gc time=0ns

  

1345

  

memory used=2.88MiB, alloc change=0 bytes, cpu time=44.00ms, real time=440.00ms, gc time=0ns

  

17144

  

memory used=2.42MiB, alloc change=0 bytes, cpu time=27.00ms, real time=103.00ms, gc time=0ns

  

1046

 (3)
> 

# uncomment to understand why the size of eig_4 is much larger than the size of eig_5
eig_4:
eig_5:
> 

restart
> 

with(LinearAlgebra):
> 

for N from 2 to 5 do
  M       := RandomMatrix(N, generator=-2.0^31..2.0^31);
  eig_||N := CodeTools:-Usage( Eigenvalues(M) ):
  print(length(eig_||N)):
end do:

memory used=1.84MiB, alloc change=0 bytes, cpu time=47.00ms, real time=1.25s, gc time=0ns

  

30

  

memory used=74.70KiB, alloc change=0 bytes, cpu time=1000.00us, real time=48.00ms, gc time=0ns

  

32

  

memory used=74.98KiB, alloc change=0 bytes, cpu time=1000.00us, real time=0ns, gc time=0ns

  

34

  

memory used=75.34KiB, alloc change=0 bytes, cpu time=1000.00us, real time=0ns, gc time=0ns

  

36

 (4)
> 

 


 

Download Eigenvalues.mw


Eigenvalues.mw

@hajnayeb  Sorry for the late, Th...

mmcdara 7891
August 14 2020
0 0

@hajnayeb 

Sorry for the late,

The last parameter in procedure KG (1 presently) corresponds to the standard deviation of the gaussian process (your white noise if the third parameter is set to a small value).
 

@vv You're right I was over opt...

mmcdara 7891
August 13 2020
0 0

@vv 
You're right I was over optimistic thinking  that could exist only one solution.
 

UNDERSTOOD...

mmcdara 7891
August 13 2020
0 0

UNDERSTOOD

 

The problem comes from the "leading" term X(t)*diff(X(t), t$2).
Maple isolates the differential termes of higher order and transforms the ode into 
diff(X(t), t$2) =( diff(X(t),t)/(1+diff(X(t),t)^2) + F) / X(t) ... wich, given ths ics,  equals (1/2-1/2)/0 = "0/0" 
Not a sinularity but an undefined form: maybe I would have waste less time with a more explicity error message?

Starting from 0 with X(0)="a small value" fixes the problem.

Now another issue: how can I found a numerical solution closer to the exact one on the range 0..T for T larger than 1 (the true solution converges asymptotically to 0 as T -> infinity)?
 


 

> 

restart
> 

interface(version)

`Standard Worksheet Interface, Maple 2015.2, Mac OS X, December 21 2015 Build ID 1097895`

 (1)
> 

with(plots):
> 

# Source term

F := t*(-600*t/(100*t^2+1)^2+80000*t^3/(100*t^2+1)^3)/(100*t^2+1)-(1/(100*t^2+1)-200*t^2/(100*t^2+1)^2)/(1+(1/(100*t^2+1)-200*t^2/(100*t^2+1)^2)^2);

# plot(F, t=0..0.5);

t*(-600*t/(100*t^2+1)^2+80000*t^3/(100*t^2+1)^3)/(100*t^2+1)-(1/(100*t^2+1)-200*t^2/(100*t^2+1)^2)/(1+(1/(100*t^2+1)-200*t^2/(100*t^2+1)^2)^2)

 (2)
> 

# Ode

ode := X(t)*diff(X(t), t$2)-diff(X(t),t)/(1+diff(X(t),t)^2) - 'F';

X(t)*(diff(diff(X(t), t), t))-(diff(X(t), t))/(1+(diff(X(t), t))^2)-F

 (3)
> 

# Initial conditions


eps := 1e-10:
ics := X(0) = eps, D(X)(0) = 1

X(0) = 0.1e-9, (D(X))(0) = 1

 (4)
> 

# The solution must be this one

U := t -> t/((t*10)^2+1)

proc (t) options operator, arrow; t/(100*t^2+1) end proc

 (5)
> 

# Check ode and ics

eval(ode, X(t)=U(t));
U(0);
D(U)(0);

