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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.

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1 Answer 1

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  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.

side-note:

for a natural cubic spline you specify $p''_0(x_0) = 0$ and $p''_n(x_{n+1}) = 0$.

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