(Solved): Construct a cubic spline approximation through the original three data points in Q1. The left endpo ...

Construct a cubic spline approximation through the original three data points in Q1. The left endpoint \\( \\left(x_{0}, y_{0}\\right) \\) has the derivative condition \\( S_{0}^{\\prime}\\left(x_{0}\\right)=2 \\) and the right end \\( \\left(x_{2}, y_{2}\\right) \\) must be treated as a natural spline. Rather than derive all the equations from scratch you may adapt the matrices corresponding to these conditions. Alternatively, your starting point can be the general spline function on the interval \\( \\left[x_{j}, x_{j+1}\\right] \\), \\[ \\begin{aligned} S_{j}(x)= & \\frac{1}{6} M_{j} \\frac{\\left(x_{j+1}-x\\right)^{3}}{h_{j}}+\\frac{1}{6} M_{j+1} \\frac{\\left(x-x_{j}\\right)^{3}}{h_{j}} \\\\ & \\quad-\\frac{1}{6} M_{j} h_{j}\\left(x_{j+1}-x\\right)-\\frac{1}{6} M_{j+1} h_{j}\\left(x-x_{j}\\right)+\\frac{y_{j}}{h_{j}}\\left(x_{j+1}-x\\right)+\\frac{y_{j+1}}{h_{j}}\\left(x-x_{j}\\right) . \\end{aligned} \\] In this form we know the respective splines pass through their knot points. We also know the splines and their second derivatives are continuous at the internal knot points. What remains is to make the first derivatives continuous at the internal knot points. Coupled with additional information at \\( x_{0} \\) and \\( x_{n} \\) we can solve the system of equations for \\( M_{i}, i=0, \\ldots, n \\).