# What are the various methods in adding an additional constraint to the quadratic spline interpolation problem

I am taking a class on numerical analysis. While the professor was deriving the theory behind quadratic splines, the professor mentioned that a quadratic spline function has the form: $$p_{i}(x)=a_{i}x^{2}+b_{i}x+c_{i}\quad\forall i=0,1,n$$ For a dataset with $$n+1$$ points, this means that we have a total of $$3n$$ unknowns $$a_{i}, b_{i}$$ and $$c_{i}$$. Additionally, we are to enforce constraints on the continuity of the interval nodes between every two consecutive points the same thing also applies for the first derivative so we have $$(n-1)+(n-1)=2(n-1)$$ constraints. In addition to the interpolation points $$p_{i}(x)=y_{i}$$ for all $$i=0,1,n$$ we have $$n+1$$ conditions to be added to $$2(n-1)$$ meaning we have a total of $$2(n-1)+n+1=3n-1$$ constraints.

This means we need to add an additional constraint in order to align the number of unknowns with number of constraints. One approach to do this is by setting $$P'(x_{0})=0$$ where $$P(x)$$ is the continuous curve that connects all $$p_{i}(x)$$ computed. Note that this type of constraint induces a natural quadratic spline.

My professor also mentioned other constraint methods without giving any details:

1. End Slope Spline
2. Periodic Spline
3. Not-a-Knot Spline

My question is can anyone provide me with some insight on each of the three methods. I would be extremely thankful as it will help me expand my knowledge further beyond the scope of this course.

1. End slope for quadratic splines is a generalization of the natural quadratic spline condition: $$p'_0(x_0) = v_0$$. Alternatively, you could specify instead $$p'_n(x_{n+1}) = v_{n+1}$$. Since $$\mathcal{C}_1$$ quadratic splines only have 1 unspecified degree of freedom, you generally can't simultaneously require both of these to be true. For cubic splines, this condition is often called a "clamped" spline as you specify both of these simultaneously. I'm not entirely sure what the generalization beyond cubic splines would be.
2. Periodic quadratic splines add the constraint $$p'_0(x_0) = p'_n(x_{n+1})$$. This constraint makes the most sense when $$y_0 = y_{n+1}$$, since then when you try to connect the spline's end points together you get an overall $$\mathcal{C}_1$$ spline. For higher order splines where you have more unspecified degrees of freedom you can require the higher derivatives are also equal across the periodic point.
3. I don't think I've heard of the not-a-knot condition applied to quadratic splines before, however for cubic splines this enforces $$p'''_0(x_1) = p'''_1(x_1)$$ and $$p'''_{n-1}(x_n) = p'''_n(x_n)$$ ($$C_2$$ cubic splines have 2 unspecified degrees of freedom, so you need two extra equations). This results in $$p_0(x) = p_1(x)$$ and $$p_{n-1}(x) = p_{n}(x)$$. I suppose the closest you can come to this with a quadratic spline is to enforce $$p''_0(x_1) = p''_1(x_1)$$ which will result in $$p_0(x) = p_1(x)$$, or alternatively you can try to enforce $$p''_{n-1}(x_n) = p''_{n}(x_n)$$, which results in $$p_{n-1}(x) = p_{n}(x)$$. You generally cannot require both to be true if you want to maintain a $$C_1$$ quadratic spline. I'm also not entirely sure what the generalization of this for higher order splines would be.
for a natural cubic spline you specify $$p''_0(x_0) = 0$$ and $$p''_n(x_{n+1}) = 0$$.