1. Introduction to the Question
In the question ["How is it Possible to Optimize the Shichman-Hodges Slope Parameters from the Left and Right using Least Square Linear Regression Approaches"][1], in Section 6, it turns out that the regression is minimization of for a series of tabulated values of $V_{DS\text{ }SAT\text{ }j}$ versus $λ$ as follows:$$ 1<j\le i_{max}\text{, where }i_{max}\text{is the maximum number row for the data points} \tag{6.1}$$
$$ I_{SD\text{ }1}=α_j V_{SD\text{ }1} \left(2*V_{SD\text{ }SAT\text{ }j}-V_{SD\text{ }1} \right) $$ $$ \underset{\text{implies}}{\longrightarrow} α_j=\frac{I_{SD\text{ }1}}{V_{SD\text{ }1} \left(2*V_{SD\text{ }SAT\text{ }j}-V_{SD\text{ }1} \right)} \tag{6.2}$$
$$w= \sum_{i=1}^{i=i_{linear\text{ }max}}{\left(I_{SD}-I_{SD\text{ modeled}}\right)^2} + \sum_{i=i_{saturation\text{ }min}}^{i=i_{saturation\text{ }max}}{\left(I_{SD}-I_{SD\text{ modeled}}\right)^2} \tag{6.3}$$
$$\frac{d w}{d λ}= \frac{ d \sum_{i=1}^{i=i_{linear\text{ }max}}{\left(I_{SD}-I_{SD\text{ modeled}}\right)^2} + \sum_{i=i_{saturation\text{ }min}}^{i=i_{saturation\text{ }max}}{\left(I_{SD}-I_{SD\text{ modeled}}\right)^2} } {d λ} = 0 \tag{6.4}$$
Referring to the mathematical model in the Reference, Section 5:
In the linear region, let:
$$ I_{SD}=α V_{SD} \left(2*V_{SD\text{ }SAT}-V_{SD}\right) \left(1+λ*(V_{SD}-V_{DS\text{ }1})\right) \tag{5.1}$$
Because of the offset $\left(V_{SD}-V_{DS\text{ }1}\right)$, at the first data point $V_{DS\text{ }1}$ the Equation can be simplified there:
$$ I_{SD\text{ }1}=α V_{SD\text{ }1} \left(2*V_{SD\text{ }SAT}-V_{SD\text{ }1} \right) \tag{5.2}$$
Then the fit can be varied for values of $V_{SD\text{ }SAT}$.
The saturation region equation can be written as:
$$ I_{SD}=α \left(V_{SD\text{ }SAT}\right)^2 \left(1+λ*(V_{SD}-V_{DS\text{ }1})\right) \tag{5.3}$$
How is it possible to do this symbolically (with enough detail for the reader to be able to follow from line to line the result)?
Note: Example data is provided in [Section 1 of the Reference] to make this question more concrete. But this question is just focused on symbolically minimizing of the least squares across the entire fit with respect to the one variable $λ$ with all of the other parameters set constant. Use of that data should not be necessary to answer this question.
I do not expect that this question requires anything more than symbolic math and calculus to perform the derivatives. But the answer is long enough, and the question needs enough detail, that I feel that this question should be a stand-alone question aside from the main reference.
2. Additional Hint as to the Answer
There are really two separate ways to approach this problem, one being full linear regression or the other above being regression through a data point, namely here to start with the derived $\alpha_j$. The advantage of this choice of the $\alpha_j$ data point is that it represents the best-available accuracy for low current flow $I_{DS}$ with an intention to extrapolate to current values $0^{Amp}\le I_{DS} \le 1^{Amp}$.
First, it is important to clarify the indices $i$ and $j$. For each $j$ point $V_{SD\text{ }SAT\text{ }j}=V_{SD\text{ }j}$. Then once $j$ has been fixed, the least square error regression is taken. And the square error is tabulated according to $j$. Then the curve that has minimal square error is plotted and further evaluated.
3. My Additional Homework Towards that Answer
For a particular $j$, then the model for $I_{SD\text{ }SAT}$ needs to be defined in the Linear and Saturation Regions so that the sum in Section 1, Equation 6.3 can be taken. After that, the derivative in Section 1, Equation 6.4 can be taken and accordingly set to zero in order to find the minimum sum of the square errors.
