How to derive Equation 6 in this paper?

[Zheng222]

@HanshuYAN Thanks for your open-source code.

1. Could you tell me how to derive Equation 6 in this paper?

2. In the following code snippet, why is lambda_t and eta_t equal to the right equation?
   lambda_t = ( lambda_s * (t_e - t_s) ) / ( lambda_s *(t_c - t_s) + (t_e - t_c) )
eta_t = ( eta_s * (t_e - t_c) ) / ( lambda_s *(t_c - t_s) + (t_e - t_c) )

[shyoshyo]

the following three assumptions in paper are sufficient to derive all formula you have mentioned:

• 
• 
• 

solve these three equations for $v, z_{t_{k-1}}, z_{t_{k}}$(in other words, express $v, z_{t_{k-1}}, z_{t_{k}}$ in terms of other variables):

import sympy
t, t_k1, t_k, z_t, z_t_k, z_t_k1, eps, lambda_k, eta_k,v = sympy.symbols('t t_{k-1} t_k z_t z_t_k z_t_{k-1} eps lambda_k eta_k v')

eq_1 = (z_t_k1 - z_t_k) / (t_k1 - t_k) - v
eq_2 = (z_t_k1 - lambda_k * z_t_k) / eta_k - eps
eq_3 = (t - t_k1) / (t_k - t_k1) * z_t_k + (t_k - t) / (t_k - t_k1) * z_t_k1 - z_t

print(sympy.solve([eq_1, eq_2, eq_3], [v, z_t_k, z_t_k1]))

# prints: {
#     v: (eps*eta_k + lambda_k*z_t - z_t)/(lambda_k*t - lambda_k*t_k - t + t_{k-1}),
#     z_t_k: (-eps*eta_k*t + eps*eta_k*t_k - t_k*z_t + t_{k-1}*z_t)/(lambda_k*t - lambda_k*t_k - t + t_{k-1}),
#     z_t_{k-1}: (-eps*eta_k*t + eps*eta_k*t_{k-1} - lambda_k*t_k*z_t + lambda_k*t_{k-1}*z_t)/(lambda_k*t - lambda_k*t_k - t + t_{k-1})
# }

where:

• result of $v$ corresponds to Eq (6) in paper
• the coefficients of $z_t$ & $\epsilon$ in the result of $z_{t_{k-1}}$ correspond to lambda_t, eta_t in the code snippet, where t_c refers to $t$， t_e refers to $t_{k-1}$, t_s refers to $t_k$, eta_s refers to $\eta_k$, and so on