RG fixed points for 3 junction terminal - characterised by reflection and transmission amplitudes

32 days ago by d3banjan

n = 3 R = PolynomialRing(RR,[['r','t'][cmp(int(i/n),i%n)]+'_'+str(1+int(i/n))+str(1+i%n) for i in range(n^2)]) S = matrix(R,3,R.gens()) #print S*S from operator import add #print [S[int(k/n)][k%n] for k in range(n^2)] #print flatten([[[(i,int(k/n),k%n,j,1*(int(k/n)!=k%n))for k in range(n^2)]for i in range(n)]for j in range(n)]) def Sh(i,j): return reduce(add,[S[i][k//n]*S[i][k%n]*S[j][k%n]*S[j][k//n]*((k//n)!=k%n) for k in range(n^2)]) Shot = matrix(R,n,n, Sh) show(Shot) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr}
2.00000000000000 r_{11}^{2} t_{12}^{2} + 2.00000000000000 r_{11}^{2} t_{13}^{2} + 2.00000000000000 t_{12}^{2} t_{13}^{2} & 2.00000000000000 r_{11} t_{12} t_{21} r_{22} + 2.00000000000000 r_{11} t_{13} t_{21} t_{23} + 2.00000000000000 t_{12} t_{13} r_{22} t_{23} & 2.00000000000000 r_{11} t_{12} t_{31} t_{32} + 2.00000000000000 r_{11} t_{13} t_{31} r_{33} + 2.00000000000000 t_{12} t_{13} t_{32} r_{33} \\
2.00000000000000 r_{11} t_{12} t_{21} r_{22} + 2.00000000000000 r_{11} t_{13} t_{21} t_{23} + 2.00000000000000 t_{12} t_{13} r_{22} t_{23} & 2.00000000000000 t_{21}^{2} r_{22}^{2} + 2.00000000000000 t_{21}^{2} t_{23}^{2} + 2.00000000000000 r_{22}^{2} t_{23}^{2} & 2.00000000000000 t_{21} r_{22} t_{31} t_{32} + 2.00000000000000 t_{21} t_{23} t_{31} r_{33} + 2.00000000000000 r_{22} t_{23} t_{32} r_{33} \\
2.00000000000000 r_{11} t_{12} t_{31} t_{32} + 2.00000000000000 r_{11} t_{13} t_{31} r_{33} + 2.00000000000000 t_{12} t_{13} t_{32} r_{33} & 2.00000000000000 t_{21} r_{22} t_{31} t_{32} + 2.00000000000000 t_{21} t_{23} t_{31} r_{33} + 2.00000000000000 r_{22} t_{23} t_{32} r_{33} & 2.00000000000000 t_{31}^{2} t_{32}^{2} + 2.00000000000000 t_{31}^{2} r_{33}^{2} + 2.00000000000000 t_{32}^{2} r_{33}^{2}
\end{array}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr}
2.00000000000000 r_{11}^{2} t_{12}^{2} + 2.00000000000000 r_{11}^{2} t_{13}^{2} + 2.00000000000000 t_{12}^{2} t_{13}^{2} & 2.00000000000000 r_{11} t_{12} t_{21} r_{22} + 2.00000000000000 r_{11} t_{13} t_{21} t_{23} + 2.00000000000000 t_{12} t_{13} r_{22} t_{23} & 2.00000000000000 r_{11} t_{12} t_{31} t_{32} + 2.00000000000000 r_{11} t_{13} t_{31} r_{33} + 2.00000000000000 t_{12} t_{13} t_{32} r_{33} \\
2.00000000000000 r_{11} t_{12} t_{21} r_{22} + 2.00000000000000 r_{11} t_{13} t_{21} t_{23} + 2.00000000000000 t_{12} t_{13} r_{22} t_{23} & 2.00000000000000 t_{21}^{2} r_{22}^{2} + 2.00000000000000 t_{21}^{2} t_{23}^{2} + 2.00000000000000 r_{22}^{2} t_{23}^{2} & 2.00000000000000 t_{21} r_{22} t_{31} t_{32} + 2.00000000000000 t_{21} t_{23} t_{31} r_{33} + 2.00000000000000 r_{22} t_{23} t_{32} r_{33} \\
2.00000000000000 r_{11} t_{12} t_{31} t_{32} + 2.00000000000000 r_{11} t_{13} t_{31} r_{33} + 2.00000000000000 t_{12} t_{13} t_{32} r_{33} & 2.00000000000000 t_{21} r_{22} t_{31} t_{32} + 2.00000000000000 t_{21} t_{23} t_{31} r_{33} + 2.00000000000000 r_{22} t_{23} t_{32} r_{33} & 2.00000000000000 t_{31}^{2} t_{32}^{2} + 2.00000000000000 t_{31}^{2} r_{33}^{2} + 2.00000000000000 t_{32}^{2} r_{33}^{2}
\end{array}\right)
def kron(i,j): if i==j: return 1 else: return 0 aa = [var('alpha_%d' % (i+1)) for i in range(n)] show(aa) #F=matrix(R,n,n,) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\alpha_{1}, \alpha_{2}, \alpha_{3}\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\alpha_{1}, \alpha_{2}, \alpha_{3}\right]
help(sage.matrix.matrix_generic_dense.Matrix_generic_dense) 
       