0

  

0

  

1

 (6)
> 

# Plots

meths  := [rkf45, dverk78, rosenbrock, lsode]:
colors := table([rkf45=red, dverk78=gold, rosenbrock=green, lsode=cyan]):

for meth in meths do
  sol_||meth := dsolve({ode, ics}, numeric, method=meth):
end do:

display(
  plot(U(t), t=0..1, color=blue, legend=exact),
  seq( odeplot(sol_||meth, [t, X(t)], t=0..1, color=red, linestyle=3, color=colors[meth], legend=meth), meth in meths )
);

 
> 

# Starting from t=0.4 doesn't help to obtain good numerical solutions

ics_2 := X(0.4) = U(0.4), D(X)(0.4) = eval(diff(U(t), t), t=0.4);

for meth in meths do
  sol_||meth := dsolve({ode, ics_2}, numeric, method=meth):
end do:

display(
  plot(U(t), t=0.4..1, color=blue, legend=exact),
  seq( odeplot(sol_||meth, [t, X(t)], t=0.4..1, color=red, linestyle=3, color=colors[meth], legend=meth), meth in meths )
);

X(.4) = 0.2352941176e-1, (D(X))(.4) = -0.5190311419e-1

  

Warning, could not obtain numerical solution at all points, plot may be incomplete

  

Warning, cannot evaluate the solution further right of .72054998, probably a singularity

  

Warning, could not obtain numerical solution at all points, plot may be incomplete

  

 
> 

 


 

Download Unsuccessful_dsolve_2.mw

I aggree with what said before. Here ar...

mmcdara 7891
August 12 2020
0 0

I aggree with what said before.

Here are just a few "statistics" about the 15 by 15 polynomial system you want to find a Grobner basis of.

Polynoms 14 and 15 have 14080 terms and require each 704241 words to be represented.
Their total degrees are both 18 and the most complex monomial they contain is the product of 11 indeterminates.

So it doesn't seem strange to me that Maple might face enormous difficulties to compute a Grobner basis and that it ends up ends by  some crash.
I even doubt that CAS specialized in GB computations could handle this?
 

> 

restart;
> 

with(LinearAlgebra):
> 

F0:=(w*f00+v*f01)/(v+w):
F1:=(u*f10+w*f11)/(w+u):
F2:=(v*f20+u*f21)/(u+v):
T:=(u)^(3)*p0+(v)^(3)*p1+(w)^(3)*p2 + 3*u*v*(u+v)*(u*e01+v*e10) + 3*v*w*(v+w)*(v*e11+w*e20) + 3*w*u*(w+u)*(w*e21+u*e00) + 12*u*v*w*(u*F0+v*F1+w*F2):
coefList:=[p0,p1,p2,e00,e01,e10,e11,e20,e21,f00,f01,f10,f11,f20,f21]:
nops(coefList):
> 

for i from 1 to nops(coefList) do
 ind:=coefList[i];
  BF||ind:=subs(w=1-u-v, factor(coeff(T,coefList[i],1)));
end do:
> 

for i from 1 to nops(coefList) do
 ind:=coefList[i];
 DDBF||ind:=factor(diff(BF||ind,u,v));
end do:
> 

QR:=x-> t1*subs(u=u1,v=v1,x) + t2*subs(u=u2,v=v2,x) + t3*subs(u=u3,v=v3,x) + (t4)*subs(u=u4,v=v4,x)+t5*subs(u=u5,v=v5,x):
> 

for i from 1 to nops(coefList) do
 ind:=coefList[i];
 eq[i]:=numer(simplify(int(DDBF||ind, u=0..1-v, v=0..1) - QR(DDBF||ind)));
end do:
> 

sys := [seq(eq[i],i=1..nops(coefList))]:
nops(sys);
nops(coefList);

