Chapter 28, Problem 2P

a.

Program Plan Intro

To express each coefficient $a_i,b_i,c_{i}and{d}_i$ in terms of $y_i,y_{i+1},D_{i}and{D}_{i+1}$ and find the computation time for the 4n coefficients from point-value pairs and first derivatives.

b.

Program Plan Intro

To show that for $i=1,2,\mathrm{...},n-1$ , $D_{i-1}+4D_i+D_{i+1}=3\left(y_{i+1}-y_{i-1}\right)$ using continuity constraints.

c.

Program Plan Intro

To show that $2D_0+D_1=3\left(y_1-y_0\right)$ and $D_{n-1}+2D_n=3\left(y_n-y{n-1}_{}\right)$

d.

Program Plan Intro

To rewrite equations 28.21 − 28.23 as matrix equation involving the vector $D=\langle D_0,D_1,\mathrm{...}{D}_n\rangle$ of unknowns and name the attributes of matrix in the formed equation.

e.

Program Plan Intro

To argue that natural cubic spline can interpolate a set of n+1 point-values pairs in $O\left(n\right)$ time.

f.

Program Plan Intro

To mention the process of determining a natural cubic spline that interpolates a set of n + 1 points $\left(x_i,y_i\right)$ satisfying $x_0<x_1<\mathrm{...}<x_n$ .

To mention the running time and matrix equation that must be solved by the above algorithm.