xx = [var('alpha_%d'% (i+1)) for i in range(n)] n = 3 R = PolynomialRing(RR,[['r','t'][cmp(int(i/n),i%n)]+'_'+str(1+int(i/n))+str(1+i%n) for i in range(n^2)]) S = matrix(R,3,R.gens()) def kron(i,j): if i==j: return 1 else: return 0 def func(S,xx): return [[(-1/2)*xx[i]*S[i][j]*kron(i,j) for i in range(n)]for j in range(n)] f = matrix(func(S,xx)) conjugate(f)*f.simplify() 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr}
\frac{1}{4} \, \alpha_{1} r_{11} \overline{\alpha_{1}} \overline{r_{11}} & 0 & 0 \\
0 & \frac{1}{4} \, \alpha_{2} r_{22} \overline{\alpha_{2}} \overline{r_{22}} & 0 \\
0 & 0 & \frac{1}{4} \, \alpha_{3} r_{33} \overline{\alpha_{3}} \overline{r_{33}}
\end{array}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr}
\frac{1}{4} \, \alpha_{1} r_{11} \overline{\alpha_{1}} \overline{r_{11}} & 0 & 0 \\
0 & \frac{1}{4} \, \alpha_{2} r_{22} \overline{\alpha_{2}} \overline{r_{22}} & 0 \\
0 & 0 & \frac{1}{4} \, \alpha_{3} r_{33} \overline{\alpha_{3}} \overline{r_{33}}
\end{array}\right)
mat = matrix([[0,1],[1,0]]) u = exp(i*mat) show(((u.conjugate_transpose())*u).n()) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr}
1.00000000000000 & 0 \\
0 & 1.00000000000000
\end{array}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr}
1.00000000000000 & 0 \\
0 & 1.00000000000000
\end{array}\right)
#fixed points n=3 #print latex(matrix([[1*(i==j)for i in range(n)]for j in range(n)])) def f(i,j,k): if (i!=k)and(i!=j): return 1*(i!=j) elif (k==i)and (i==j): return 1 else : return 0 #print latex(matrix([[f(i,j,2) for i in range(n)] for j in range(n)])) matrix([[1*(i==((j+1)%n))for i in range(n)]for j in range(n)]) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr}
0 & 1 & 0 \\
0 & 0 & 1 \\
1 & 0 & 0
\end{array}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr}
0 & 1 & 0 \\
0 & 0 & 1 \\
1 & 0 & 0
\end{array}\right)
## thus it begins! # Define S, F and the Governing Equations xx = [var('alpha_%d'% (i+1)) for i in range(n)] n = 3 R = PolynomialRing(RR,[['r','t'][cmp(int(i/n),i%n)]+'_'+str(1+int(i/n))+str(1+i%n) for i in range(n^2)]) S = matrix(R,3,R.gens()) def kron(i,j): if i==j: return 1 else: return 0 def func(S,xx): return [[(-1/2)*xx[i]*S[i][j]*kron(i,j) for i in range(n)]for j in range(n)] f = matrix(func(S,xx)) dSdl = S*f.conjugate_transpose()*S - f # use subs via a dict as below ##print diff_S({S[0][0]:0}) 
       
