In terms of the dimensionless parameters \begin{align}\label{eq:dim:1} U_1 \equiv u_1/u_0\,,\qquad D \equiv d/\ell\,,\qquad W \equiv w/\ell\,, \end{align} the constraints on the thumb function are \begin{align}\label{eq:matrix} \begin{bmatrix} 1& D+\tfrac{W}{2} &(D+\tfrac{W}{2})^2& (D+\tfrac{W}{2})^3\\ 1& D-\tfrac{W}{2} &(D-\tfrac{W}{2})^2& (D-\tfrac{W}{2})^3\\ 0 & 1 & 2(D+\tfrac{W}{2}) & 3(D+\tfrac{W}{2})^2 \\ 0 & 1 & 2(D-\tfrac{W}{2}) & 3(D-\tfrac{W}{2})^2 \\ \end{bmatrix} \begin{bmatrix} s_0\\ s_1\\ s_2\\ s_3 \end{bmatrix} = \begin{bmatrix} U_1\\ U_1\\ -\frac{U_1}{1 - D-{W}/{2}}\\ \frac{U_1}{D-{W}/{2}} \end{bmatrix}. \end{align}
Inversion of this \(4\times4\) system yields \(s\).