3.1 Definition of $I_{SD\text{ SAT j}}$ in the Linear $I_{SD\text{ SAT Linear j}}$ and Saturation $I_{SD\text{ SAT Saturation Region j}}$ Regions
The quantity $I_{SD\text{ }SAT}$ needs to be determined in the Linear and Saturation Regions so that the sum of the square errors can be taken together across the entire domain.
$$ I_{SD\text{ }SAT\text{ }Linear\text{ }j}= α_j V_{SD\text{ }SAT\text{ }j} \left(2*V_{SD\text{ }SAT\text{ }j}-V_{SD\text{ }SAT\text{ }j}\right) \left(1+λ_j*(V_{SD\text{ }SAT\text{ }j}-V_{SD\text{ }1})\right) $$ $$= α_j \left(V_{SD\text{ }SAT\text{ }j}\right)^2* \left(1+λ_j*(V_{SD\text{ }SAT\text{ }j}-V_{SD\text{ }1})\right) \tag{3.1.1}$$ $$ I_{SD\text{ }SAT\text{ Saturation Region }j}= α_j \left(V_{SD\text{ }SAT\text{ }j}\right)^2* \left(1+λ_j*(V_{SD\text{ }SAT\text{ }j}-V_{SD\text{ }1})\right) \tag{3.1.2}$$
3.2 Definition of $I_{SD\text{ i j}}$ in the Linear $I_{SD\text{ Linear i j}}$ Region $1 < i\le j$
$$ I_\text{ SD Linear Region i j }= α_j V_\text{ SD i } \left( 2*V_\text{ SD SAT j }-V_\text{ SD i } \right) \left( 1+λ_j* \left( V_\text{ SD i } - V_\text{ SD 1 } \right) \right) \tag{3.2} $$
3.3 Definition of $I_{DS\text{ }SAT}$ in Saturation $I_{DS\text{ }SAT\text{ }Saturation}$ Region $j \le i \le i_\text{ max }$
$$I_\text{ SD Saturation Region i j }= α_j \left(V_\text{ SD SAT j }\right)^2* \left(1+λ_j*(V_\text{ SD i }-V_{SD\text{ }1})\right) \tag{3.3}$$
3.4 Sum of $I_{DS\text{ }SAT}$ From the Linear $I_{DS\text{ }SAT\text{ }Linear}$ and Saturation Region $I_{DS\text{ }SAT\text{ }Saturation}$
From Section 1, Equation 6.4, another step can be taken to simplify the calculation, namely:
$$\frac{d w}{d λ_j}= \frac{ d \sum_{i=1}^{i=i_{linear\text{ }max}}{\left(I_{SD}-I_{SD\text{ modeled}}\right)^2} + \sum_{i=i_{saturation\text{ }min}}^{i=i_{saturation\text{ }max}}{\left(I_{SD}-I_{SD\text{ modeled}}\right)^2} } {d λ_j} = 0 \tag{3.4}$$
$$ \left(-2\right)*\sum_{i=1}^{i=j} { \left(I_\text{ SD i}-I_\text{ SD linear region modeled i j}\right) *\frac{d I_\text{ SD linear region modeled i j}}{dλ_\text{ j }} }+$$ $$ +\left(-2\right)*\sum_{i=j}^{i\le j_\text{ max }} { \left(I_\text{ SD i}-I_\text{ SD saturation region modeled i j}\right) *\frac{d I_\text{ SD saturation region modeled i j}}{dλ_\text{ j }} }$$ $$=0\tag{3.4.1}$$
$$ \sum_{i=1}^{i=j} { \left(I_\text{ SD i}-I_\text{ SD linear region modeled i j}\right) *\frac{d I_\text{ SD linear region modeled i j}}{dλ_\text{ j }} }+$$ $$ +\sum_{i=j}^{i\le j_\text{ max }} { \left(I_\text{ SD i}-I_\text{ SD saturation region modeled i j}\right) *\frac{d I_\text{ SD saturation region modeled i j}}{dλ_\text{ j }} }=0\tag{3.4.2}$$
$$ \frac{ d I_\text{ SD linear region modeled i j} } {dλ_\text{ j }} =$$ $$=\frac{d \left( α_j V_\text{ SD i } \left( 2*V_\text{ SD SAT j }-V_\text{ SD i } \right) \left( 1+λ_j* \left( V_\text{ SD i } - V_\text{ SD 1 } \right) \right) \right) }{d λ_j} $$ $$ = α_j V_\text{ SD i } \left( 2*V_\text{ SD SAT j }-V_\text{ SD i } \right) \left( \left( V_\text{ SD i } - V_\text{ SD 1 } \right) \right) \tag{3.4.3}$$
$$ \frac{d I_\text{ SD saturation region modeled i j } } {dλ_\text{ j }}=$$ $$=\frac{d ( α_j \left(V_{SD\text{ }SAT\text{ }j}\right)^2* \left(1+λ_j*(V_{SD\text{ }SAT\text{ }j}-V_{SD\text{ }1})\right))}{d λ_j} $$ $$= ( α_j \left(V_{SD\text{ }SAT\text{ }j}\right)^2* \left((V_{SD\text{ }SAT\text{ }j}-V_{SD\text{ }1})\right)) \tag{3.4.4}$$
The equations for the models in the linear and saturation regions contain constants and also linear terms in $λ_j$, so the relevant sums can be formed for $λ_j$ to be solved for in terms of the data points $V_\text{ SD i}$ and $I_\text{ SD i}$ versus $j$, tabulated, and then the least square fit can be identified.
Some more details about that can come now as an update to the question (since there is still more needed to solve for) or alternatively as part of an answer that is complete, including table examples.