(Solved): use and show matlab file3c onlyalgorithm 3.4 3. Construct the natural cubic spline for the following ...

3. Construct the natural cubic spline for the following data. a. b. c. d. To construct the cubic spline interpolant $S$ for the function $f$, defined at the numbers $x_0<x_1<?<x_n$, satisfying $S^{??}\left(x_0\right)=S^{??}\left(x_n\right)=0$ : INPUT $n:x_0,x_1,\dots ,x_n:a_0=f\left(x_0\right),a_1=f\left(x_1\right),\dots ,a_n=f\left(x_n\right)$. OUTPUT $a_j,b_j,c_j,d_j$ for $j=0,1,\dots ,n?1$. (Note: $S(x)=S_j(x)=a_j+b_j\left(x?x_j\right)+c_j{\left(x?x_j\right)}^2+d_j{\left(x?x_j\right)}^3$ for $x_j?x?x_{j+1}$.) Step 1 For $i=0,1,\dots ,n?1$ set $h_i=x_{i+1}?x_i$. Step 2 For $i=1,2,\dots ,n?1$ set ${?}_i=\frac{3}{h_i}\left(a_{i+1}?a_i\right)?\frac{3}{h_{i?1}}\left(a_i?a_{i?1}\right).$ Step 3 Set $l_0=1;$ (Steps 3, 4, and 5 and part of Step 6 solve a tridiagonal linear system using a method described in Algorithm 6.7.) $\begin{array}{l}{?}_0=0;\\ z_0=0.\end{array}$ Step 4 For $i=1,2,\dots ,n?1$ $\begin{array}{rl}\text{ set }{l}_i& =2\left(x_{i+1}?x_{i?1}\right)?h_{i?1}{?}_{i?1}\\ {?}_i& =h_i/l_i\\ z_i& =\left({?}_i?h_{i?1}{z}_{i?1}\right)/l_i\end{array}$ Step 5 Set $I_n=1$; $\begin{array}{l}{z}_n=0;\\ c_n=0.\end{array}$ Step 6 For $j=n?1,n?2,\dots ,0$ $\begin{array}{rl}\text{ set }{c}_j& =z_j?{?}_j{c}_{j+1}:\\ b_j& =\left(a_{j+1}?a_j\right)/h_j?h_j\left(c_{j+1}+2c_j\right)/3;\\ d_j& =\left(c_{j+1}?c_j\right)/\left(3h_j\right).\end{array}$ Step 7 OUTPUT $(a_j,b_j,c_j,d_j$ for $j=0,1,\dots ,n?1)$; STOP.