 
       
n = 3 vnames = [('r' if i==j else 't') +'_%d%d' % (i,j) for i in [1..n] for j in [1..n]] R = PolynomialRing(RR, vnames) S = matrix(R, 3, R.gens()) f = S mat = [[1*(i==((j+1)%n)) for i in [0..n-1]]for j in [0..n-1]] #parent(mat) #create dict SubDict = dict([(S[p//n][p%n],mat[p//n][p%n]) for p in [0..((n^2)-1)]]) print f.subs(SubDict) #yay # generate list of fixed point FPList #define a function dSdl 
        
[               0 1.00000000000000                0]
[               0                0 1.00000000000000]
[1.00000000000000                0                0]
[               0 1.00000000000000                0]
[               0                0 1.00000000000000]
[1.00000000000000                0                0]
#type(r_11) var('a_45') [a_45==0] var('r_11') S[0][0]==r_11 
        
\newcommand{\Bold}[1]{\mathbf{#1}}r_{11} = r_{11}
\newcommand{\Bold}[1]{\mathbf{#1}}r_{11} = r_{11}
def StabDirections(S_fp,dSdl,delta,n): PertDirList = [matrix([[delta*((i*n+j)==p) for i in range(n)] for j in range(n)]) for p in range(n^2)] return [(dSdl(S_fp + Pert) - dSdl(S_fp))/delta for Pert in PertDirList] def PrettyPrintThroughFPList(): return none n=3 FPList = [matrix([[1*(i==j)for i in range(n)]for j in range(n)])] def f(i,j,k): if (i!=k)and(j!=k): return 1*(i!=j) elif (k==i)and (i==j): return 1 else : return 0 [FPList.append(matrix([[f(i,j,k) for i in range(n)] for j in range(n)])) for k in range(n)] [FPList.append(matrix([[1*(i==((j+r)%n))for i in range(n)]for j in range(n)])) for r in [1,2]] FPList xx = [var('alpha_%d'% (i+1)) for i in range(n)] def kron(i,j): if i==j: return 1 else: return 0 a = 1/sum([1/i for i in xx ]) def listfunc(xx): return [[ -a*kron(i,j)/xx[i] + sqrt((1-a/xx[i])*(1-a/xx[j]))*(1-kron(i,j)) for i in range(n)]for j in range(n)] FPList.append(matrix(listfunc(xx))) FPList 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(\begin{array}{rrr}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right), \left(\begin{array}{rrr}
1 & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0
\end{array}\right), \left(\begin{array}{rrr}
0 & 0 & 1 \\
0 & 1 & 0 \\
1 & 0 & 0
\end{array}\right), \left(\begin{array}{rrr}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 1
\end{array}\right), \left(\begin{array}{rrr}
0 & 1 & 0 \\
0 & 0 & 1 \\
1 & 0 & 0
\end{array}\right), \left(\begin{array}{rrr}
0 & 0 & 1 \\
1 & 0 & 0 \\
0 & 1 & 0
\end{array}\right), \left(\begin{array}{rrr}
-\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{1}} & \sqrt{{\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{2}} - 1\right)} {\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{1}} - 1\right)}} & \sqrt{{\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{3}} - 1\right)} {\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{1}} - 1\right)}} \\
\sqrt{{\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{2}} - 1\right)} {\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{1}} - 1\right)}} & -\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{2}} & \sqrt{{\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{3}} - 1\right)} {\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{2}} - 1\right)}} \\
\sqrt{{\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{3}} - 1\right)} {\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{1}} - 1\right)}} & \sqrt{{\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{3}} - 1\right)} {\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{2}} - 1\right)}} & -\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{3}}
\end{array}\right)\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(\begin{array}{rrr}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right), \left(\begin{array}{rrr}
1 & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0
\end{array}\right), \left(\begin{array}{rrr}
0 & 0 & 1 \\
0 & 1 & 0 \\
1 & 0 & 0
\end{array}\right), \left(\begin{array}{rrr}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 1
\end{array}\right), \left(\begin{array}{rrr}
0 & 1 & 0 \\
0 & 0 & 1 \\
1 & 0 & 0
\end{array}\right), \left(\begin{array}{rrr}
0 & 0 & 1 \\
1 & 0 & 0 \\
0 & 1 & 0
\end{array}\right), \left(\begin{array}{rrr}
-\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{1}} & \sqrt{{\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{2}} - 1\right)} {\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{1}} - 1\right)}} & \sqrt{{\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{3}} - 1\right)} {\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{1}} - 1\right)}} \\
\sqrt{{\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{2}} - 1\right)} {\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{1}} - 1\right)}} & -\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{2}} & \sqrt{{\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{3}} - 1\right)} {\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{2}} - 1\right)}} \\
\sqrt{{\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{3}} - 1\right)} {\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{1}} - 1\right)}} & \sqrt{{\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{3}} - 1\right)} {\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{2}} - 1\right)}} & -\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{3}}
\end{array}\right)\right]
xx = [var('alpha_%d'% (i+1)) for i in range(n)] def kron(i,j): if i==j: return 1 else: return 0 a = 1/sum([1/i for i in xx ]) def func(xx): return [[ -a*kron(i,j)/xx[i] + sqrt((1-a/xx[i])*(1-a/xx[j]))*(1-kron(i,j)) for i in range(n)]for j in range(n)] matrix(func(xx)) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr}
-\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{1}} & \sqrt{{\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{2}} - 1\right)} {\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{1}} - 1\right)}} & \sqrt{{\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{3}} - 1\right)} {\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{1}} - 1\right)}} \\
\sqrt{{\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{2}} - 1\right)} {\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{1}} - 1\right)}} & -\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{2}} & \sqrt{{\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{3}} - 1\right)} {\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{2}} - 1\right)}} \\
\sqrt{{\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{3}} - 1\right)} {\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{1}} - 1\right)}} & \sqrt{{\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{3}} - 1\right)} {\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{2}} - 1\right)}} & -\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{3}}
\end{array}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr}
-\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{1}} & \sqrt{{\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{2}} - 1\right)} {\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{1}} - 1\right)}} & \sqrt{{\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{3}} - 1\right)} {\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{1}} - 1\right)}} \\
\sqrt{{\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{2}} - 1\right)} {\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{1}} - 1\right)}} & -\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{2}} & \sqrt{{\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{3}} - 1\right)} {\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{2}} - 1\right)}} \\
\sqrt{{\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{3}} - 1\right)} {\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{1}} - 1\right)}} & \sqrt{{\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{3}} - 1\right)} {\left(\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{2}} - 1\right)}} & -\frac{1}{{\left(\frac{1}{\alpha_{1}} + \frac{1}{\alpha_{2}} + \frac{1}{\alpha_{3}}\right)} \alpha_{3}}
\end{array}\right)
 
