example of under-determined polynomial interpolation

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

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

such that

$p(x_{1})=y_{1},\quad p(x_{2})=y_{2}.$

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

E:\mathcal{P}_{3}\rightarrow\mathbb{R}^{2},

given by

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

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

\begin{pmatrix}\left(x_{1}\right)^{3}&\left(x_{1}\right)^{2}&x_{1}&1\\ \left(x_{2}\right)^{3}&\left(x_{2}\right)^{2}&x_{2}&1\\ \end{pmatrix}\begin{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

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}

The general solution of our interpolation problem is therefore given as $p_{0}+q$ , where $q\in\mathcal{P}_{3}$ is a solution of the homogeneous problem

E(q)=0.

A basis of solutions for the latter is, evidently,

q_{1}(x)=(x-x_{1})(x-x_{2}),\quad q_{2}(x)=xq_{1}(x),\qquad x\in\mathbb{R}

The general solution to our interpolation problem is therefore

p(x)=\frac{x-x_{1}}{x_{2}-x_{1}}y_{1}+\frac{x-x_{2}}{x_{1}-x_{2}}y_{2}+(ax+b)(% x-x_{1})(x-x_{2}),\quad x\in\mathbb{R},

with $a,b\in\mathbb{R}$ arbitrary. The general under-determined interpolation problem is treated in an entirely analogous manner.

Title	example of under-determined polynomial interpolation
Canonical name	ExampleOfUnderdeterminedPolynomialInterpolation
Date of creation	2013-03-22 12:35:22
Last modified on	2013-03-22 12:35:22
Owner	rmilson (146)
Last modified by	rmilson (146)
Numerical id	5
Author	rmilson (146)
Entry type	Example
Classification	msc 15A06