From (5.11) and (5.12), we can solve for the \(n a\)-coefficients to find \[a_{i}=\frac{1}{3 h_{i}}\left(b_{i+1}-b_{i}\right), \quad i=0 \text { to } n-1 \nonumber \] From (5.9), we can solve for the ...From (5.11) and (5.12), we can solve for the \(n a\)-coefficients to find \[a_{i}=\frac{1}{3 h_{i}}\left(b_{i+1}-b_{i}\right), \quad i=0 \text { to } n-1 \nonumber \] From (5.9), we can solve for the \(n\) c-coefficients as follows: \[\begin{aligned} c_{i} &=\frac{1}{h_{i}}\left(f_{i}-a_{i} h_{i}^{3}-b_{i} h_{i}^{2}\right) \\ &=\frac{1}{h_{i}}\left(f_{i}-\frac{1}{3 h_{i}}\left(b_{i+1}-b_{i}\right) h_{i}^{3}-b_{i} h_{i}^{2}\right) \\ &=\frac{f_{i}}{h_{i}}-\frac{1}{3} h_{i}\left(b_{i+1}+2 b_{i}\r…