       
#analytical fixed point calculation # loaded previous constructs col = matrix([S[k//n][k%n] for k in range(n^2)]) stab = matrix([dSdl[k//n][k%n] for k in range(n^2)]) stabdiff = [[ k[0].derivative(S[j//n][j%n]) for j in range(n^2)] for k in stab] #stabdiff[0][0][0] 
        
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{2} \, r_{11}^{2} \overline{\alpha_{1}} D[0]\left({\rm conjugate}\right)\left(r_{11}\right) - r_{11} \overline{\alpha_{1}} \overline{r_{11}} + \frac{1}{2} \, \alpha_{1}
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{2} \, r_{11}^{2} \overline{\alpha_{1}} D[0]\left({\rm conjugate}\right)\left(r_{11}\right) - r_{11} \overline{\alpha_{1}} \overline{r_{11}} + \frac{1}{2} \, \alpha_{1}
 
       
[S[0][0]==0] 
        
Traceback (click to the left of this block for traceback)
...
NameError: name 'S' is not defined
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_8.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("W1NbMF1bMF09PTBd"),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/tmp/tmp9t6DtK/___code___.py", line 3, in <module>
    exec compile(u'[S[_sage_const_0 ][_sage_const_0 ]==_sage_const_0 ]
  File "", line 1, in <module>
    
NameError: name 'S' is not defined