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Also, does it relate to the degree of the polynomial in any way? I am struggling to get a high-level understanding of the characteristics of different degrees of polynomials - for example, their shape etc. In my textbook, as another example, it says that we can use cubic Hermite interpolation polynomials to construct a cubic spline. Why not a quadratic Hermite interpolation polynomial to build a quadratic spline? What would be ridiculous about the last statement?

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    $\begingroup$ What happens to the power of each term in a polynomial when you differentiate it once ? So what happens to the degree of the polynomial when you differentiate it once ? Can you see how this places a limit on the number of times a polynomial can be differentiated before you get a constant function (which is a degree $0$ polynomial) ? $\endgroup$ – gandalf61 Apr 16 at 11:35
  • $\begingroup$ I actually know this. When you point it out it is obvious but when I try to juggle everything in my head I lose the big picture and then everything falls apart. So the polynomial can be differentiated the number of times of the degree plus one. If one polynomial takes longer to be differentiated to reach 0 (i.e has a higher degree) than another what does it tell us about their respective shapes? $\endgroup$ – Tightrope Apr 16 at 11:41
  • $\begingroup$ @Tightrope Not much. It will get steeper, eventually. Like any parabola eventually gets steeper than any line, if only you walk far enough away from the origin. But that's about it, in general terms. Also, I would be impressed if anyone could tell the degree of a polynomial above degree 3 or 4 just by looking at (a part of) the graph, so you don't need to worry too much about the specifics of what shape they have. Higher degree means more possibility to wave up and down, but that's it. The graph of a higher degree polynomial doesn't have to wave up and down at all. $\endgroup$ – Arthur Apr 16 at 11:42
  • $\begingroup$ Do you ask about the value of the polynomial at all $x$ or at a certain $x$? $\endgroup$ – user Apr 16 at 11:51
  • $\begingroup$ I'm not sure how to answer your question. I am not asking about the value of the polynomial rather the degree and how often it can be differentiated. What would the value of the polynomial at all x entail? Isn't it just the polynomial itself? $\endgroup$ – Tightrope Apr 16 at 12:00
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Assuming your coefficients are real numbers (or rational, or integers, or complex), then the number of times you can differentiate a polynomial before reaching the $0$ polynomial is the degree of the polynomial plus $1$. Each time you differentiate, you get a result which has a degree $1$ lower than the one you started with, and only when you reach a degree $0$ polynomial (a constant) can you differentiate to get $0$.

And the problem with quadratic splines is that you don't have enough degrees of freedom. You can't freely set

  • Value at the start point
  • Value at the end point
  • Derivative at the start point
  • Derivative at the end point

if you only have a quadratic to work with. A quadratic has three coefficients, and thus only three pieces of information (like the value at some point, or the value of the derivative at some point) is enough to fully specify one. If you try to cram in four pieces of information (like the list above), you will most likely not be able to find a function which satisfies all of them.

A cubic, on the other hand, has $4$ coefficients, and thus has room for four separate pieces of information.

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  • $\begingroup$ I feel stupid for asking this but I want to make it more concrete. You mentioned that a cubic has four pieces of information corresponding to the number of coefficients. If I have a cubic function, say $f(x) = 2x^3 + 3x^2 + 2x -7$ what specific information does the term $2x^3$ give me? And $3x^2$? and so forth? $\endgroup$ – Tightrope Apr 16 at 11:50
  • $\begingroup$ @Tightrope Exactly what, graphically, the different coefficients correspond to is difficult to describe (except the constant term). But each piece of information locks down degrees of freedom for the coefficients. For instance, if you know that $f(0) = -7$, then you know that the constant term is $7$. If you know that $f(1) = 0$, then you know that the sum of all the coefficients is $0$. And in this way, each piece of information tells you something new about the coefficients. Once you have four pieces of information, you usually have enough to find the coefficients. $\endgroup$ – Arthur Apr 16 at 12:03
  • $\begingroup$ Another question. Let's say I want to interpolate a curving function using only two piecewise-linear interpolating polynomial of equal sub intervals. In other words the two interpolating polynomials meet in the middle. Let's say that it is important that the interpolating polynomials meet "smoothly" in the middle but that at their other endpoints (the endpoints of the function we attempt to approximate) smoothness isn't that important. $\endgroup$ – Tightrope Apr 16 at 12:15
  • $\begingroup$ Would two quadratic functions be sufficient here? Since I only require three pieces of information for each interpolating polynomial: a value at one endpoint (the one not requiring smoothness), a value at the other endpoint (the one meeting the other interpolating polynomial) and a derivative at the latter endpoint in order to facilitate the smoothness in the middle? $\endgroup$ – Tightrope Apr 16 at 12:15
  • $\begingroup$ @Tightrope Yes, in that case a quadratic polynomial on each of the two parts would be sufficient, for the reason you describe. $\endgroup$ – Arthur Apr 16 at 12:25

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