```python
from sympy import *
t, dt = symbols('t dt')
P, Q = symbols('P Q', cls=Function)

# Target expression P(t_{n+1/2})*Q(t_{n+1/2})
# Simpler: P(0)*Q(0)
# Arithmetic means of each factor:
# 1/4*(P(-dt/2) + P(dt/2))*(Q(-dt/2) + Q(dt/2))
# Arithmetic mean of the product:
# 1/2*(P(-dt/2)*Q(-dt/2) + P(dt/2)*Q(dt/2))
# Let's Taylor expand to compare

target = P(0)*Q(0)
num_terms = 6
P_p = P(t).series(t, 0, num_terms).subs(t, dt/2)
print(P_p)
P_m = P(t).series(t, 0, num_terms).subs(t, -dt/2)
print(P_m)
Q_p = Q(t).series(t, 0, num_terms).subs(t, dt/2)
print(Q_p)
Q_m = Q(t).series(t, 0, num_terms).subs(t, -dt/2)
print(Q_m)

product_mean = Rational(1,2)*(P_m*Q_m + P_p*Q_p)
product_mean = simplify(expand(product_mean))
product_mean_error = product_mean - target

factor_mean = Rational(1,2)*(P_m + P_p)*Rational(1,2)*(Q_m + Q_p)
factor_mean = simplify(expand(factor_mean))
factor_mean_error = factor_mean - target

print(('product_mean_error:', product_mean_error))
print(('factor_mean_error:', factor_mean_error))

```

    P(0) + dt*Subs(Derivative(P(_x), _x), _x, 0)/2 + dt**2*Subs(Derivative(P(_x), (_x, 2)), _x, 0)/8 + dt**3*Subs(Derivative(P(_x), (_x, 3)), _x, 0)/48 + dt**4*Subs(Derivative(P(_x), (_x, 4)), _x, 0)/384 + dt**5*Subs(Derivative(P(_x), (_x, 5)), _x, 0)/3840 + O(dt**6)
    P(0) - dt*Subs(Derivative(P(_x), _x), _x, 0)/2 + dt**2*Subs(Derivative(P(_x), (_x, 2)), _x, 0)/8 - dt**3*Subs(Derivative(P(_x), (_x, 3)), _x, 0)/48 + dt**4*Subs(Derivative(P(_x), (_x, 4)), _x, 0)/384 - dt**5*Subs(Derivative(P(_x), (_x, 5)), _x, 0)/3840 + O(dt**6)
    Q(0) + dt*Subs(Derivative(Q(_x), _x), _x, 0)/2 + dt**2*Subs(Derivative(Q(_x), (_x, 2)), _x, 0)/8 + dt**3*Subs(Derivative(Q(_x), (_x, 3)), _x, 0)/48 + dt**4*Subs(Derivative(Q(_x), (_x, 4)), _x, 0)/384 + dt**5*Subs(Derivative(Q(_x), (_x, 5)), _x, 0)/3840 + O(dt**6)
    Q(0) - dt*Subs(Derivative(Q(_x), _x), _x, 0)/2 + dt**2*Subs(Derivative(Q(_x), (_x, 2)), _x, 0)/8 - dt**3*Subs(Derivative(Q(_x), (_x, 3)), _x, 0)/48 + dt**4*Subs(Derivative(Q(_x), (_x, 4)), _x, 0)/384 - dt**5*Subs(Derivative(Q(_x), (_x, 5)), _x, 0)/3840 + O(dt**6)
    ('product_mean_error:', dt**2*Subs(Derivative(P(_x), _x), _x, 0)*Subs(Derivative(Q(_x), _x), _x, 0)/4 + dt**2*Q(0)*Subs(Derivative(P(_x), (_x, 2)), _x, 0)/8 + dt**2*P(0)*Subs(Derivative(Q(_x), (_x, 2)), _x, 0)/8 + dt**4*Subs(Derivative(P(_x), (_x, 3)), _x, 0)*Subs(Derivative(Q(_x), _x), _x, 0)/96 + dt**4*Subs(Derivative(P(_x), (_x, 2)), _x, 0)*Subs(Derivative(Q(_x), (_x, 2)), _x, 0)/64 + dt**4*Subs(Derivative(P(_x), _x), _x, 0)*Subs(Derivative(Q(_x), (_x, 3)), _x, 0)/96 + dt**4*Q(0)*Subs(Derivative(P(_x), (_x, 4)), _x, 0)/384 + dt**4*P(0)*Subs(Derivative(Q(_x), (_x, 4)), _x, 0)/384 + O(dt**6))
    ('factor_mean_error:', dt**2*Q(0)*Subs(Derivative(P(_x), (_x, 2)), _x, 0)/8 + dt**2*P(0)*Subs(Derivative(Q(_x), (_x, 2)), _x, 0)/8 + dt**4*Subs(Derivative(P(_x), (_x, 2)), _x, 0)*Subs(Derivative(Q(_x), (_x, 2)), _x, 0)/64 + dt**4*Q(0)*Subs(Derivative(P(_x), (_x, 4)), _x, 0)/384 + dt**4*P(0)*Subs(Derivative(Q(_x), (_x, 4)), _x, 0)/384 + O(dt**6))



```python

```

5cf9594a546d3ef2a02471f9851e2f545a130933
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Data Science and Machine Learning/Machine-Learning-In-Python-THOROUGH/EXAMPLES/FINITE_ELEMENTS/INTRO/EXERCICES/20_PRODUCT_ARITH_MEAN_SYMPY.ipynb
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Qwen/Qwen-72B
1. YES
2. YES