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Quadratic spline?


#1

Why don’t we learn about quadratic spline? Is it not as useful as cubic spline?


#2

This is a good question. I mentioned something in class, but I can say more here.
In short, we do not use quadratic splines for teaching purposes, because
they would require different data points from knots, which may complicate
the presentation.
To say a bit more, recall that, on n subintervals (n+1 knots),
linear C^0 splines have n+1 free parameters,
quadratic C^1 splines have n+2 parameters,
cubic C^2 splines have n+3 parameters,
and so on.
Linear splines use the n+1 knots as data points for interpolation
(and these are enough to uniquely determine them),
cubic splines use the n+1 knots as data points for interpolation
and another two end-conditions to make up n+3 equations.
Quadratic splines would need n+2 data points for interpolation.
It is hard to make these data points balanced, if the n+1 knots are used.
(One extra condition would be needed from one only end.)
Unbalanced conditions lead to instablilities.
So, quadratic splines use the n midpoints of subintervals
and the two end points, to make n+2 equations for interpolation.
This makes the presentation a bit more complicated.
So, quadratic splines are used, but not for teaching.