I have an equation set like this:
$$ R_e(T_i)R_t(T_i) + R_e(T_i)R_p(T_i) = R_sR_t(T_i) + R_sR_p + R_pR_t(T_i) \\ R_e(T_f)R_t(T_f) + R_e(T_f)R_p(T_f) = R_sR_t(T_f) + R_sR_p + R_pR_t(T_f) $$
Where, $R_t(T)$ and $R_e(T)$ are function of temperature ($T$), and $T_i < T_f$
I want to find approximate $R_p$ and $R_s$ values which satisfy this equation.
$R_s$ and $R_p$ can be assumed to be proportional and/or any number of auxiliary equations can be added for temperature values between $T_i$ and $T_f$ as shown below:
$$ R_e(T_m)R_t(T_m) + R_e(T_m)R_p(T_m) = R_sR_t(T_m) + R_sR_p + R_pR_t(T_m) \\ $$
For $\quad T_m = \dfrac{T_i+T_f}{2}$.
I tried using the Least Squares Theorem, but the term $R_sR_p$ makes the equation set non-linear; so I can't apply it.
Is there any idea you can share with me?
If you are wondering the physical background of the problem, there is a thermistor resistor whose value is $R_t(T_i)$ and $R_t(T_f)$ at temperatures $T_i$ and $T_f$ respectively. I want to obtain an $R_e$ resistor (by adding a series ($R_s$) and parallel ($R_p$) resistor) whose value will be $R_e(T_i)$ and $R_e(T_f)$ at temperatures $T_i$ and $T_f$ respectively.
To light a little bit I reformulated your equation :
$S=R_s$
$P=R_p$
And you have a system of equations of the following form :
$$A_1 = B_1*S+PS+C_1*P\\A_2 = B_2*S+PS+C_2*P$$ What if you substract the second to the first, you get rid of the non linear member. $$A_1-A_2=(B_1-B_2)*S+(C_1-C_2)*P$$ So you have the $P$ in function of $S$ and introducing it in one of the equation you have a one-unknow-two-degree equation. Or you can take a third equation and substract it to the first to get a second linear equation between $S$ and $P$
Does it work?