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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Euler Equations
The Euler equations in primitive variable form, $q = (\rho, u, p)^\intercal$ appear as
Step1:  The eigenvalues are the speeds at which information propagates with.  SymPy returns them as a
dictionary, giving the multiplicity for each eigenvalue.
Step2:  The right eigenvectors are what SymPy gives natively.  For a given eigenvalue, $\lambda$, these 
satisfy
Step3:  0-th right eigenvector
Step4:  this corresponds to the eigenvalue
Step5:  1-st right eigenvector
Step6:  this corresponds to the eigenvalue
Step7:  2-nd right eigenvector
Step8:  this corresponds to the eigenvalue
Step9:  Here they are as a matrix, $R$, in order from smallest to largest eigenvalue
Step10:  Left Eigenvectors
The left eigenvectors satisfy
Step11:  Traditionally, we normalize these such that $l^{(\mu)} \cdot r^{(\nu)} = \delta_{\mu\nu}$
Step12:  0-th left eigenvector
Step13:  1-st left eigenvector
Step14:  2-nd left eigenvector
Step15:  Entropy formulation
here we write the system in terms of $q_s = (\rho, u, s)^\intercal$, where the system is
$${q_s}_t + A_s(q_s) {q_s}_x = 0$$
and 
$$
A_s = \left (\begin{matrix}u & \rho & 0\
      \frac{c^{2}}{\rho} & u & \frac{p_{s}}{\rho}\
         0 & 0 & u\end{matrix}\right)
         $$
Step16:  left eigenvectors
Step17:  normalization
Step18:  2-d system
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Python Code:
from sympy.abc import rho
rho, u, c = symbols('rho u c')
A = Matrix([[u, rho, 0], [0, u, rho**-1], [0, c**2 * rho, u]])
A
Explanation: Euler Equations
The Euler equations in primitive variable form, $q = (\rho, u, p)^\intercal$ appear as:
$$q_t + A(q) q_x = 0$$
with the matrix $A(q)$:
$$A(q) = \left ( \begin{array}{ccc} u  & \rho     & 0 \                        
                                  0  &  u       & 1/\rho \                   
                                  0  & \gamma p & u \end{array} \right )
$$
The sound speed is related to the adiabatic index, $\gamma$, as $c^2 = \gamma p /\rho$.
We can represent this matrix symbolically in SymPy and explore its eigensystem.
End of explanation
A.eigenvals()
Explanation: The eigenvalues are the speeds at which information propagates with.  SymPy returns them as a
dictionary, giving the multiplicity for each eigenvalue.
End of explanation
R = A.eigenvects()   # this returns a tuple for each eigenvector with multiplicity -- unpack it
r = []
lam = []
for (ev, _, rtmp) in R:
    r.append(rtmp[0])
    lam.append(ev)
    
# we can normalize them anyway we want, so let's make the first entry 1
for n in range(len(r)):
    v = r[n]
    r[n] = v/v[0]
Explanation: The right eigenvectors are what SymPy gives natively.  For a given eigenvalue, $\lambda$, these 
satisfy:
$$A r = \lambda r$$
Right Eigenvectors
End of explanation
r[0]
Explanation: 0-th right eigenvector
End of explanation
lam[0]
Explanation: this corresponds to the eigenvalue
End of explanation
r[1]
Explanation: 1-st right eigenvector
End of explanation
lam[1]
Explanation: this corresponds to the eigenvalue
End of explanation
r[2]
Explanation: 2-nd right eigenvector
End of explanation
lam[2]
Explanation: this corresponds to the eigenvalue
End of explanation
R = zeros(3,3)
R[:,0] = r[1]
R[:,1] = r[0]
R[:,2] = r[2]
R
Explanation: Here they are as a matrix, $R$, in order from smallest to largest eigenvalue
End of explanation
B = A.transpose()
B
L = B.eigenvects()
l = []
laml = []
for (ev, _, ltmp) in L:
    l.append(ltmp[0].transpose())
    laml.append(ev)
    
Explanation: Left Eigenvectors
The left eigenvectors satisfy:
$$l A = \lambda l$$
SymPy doesn't have a method to get left eigenvectors directly, so we take the transpose of this expression:
$$(l A)^\intercal = A^\intercal l^\intercal = \lambda l^\intercal$$
Therefore, the transpose of the left eigenvectors, $l^\intercal$, are the right eigenvectors of transpose of $A$
End of explanation
for n in range(len(l)):
    if lam[n] == laml[n]:
        ltmp = l[n]
        p = ltmp.dot(r[n])
        l[n] = ltmp/p
Explanation: Traditionally, we normalize these such that $l^{(\mu)} \cdot r^{(\nu)} = \delta_{\mu\nu}$
End of explanation
l[0]
Explanation: 0-th left eigenvector
End of explanation
l[1]
Explanation: 1-st left eigenvector
End of explanation
l[2]
Explanation: 2-nd left eigenvector
End of explanation
ps = symbols('p_s')
As = Matrix([[u, rho, 0], [c**2/rho, u, ps/rho], [0, 0, u]])
As
As.eigenvals()
R = As.eigenvects()   # this returns a tuple for each eigenvector with multiplicity -- unpack it
r = []
lam = []
for (ev, _, rtmp) in R:
    r.append(rtmp[0])
    lam.append(ev)
    
# we can normalize them anyway we want, so let's make the first entry 1
for n in range(len(r)):
    v = r[n]
    r[n] = v/v[0]
r[0], lam[0]
r[1], lam[1]
r[2], lam[2]
Explanation: Entropy formulation
here we write the system in terms of $q_s = (\rho, u, s)^\intercal$, where the system is
$${q_s}_t + A_s(q_s) {q_s}_x = 0$$
and 
$$
A_s = \left (\begin{matrix}u & \rho & 0\
      \frac{c^{2}}{\rho} & u & \frac{p_{s}}{\rho}\
         0 & 0 & u\end{matrix}\right)
         $$
End of explanation
Bs = As.transpose()
L = B.eigenvects()
l = []
laml = []
for (ev, _, ltmp) in L:
    l.append(ltmp[0].transpose())
    laml.append(ev)
    
Explanation: left eigenvectors
End of explanation
for n in range(len(l)):
    if lam[n] == laml[n]:
        ltmp = l[n]
        p = ltmp.dot(r[n])
        l[n] = ltmp/p
simplify(l[0])
l[1]
l[2]
Explanation: normalization
End of explanation
rho, u, v, c = symbols('rho u v c')
A = Matrix([[u, rho, 0, 0], [0, u, 0, rho**-1], [0,0, u, 0], [0, c**2 * rho, 0, u]])
A
A.eigenvals()
R = A.eigenvects()   # this returns a tuple for each eigenvector with multiplicity -- unpack it
r = []
lam = []
for (ev, _, rtmp) in R:
    for rv in rtmp:
        r.append(rv)
        lam.append(ev)
    
# we can normalize them anyway we want, so let's make the first entry 1
for n in range(len(r)):
    v = r[n]
    if not v[0] == 0:
        r[n] = v/v[0]
r[0], lam[0]
r[1], lam[1]
r[2], lam[2]
r[3], lam[3]
Explanation: 2-d system
End of explanation
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