chaos world

Polynomial Interpolattion through n points(2 dimensions)

Think about a series of different points in XOY plane, assume $(a_1,b_1),(a_2,b_2),…,(a_n,b_n)$, is there a curve go through all these points? How many curves can satisfy?

On this problem, we can assume there exist a solution as below:

$$f(x)=c_0+c_1x+c_2x^2+{cdots}+c_{n-1}x^{n-1}$$

Then replace $x$ with $a_i$ and $f(x)$ with $b_i$, we can get a group of equations like this:

$$begin{cases} b_1=c_0+c_1a_1+c_2{a_1}^2+{cdots}+c_{n-1}{a_1}^{n-1}\ b_2=c_0+c_1a_2+c_2{a_2}^2+{cdots}+c_{n-1}{a_2}^{n-1}\ {vdots}quad{cdots}\ b_1=c_0+c_1a_n+c_2{a_n}^2+{cdots}+c_{n-1}{a_n}^{n-1} end{cases}$$

In order to solve $c_0,c_1,…,c_{n-1}$ from the equations group, we should first extract the coefficient matrix from it as following:

$$% <![CDATA[ begin{bmatrix} 1&{a_1}&{cdots}&{a_1^{n-1}}\ 1&{a_2}&{cdots}&{a_2^{n-1}}\ {vdots}&{vdots}&{ddots}&{vdots}\ 1&{a_n}&{cdots}&{a_n^{n-1}} end{bmatrix} %]]>$$

If you transpose it, you can get the Vandermonde matrix:

$$% <![CDATA[ begin{bmatrix} 1&1&{cdots}&1\ {a_1}&{a_2}&{cdots}&{a_n}\ {vdots}&{vdots}&{ddots}&{vdots}\ {a_1^{n-1}}&{a_2^{n-1}}&{cdots}&{a_n^{n-1}} end{bmatrix} %]]>$$

its determinant is equal to

$$% <![CDATA[ prod_{1le j < i le n}{(a_i-a_j)} %]]>$$

So the determinant is not equal to $0$ while the $a_i$ is different from each other. Now we can derive a conclusion that there is one and only one solution $[ c_0,c_1,cdots,c_{n-1}]$ for the equations group from the determinant.

There is only one $n-1$ times polynomial function can satisfy interpolation of $n$ diffirent points on $2$ dimension plane.