Finding an equation given points

Question

I have a list of dozens of points and would like to find a way of finding an equation that passes through every point. I am NOT looking for linear regression

Answer 1

Two points $(x_1, y_1)$ and $(x_2, y_2)$ uniquely determine a straight line and the equation is given by

$ y - y_1 = m (x - x_1) $

where

$m = \mbox{slope} = {y_2 - y_1 \over x_2 - x_1}$

This is a general formula..

Now, if you have dozens of points and look for a straight line that passes through every one of them, then it is not possible in most of the situations unless we are in a special case when all these points lie in a straight line (collinear points).

If the required polynomial is not required to be a straight line, then you generalize the above calculation..

Suppose that you have a data of $N$ points.

Then we can try to find a polynomial of degree $N - 1$ given by

$$p(x) = a_0 x^{{N - 1}} + \cdots + a_{N - 2} x + a_{N - 1}$$.

(Note that this has a total of $N$ unknowns.)

If this polynomial passes through all the $N$ points $(x_i, y_i)$, it is called an interpolating polynomial and we have a set of $N$ linear equations in $N$ unknowns ($a_0, a_1, \ldots, a_{N - 1}$)

Set this up as a linear system of equations.

A necessary and sufficient condition for a unique solution of this system (unique interpolating polynomial) is that the coefficient matrix has full rank (or nonsingular).

Answer 2

Well, if you do not mind have a polynomial of dozens of degrees and you have points $\{(x_i,y_i)\vert 1\le i\le m\}$

$$ f(x)=\sum_{i=1}^m\left(\prod_{j\ne i}^m\frac{x-x_j}{x_i-x_j}\right)y_i $$

Then when $x=x_i$ the coefficient of $y_i$ will be $1$.

But the coefficient for each $y_j$ where $j\ne i$ will have the term $x_i-x_i$ in the numerator, giving a coefficient of $0$.

Thus, for each $i$, $f(x_i)=y_i$.

But your function is a polynomial of degree $m-1$, which you say, is dozens of degrees.

Answer 3

Import your point coordinates into Excel, and use the Trend Line function. This will fit various types of equations to the points. The fitted curve won’t go through the points exactly, but if you have dozens of points then that’s probably not what you want, anyway.