15

  

15

 (1)
> 

# number of terms in each eq[i]

seq(nops(eq[i]),i=1..nops(coefList));

# number of words used to represent each eq[i]
seq(length(eq[i]),i=1..nops(coefList));

# total degree of each eq[i]

seq(degree(eq[i]),i=1..nops(coefList));

# total degree of each eq[i]

seq(degree(eq[i]),i=1..nops(coefList));

# maximum number of indeterminates in the monomials of each eq[i]

ni := NULL:
for i from 1 to nops(coefList) do
  nj := 0:
  for j from 1 to nops(eq[i]) do
    nj := max(nj, numelems(indets(op(j, eq[i]), name)))
  end do;
  ni := ni, nj
end do:
ni;

1, 1, 16, 16, 11, 11, 16, 30, 30, 3645, 2268, 2268, 3645, 14080, 14080

  

0, 0, 170, 255, 180, 180, 255, 416, 416, 115876, 73669, 73669, 115876, 704241, 704241

  

-infinity, -infinity, 2, 3, 3, 3, 3, 3, 3, 13, 13, 13, 13, 18, 18

  

-infinity, -infinity, 2, 3, 3, 3, 3, 3, 3, 13, 13, 13, 13, 18, 18

  

0, 0, 2, 3, 3, 3, 3, 3, 3, 7, 7, 7, 7, 11, 11

 (2)
> 

 


 

Download GB.mw

@Adam Ledger Watchout, the output (...

mmcdara 7891
August 11 2020
0 0

@Adam Ledger 

Watchout, the output (5) is not the one of CF_LS  but print's.
I have removed the print instruction, replaced the colon at the end of the line CF_LS :=... by a semicolon, and right clicked on the output.
Here is the result.


PS: I'm not familiar with the 2D input mode and I never use it. 

> 

restart:
> 

with(numtheory):
> 

S[0] := proc (N, K) options operator, arrow; map(simplify, {seq(seq(seq(piecewise((a^varphi(b))^(1/(c+1))-floor((a^varphi(b))^(1/(c+1))) = 0, [a, b, c], NULL), a = 1 .. N), b = 1 .. N), c = 1 .. K)}, 'radical') end proc

proc (N, K) options operator, arrow; map(simplify, {seq(seq(seq(piecewise((a^numtheory:-varphi(b))^(1/(c+1))-floor((a^numtheory:-varphi(b))^(1/(c+1))) = 0, [a, b, c], NULL), a = 1 .. N), b = 1 .. N), c = 1 .. K)}, 'radical') end proc

 (1)
> 

T := proc (N, K) options operator, arrow; {seq(seq(seq([a, b, c], a = 1 .. N), b = 1 .. N), c = 1 .. K)} end proc

proc (N, K) options operator, arrow; {seq(seq(seq([a, b, c], a = 1 .. N), b = 1 .. N), c = 1 .. K)} end proc

 (2)
> 

S[1] := proc (N, K) options operator, arrow; `minus`(T(N, K), S[0](N, K)) end proc

proc (N, K) options operator, arrow; `minus`(T(N, K), S[0](N, K)) end proc

 (3)
> 

CardRatio := proc (N, K) options operator, arrow; nops(S[0](N, K))/nops(S[1](N, K)) end proc

proc (N, K) options operator, arrow; nops(S[0](N, K))/nops(S[1](N, K)) end proc

 (4)
> 

CF_LS := [
           CurveFitting[LeastSquares]([seq([k, CardRatio(2, k)], k = 1 .. 10)], K),
           CurveFitting[LeastSquares]([seq([k, CardRatio(3, k)], k = 1 .. 10)], K),
           CurveFitting[LeastSquares]([seq([k, CardRatio(4, k)], k = 1 .. 10)], K)
         ];

[1, 44268857/45401356-(532409481/9988298320)*K, 24308311919/13309971675-(135902619982/773879781675)*K]

 (5)
> 

smartplot( (5) );

 
> 

# Maybe it would be better to do somethink like this?

colors := [red, green, blue]:

plots:-display(
  seq( plot(CF_LS[k], color=colors[k], legend=cat("Cardio(", k+1, ")")), k=1..3)
)

 
> 

 


 

Download numtheory.mw

@Preben Alsholm No code missing, ph...

mmcdara 7891
August 11 2020
0 0

@Preben Alsholm 

No code missing, phi is the totient function, @Adam Ledger has just omited to say that.

