I have several 2-dimensional points $P=\{(x_i,y_i)\}_{i=1}^{28}$ that I calculated with an algorithm. The (natural) values $x_i$ are its input and the (rational) values $y_i$ are its output. I'm struggling with finding a closed-form formula $f(x)$ using properties of the algorithm, or insights from the problem, so my aim now is to find the function $g(x)$ for which $g(x_i)=y_i$ for $1\le x\le 28$ (and hope that $g(x)$ gives the value of the algorithm for $x>28$). I've got a feeling that knowing something about $g(x)$ found by (say) interpolation can help me work out the closed-form formula.
There is something I know for sure about $f(x)$ and is that it has an upper bound $f(x) \le (x^2 - 1)/3$. I can find lower bounds by plotting but that hasn't helped me so far (neither has helped finding other tighter upper bounds).
My attemps:
- The curve that join the points in $P$ seems to make a parabola, however, the solution $(a,b,c)$ to the system of equations $$\left\{ \begin{array}{rcr} ax_i^2 + bx_i + c &=& y_i\\ ax_j^2 + bx_j + c &=& y_j\\ ax_k^2 + bx_k + c &=& y_k\\ \end{array} \right.$$ is different for every different $(i,j,k)$. I assume that if the points in $P$ were points of a parabola of the form $ax^2 + bx + c = y$ I would get the same $(a,b,c)$ for every $(i,j,k)$.
- I tried calculating the coefficients of a conic in generalized form $ax^2 + bxy + cy^2 + mx + ny + k = 0$ but the system of 6 equations (using $a,b,c,m,n,k$) is homogeneous and the coefficient matrix is not singular so I always get the trivial solution. If I assume one of $a,b,c$ to be $1$, the solution to the system yields two straight lines.
I would like to receive advice on how to obtain the function $g(x)$. Also, what is the likelihood that $g(x)$ is not a polynomial (knowing that the $y_i$ are rational)? I mean, it could be some $a(x)x^2 + bx + c = y$ where $a(x)$ is a non-constant value as a function of $x$, say $a(x)=1/(2x + 1)$ (unfortunately, I can't compute any more values to determine whether $f(x)$ is asymptotically linear when $x\rightarrow \infty$). In this case, what methods are there to determine that one or more coefficients $a,b,c$ depend on $x$?
Thank you for your time.
EDIT
Some users asked to see a plot of the function. Here is the plot of the points $\{(x_i,y_i)\}_{i=1}^{28}$ in blue and of an upper bound $(x^2 - 1)/3$ in red.
They've also asked for details on the algorithm. I'm afraid I can't disclose many details but I can say it constructs trees that minimize a given function (which I'm certainly not going to put here). Also, while the values $y_i$ are rational and can be calculated exactly using the algorithm, and their values can be approximated using the graph, I'm afraid I can't disclose their exact values either.
EDIT (2020/08/11)
So, I managed to gather more values of the function, now reaching up to $n=140$. The values $(x,f(x))$ do not seem to shed light on the shape of $f(x)$ (whether it is a parabola, or something else...), so I decided to plot the derivative: $f'(x)=f(x)-f(x-1)$. This is what I get.
The points represent values of $f'(x)$. I thought that now, given that the function shows apparent repeating patterns, in order to find a closed-form formula for $f(x)$ I would go about proving that $f'(x)$ has those patterns. Any comments on this? Is it a good idea?