Computing splines using Hermite interpolants

Question

The form for a cubic Hermite interpolant has the form $p_i(x)=a_i+b_i(x-x_i)+c_i(x-x_i)^2+d_i(x-x_i)^3$ according to the following conditions:

$p_i(x_i) = y_i$

$p_i(x_{i+1}) = y_{i+1}$

$p_i'(x_i)=s_i$

$p_i(x_{i+1})=s_{i+1}$

Then the solution to $a_i,b_i,c_i,d_i$ is:

$a_i=y_i$

$b_i=s_i$

$c_{i}=\frac{3y_{i}'-2s_i-s_{i+1}}{\Delta x_i}$

$d_{i}=\frac{s_{i+1}+s_i-2y_i'}{(\Delta x_i)^2}$

I'm not too sure how we obtained the equations for $a_i,b_i,c_i,d_i$.

If we were to instead find the quadratic spline, what would be the values for $a_i,b_i,c_i$?

Answer 1

You obtained $a_i, b_i, c_i$ and $d_i$ by solving 4 linear equations derived from the 4 boundary conditions in your original post.

If we were to find a quadratic spline, then in general you cannot find a solution since a quadratic spline in general cannot satisfy the 4 boundary conditions at the same time.