Polynomial interpolation
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:
\[\begin{bmatrix} 1&{a_1}&{\cdots}&{a_1^{n-1}}\\ 1&{a_2}&{\cdots}&{a_2^{n-1}}\\ {\vdots}&{\vdots}&{\ddots}&{\vdots}\\ 1&{a_n}&{\cdots}&{a_n^{n-1}} \end{bmatrix}\]If you transpose it, you can get the Vandermonde matrix:
\[\begin{bmatrix} 1&1&{\cdots}&1\\ {a_1}&{a_2}&{\cdots}&{a_n}\\ {\vdots}&{\vdots}&{\ddots}&{\vdots}\\ {a_1^{n-1}}&{a_2^{n-1}}&{\cdots}&{a_n^{n-1}} \end{bmatrix}\]its determinant is equal to
\[\prod_{1\le j < i \le n}{(a_i-a_j)}\]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.
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