Cubic Spline
Overview
The cubic spline splits the domain into a set of knot points. The spline interpolates the values between the knot points
with a cubic polynomial as follows:
{% S_k(x) = d_k (x-x_k)^3 + c_k(x-x_k)^2 + b_k(x-x_k) + a_k %}
Boundary Conditions
Each cubic polynomial must match the value of the function at the knot points. Each polynomial must also match the polynomials that
it is adjacent to in its first and second derivatives. For the first and last polynomial, there is one side which is not adjacent
to another polynomial, so we cant match derivatives at the left and right boundaries. However, given these boundaries, there
are still 2 degrees of freedom, so we have to set some additional constraints. The choice of constraints determines the type
of cubic spline we use.
- Natural : the natural boundary conditions sets the second derivatives of the two boundary points to zero
- Clamped : clamped cubic spline sets the first derivatives at the boundaries to preset values.
Examples
