How to determine the values of the second derivatives $f_{i}$ and $f_{i+1}$? We use the continuity assumption of the first derivatives $$s'_{i-1}(x_i)= s'_i(x_i),$$ and set $x=x_i$. Defining $h_i=x_{i+1}-x_i$ we obtain finally the following expression $$h_{i-1}f_{i-1}+2(h_{i}+h_{i-1})f_i+h_if_{i+1}= \frac{6}{h_i}(y_{i+1}-y_i)-\frac{6}{h_{i-1}}(y_{i}-y_{i-1}),$$ and introducing the shorthands $u_i=2(h_{i}+h_{i-1})$, $v_i=\frac{6}{h_i}(y_{i+1}-y_i)-\frac{6}{h_{i-1}}(y_{i}-y_{i-1})$, we can reformulate the problem as a set of linear equations to be solved through e.g., Gaussian elemination