So Adam's code run correctly after loading the numtheory package but I don't understand what Adam  wants?
 

> 

restart:
> 

with(numtheory):
> 

S[0] := proc (N, K) options operator, arrow; map(simplify, {seq(seq(seq(piecewise((a^varphi(b))^(1/(c+1))-floor((a^varphi(b))^(1/(c+1))) = 0, [a, b, c], NULL), a = 1 .. N), b = 1 .. N), c = 1 .. K)}, 'radical') end proc

proc (N, K) options operator, arrow; map(simplify, {seq(seq(seq(piecewise((a^numtheory:-varphi(b))^(1/(c+1))-floor((a^numtheory:-varphi(b))^(1/(c+1))) = 0, [a, b, c], NULL), a = 1 .. N), b = 1 .. N), c = 1 .. K)}, 'radical') end proc

 (1)
> 

T := proc (N, K) options operator, arrow; {seq(seq(seq([a, b, c], a = 1 .. N), b = 1 .. N), c = 1 .. K)} end proc

proc (N, K) options operator, arrow; {seq(seq(seq([a, b, c], a = 1 .. N), b = 1 .. N), c = 1 .. K)} end proc

 (2)
> 

S[1] := proc (N, K) options operator, arrow; `minus`(T(N, K), S[0](N, K)) end proc

proc (N, K) options operator, arrow; `minus`(T(N, K), S[0](N, K)) end proc

 (3)
> 

CardRatio := proc (N, K) options operator, arrow; nops(S[0](N, K))/nops(S[1](N, K)) end proc

proc (N, K) options operator, arrow; nops(S[0](N, K))/nops(S[1](N, K)) end proc

 (4)
> 

CF_LS := {
           CurveFitting[LeastSquares]([seq([k, CardRatio(2, k)], k = 1 .. 10)], K),
           CurveFitting[LeastSquares]([seq([k, CardRatio(3, k)], k = 1 .. 10)], K),
           CurveFitting[LeastSquares]([seq([k, CardRatio(4, k)], k = 1 .. 10)], K)
         }:
print~(CF_LS):

1

  

44268857/45401356-(532409481/9988298320)*K

  

24308311919/13309971675-(135902619982/773879781675)*K

 (5)
> 

 

 

Download numtheory.mw

That's impossible : -2/(15*h) -1/(6*...

mmcdara 7891
August 10 2020
0 0

That's impossible : -2/(15*h) -1/(6*h) + 3/(10*h) = 0, thus the result you should want is 0, not 4*epsilon/(15*h)


 

> 

restart
> 

expr := -2*f(x-2*h)/15/h-f(x)/6/h+3*f(x+3*h)/10/h+2*ff(x-2*h)/15/h+ff(x)/6/h-3*ff(x+3*h)/10/h

-(2/15)*f(x-2*h)/h-(1/6)*f(x)/h+(3/10)*f(x+3*h)/h+(2/15)*ff(x-2*h)/h+(1/6)*ff(x)/h-(3/10)*ff(x+3*h)/h

 (1)
> 

MyRules := { seq(f(x+i*h)=ff(x+i*h)+epsilon, i=-2..3) };

{f(x) = ff(x)+epsilon, f(-h+x) = ff(-h+x)+epsilon, f(h+x) = ff(h+x)+epsilon, f(2*h+x) = ff(2*h+x)+epsilon, f(x-2*h) = ff(x-2*h)+epsilon, f(x+3*h) = ff(x+3*h)+epsilon}

