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