What is the difference, if any, between spline interpolation and piecewise polynomial interpolation?
[Math] Spline interpolation versus polynomial interpolation
interpolationspline
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[Math] comparison of piecewise linear interpolation, cubic interpolation, cubic spline interpolation
In general, cubic interpolation is better than linear interpolation in most aspects such as smoothness of the function and higher accuracy in approximating the original function. However, there is at least one aspect where linear interpolation is better: the linear interpolation will not produce the "overshoot" situation. For example, if you have a function that always produce positive y values, using cubic interpolation to approximate this function might give you a function that "overshoots" to negative y territory even when all the interpolated y values are positive. This will not happen with linear interpolation.
As for "cubic spline interpolation", I am not aware of any context where "cubic interpolation" and "cubic spline interpolation" are considered as different.
The problem was that the data looked kind of like this:
For this data, when an attempt was made to fit $y=a_ix^3+b_ix^2+c_ix+d_i$ between the spline points and it didn't work well because the data can't be expressed as a function of $x$; instead there can be multiple values of $y$ for each value of $x$. There are spline algorithms that can handle this, but if it is desired to use an equation as noted above, in this case we can instead fit $x=a_iy^3+b_iy^2+c_iy+d_i$ in each interval. For these data we anticipate good results just by looking at the data in that the graph seems to depict $x$ as a smooth function of $y$.
Best Answer
Expanding a little on what Arkamis said ...
Some people would define a spline to be any piecewise polynomial function. For example, deBoor's book uses this definition, and it's one of the definitive works on the subject. With that definition, there is no difference between the two kinds of interpolation you mentioned, of course.
Other people (like here) define a spline of degree $m$ to be a piecewise polynomial with $m-1$ continuous derivatives. So, with this definition, cubic splines (degree 3) would be $C_2$. These folks sometimes refer to less continuous piecewise polynomials as "subsplines" or "defective splines". In fact, a spline of degree $m$ that has a continuous $(m-k)$-th derivative is said to have defect $k$.