 (2)
> 

subs(MyRules, expr);

-(2/15)*(ff(x-2*h)+epsilon)/h-(1/6)*(ff(x)+epsilon)/h+(3/10)*(ff(x+3*h)+epsilon)/h+(2/15)*ff(x-2*h)/h+(1/6)*ff(x)/h-(3/10)*ff(x+3*h)/h

 (3)
> 

expand(%);

0

 (4)
> 

 


 

Download MyRules.mw

The exact solution can be obtained by ha...

mmcdara 7891
August 09 2020
0 0

@Gillee @acer

 

I realized afterwards that I had forgotten to explain some technical details in my first answer to Gillee.
This omission is corrected in the attached file.

It shows step by step how the exact solution can be obtained by hand without using Maple.


 

> 

restart:
> 

with(Statistics):
> 

# Your problem restated in the correct bayesian form:

# Let X a binomial random variable of parameters N=16 and p.
# p is an unknown value yout want to assess given some outcomes of X.
# The bayesian paradigm then consists in
#   1/ modelling your lack of knowledge about p by saying that p is a realization of some
#      random variable P
#   2/ deciding of some prior distribution Prior(P=q) for P
#   3/ writting the posterior distribution Post(P=q) of P by using Bayes' formula
#
# Thus: Post(P=q | X=K) = Prob(X=K | P=q) * Prior(P=q)
# where the notation Prob(A | B) means "probability of the event A given the event B"
# (you can replace "probability" by "density of probability" in case of continuous random variables)
#
# Now the main result:
#    Let L some likelihood, F the prior distribution of a random variable Y, and G the posterior
#    distribution of Y with respect to L.
#    If F belongs to the family of the conjugate priors of L, then G belonfgs to the same family.
#
# Application:
#    The conjugate prior of the binomial distribution is the family of Beta distributions.
#    Then if you chose Prior(P=q) as a Beta distribution of parameters (a, b), then Post(P=q)
#    will be a Beta distribution of parameters (a', b').
#
# Classical result: Post(P=q | X=K) = Beta(K+a, N-K+b)
#
# How to use this result?
#    It's well known (or not?) that the uniform distribution of support [0, 1] is a Beta
#    distribution of parameters (a=1, b=1).
# Then:
#    Post(P=q | X=K) = Beta(K+1, N-K+1)
#
# With N=16 and K=6 one obtains that
#         the posterior P is a Beta distribution Beta(7, 11)
#

# VERIFICATION (fast acer's code)

randomize():

n_draw      := 10^5:
prior_rate  := Sample(Uniform(0, 1), n_draw):
n_trials    := 16:
dummyV      := Vector(1,datatype=float[8]):
RV          := RandomVariable(Binomial(n_trials,pat)):
> 

subscribers := CodeTools:-Usage( map(proc(u) option hfloat;
> 

                                       global pat;
> 

                                       pat := u;
> 

                                       Sample(RV,dummyV);
> 

                                       dummyV[1];
> 

                                     end proc,
> 

                                     prior_rate) ):
> 


post_rate   := map(i -> if subscribers[i]=6. then prior_rate[i] end if, [$1..n_draw]):
 

memory used=0.66GiB, alloc change=38.01MiB, cpu time=9.69s, real time=9.71s, gc time=267.72ms

  
> 

plots:-display(
   Histogram(post_rate, view=[0..1, default], minbins=100, style='polygon'),
   plot(PDF(BetaDistribution(7, 11), t), t=0..1, color=red, thickness=3)
);

 
> 

 


 

Download FinalAnswer.mw

@Carl Love @acer Hi,Are errors...

mmcdara 7891
August 08 2020
0 0

@Carl Love @acer

 

Hi,
Are the errors catchable byt try..catch..end try the same that are catchable by traperror?
When could it be more interesting to use traperror instead of try..catch..end try?

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