PlanetMath: example of under-determined polynomial interpolation

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example of under-determined polynomial interpolation

(Example)

Consider the following interpolation problem:

Given $x_1,y_1,x_2,y_2\in \mathbb{R}$ with $x_1\neq x_2$ to determine all cubic polynomials

$\displaystyle p(x)= ax^3+bx^2+cx+d,\quad x,a,b,c,d\in \mathbb{R}$

such that

$\displaystyle p(x_1) = y_1,\quad p(x_2) = y_2.$

This is a linear problem. Let $\mathcal{P}_3$ denote the vector space of cubic polynomials. The underlying linear mapping is the multi-evaluation mapping

$\displaystyle E:\mathcal{P}_3\rightarrow\mathbb{R}^2,$

given by

$\displaystyle p\mapsto \begin{pmatrix} p(x_1)\\ p(x_2) \end{pmatrix},\quad p\in \mathcal{P}_3$

The interpolation problem in question is represented by the equation

$\displaystyle E(p) = \begin{pmatrix} y_1 \\ y_2 \end{pmatrix}$

where $p\in \mathcal{P}_3$ is the unknown. One can recast the problem into the traditional form by taking standard bases of $\mathcal{P}_3$ and $\mathbb{R}^2$ and then seeking all possible $a,b,c,d\in\mathbb{R}$ such that

$\displaystyle \begin{pmatrix} \left(x_1\right)^3 & \left(x_1\right)^2 & x_1 & 1... ...{pmatrix} a\\ b\\ c\\ d \end{pmatrix}= \begin{pmatrix} y_1 \\ y_2 \end{pmatrix}$

However, it is best to treat this problem at an abstract level, rather than mucking about with row reduction. The Lagrange interpolation formula gives us a particular solution, namely the linear polynomial

$\displaystyle p_0(x) = \frac{x-x_1}{x_2-x_1}y_1 + \frac{x-x_2}{x_1-x_2} y_2,\quad x\in \mathbb{R}$