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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:
 Example of integrating a network
This notebook illustrates how to create a python network and integrate
it with the scipy library.
Step1:  We'll start again with the basic CNO network explored earlier.
Step2:  A PythonNetwork is based on a RateCollection but has methods to write the RHS of the system of ODEs.
Step3:  For example, this network knows how to write the full term for a reaction that goes into the $dY/dt$ equation of the ODE system.
Here we pick one of the rates that is part of the network an explore it.
Step4:  and the code needed to evaluate that rate (the T-dependent part) as
Step5:  The temperature-dependent rate evaluation functions take a Tfactor object, which precomputes most of the commonly-used temperature factors in the rates.
The write_network() method will output the python code needed to define the RHS of a network for integration with the SciPy integrators.
Since python code can be slow, we use Numba to do just-in-time compilation of the functions to speed things up.
Step6:  We can now import the network that was just created and integrate it using the SciPy ODE solvers
Step7:  Integrating the network
We can use the stiff ODE integration solvers that are part of Scipy to integrate this system now
Step8:  Initialize the thermodynamic conditions and initial composition.  We express the composition as molar fractions, Y0.
Step9:  Now we integrate.  We use the BDF method, since reaction networks are in general stiff
Step10:  A network plot
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Python Code:
import pynucastro as pyna
Explanation: Example of integrating a network
This notebook illustrates how to create a python network and integrate
it with the scipy library.
End of explanation
files = ["c12-pg-n13-ls09", 
         "c13-pg-n14-nacr",
         "n13--c13-wc12",
         "n13-pg-o14-lg06",
         "n14-pg-o15-im05",
         "n15-pa-c12-nacr",
         "o14--n14-wc12",
         "o15--n15-wc12"]
Explanation: We'll start again with the basic CNO network explored earlier.
End of explanation
pynet = pyna.PythonNetwork(files)
Explanation: A PythonNetwork is based on a RateCollection but has methods to write the RHS of the system of ODEs.
End of explanation
r = pynet.rates[1]
print(r)
print(pynet.ydot_string(r))
Explanation: For example, this network knows how to write the full term for a reaction that goes into the $dY/dt$ equation of the ODE system.
Here we pick one of the rates that is part of the network an explore it.
End of explanation
print(pynet.function_string(r))
Explanation: and the code needed to evaluate that rate (the T-dependent part) as:
End of explanation
pynet.write_network("cno_test_integrate.py")
%cat cno_test_integrate.py
Explanation: The temperature-dependent rate evaluation functions take a Tfactor object, which precomputes most of the commonly-used temperature factors in the rates.
The write_network() method will output the python code needed to define the RHS of a network for integration with the SciPy integrators.
Since python code can be slow, we use Numba to do just-in-time compilation of the functions to speed things up.
End of explanation
import cno_test_integrate as cno
Explanation: We can now import the network that was just created and integrate it using the SciPy ODE solvers
End of explanation
from scipy.integrate import solve_ivp
import numpy as np
Explanation: Integrating the network
We can use the stiff ODE integration solvers that are part of Scipy to integrate this system now
End of explanation
rho = 150
T = 1.5e7
X0 = np.zeros(cno.nnuc)
X0[cno.ip] = 0.7
X0[cno.ihe4] = 0.28
X0[cno.ic12] = 0.02
Y0 = X0/cno.A
Explanation: Initialize the thermodynamic conditions and initial composition.  We express the composition as molar fractions, Y0.
End of explanation
tmax = 1.e20
sol = solve_ivp(cno.rhs, [0, tmax], Y0, method="BDF",
                dense_output=True, args=(rho, T), rtol=1.e-6, atol=1.e-6)
Explanation: Now we integrate.  We use the BDF method, since reaction networks are in general stiff
End of explanation
import matplotlib.pyplot as plt
fig = plt.figure()
ax = fig.add_subplot(111)
for i in range(cno.nnuc):
    ax.loglog(sol.t, sol.y[i,:] * cno.A[i], label=f"X({cno.names[i].capitalize()})")
ax.set_xlim(1.e10, 1.e20)
ax.set_ylim(1.e-8, 1.0)
ax.legend(fontsize="small")
fig.set_size_inches((10, 8))
Explanation: A network plot
End of explanation
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