$$ x = \left[ \begin{array}- 1 & \frac{2}{3} & 1 & 2 \end{array} \right] $$

$$ y = \left[ \begin{array}- 4 & -4 & -\frac{7}{2} & -\frac{1}{2} \end{array} \right] $$

$$ h_i=x_{i+1}-x_i\quad\text{for all }x_i\in{x} $$

$$ h_{1} = \frac{2}{3} $$

$$ h_{2} = \frac{1}{3} $$

$$ h_{3} = 1 $$

$$ \frac{h_i}{6}a_{i-1} +\frac{h_i+h_{i+1}}{3}a_i +\frac{h_{i+1}}{6}a_{i+1} =\frac{y_{i+2}-y_{i+1}}{h_{i+1}} -\frac{y_{i+1}-y_i}{h_i} \quad(i=1,...,n-2) $$

$$ \text{i = 1: } \quad\frac{1}{9}a_0+\frac{1}{3}a_1+\frac{1}{18}a_2 =\frac{-\frac{7}{2}-\left(-4\right)}{\frac{1}{3}}-\frac{-4-4}{\frac{2}{3}} =\frac{27}{2} $$

$$ \text{i = 2: } \quad\frac{1}{18}a_1+\frac{4}{9}a_2+\frac{1}{6}a_3 =\frac{-\frac{1}{2}-\left(-\frac{7}{2}\right)}{1}-\frac{-\frac{7}{2}-\left(-4\right)}{\frac{1}{3}} =\frac{3}{2} $$

$$ \text{Let } a_0=a_{n}=0 $$

Set up matrix:

$$ \left[ \begin{array}{cccc|c} 0 & \frac{1}{3} & \frac{1}{18} & 0 & \frac{27}{2} \\ 0 & \frac{1}{18} & \frac{4}{9} & 0 & \frac{3}{2} \\ \end{array} \right] $$

Reduced Row Echelon Form:

$$ \left[ \begin{array}{cccc|c} 0 & 1 & 0 & 0 & 40.7872 \\ 0 & 0 & 1 & 0 & 1.7234 \\ \end{array} \right] $$

$$ a_0 = 0 \quad a_1 = 40.7872 \quad a_2 = -1.7234 \quad a_3 = 0 $$

$$ b_i=\frac{y_i}{h_i}-\frac{a_{i-1}h_i}{6} $$

$$ b_1=\frac{4}{\frac{2}{3}}-\frac{0}{6}=6 $$

$$ b_2=\frac{-4}{\frac{1}{3}}-\frac{40.7872\cdot \frac{1}{3}}{6}=-14.2660 $$

$$ b_3=-\frac{7}{2}-\frac{-1.7234\cdot 1}{6}=-3.2128 $$

$$ c_i=\frac{y_{i+1}}{h_i}-\frac{a_ih_i}{6} $$

$$ c_1=\frac{-4}{\frac{2}{3}}-\frac{40.7872\cdot \frac{2}{3}}{6}=-10.5319 $$

$$ c_2=\frac{-\frac{7}{2}}{\frac{1}{3}}-\frac{-1.7234\cdot \frac{1}{3}}{6}=-10.4043 $$

$$ c_3=-\frac{1}{2}-\frac{0}{6}=-\frac{1}{2} $$

$$ a_0\frac{\left(x_2-x\right)^3}{6h_1}+a_1\frac{\left(x-x_1\right)^3}{6h_1}+b_1\left(x_2-x\right)+c_1\left(x-x_1\right) $$