# example of under-determined polynomial interpolation

Consider the following
interpolation^{} problem:

Given ${x}_{\mathrm{1}}\mathrm{,}{y}_{\mathrm{1}}\mathrm{,}{x}_{\mathrm{2}}\mathrm{,}{y}_{\mathrm{2}}\mathrm{\in}\mathrm{R}$ with ${x}_{\mathrm{1}}\mathrm{\ne}{x}_{\mathrm{2}}$ to determine all cubic polynomials

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

$$p({x}_{1})={y}_{1},p({x}_{2})={y}_{2}.$$

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

$$E:{\mathcal{P}}_{3}\to {\mathbb{R}}^{2},$$ |

given by

$$p\mapsto \left(\begin{array}{c}\hfill p({x}_{1})\hfill \\ \hfill p({x}_{2})\hfill \end{array}\right),p\in {\mathcal{P}}_{3}$$ |

The interpolation problem in question is represented by the equation

$$E(p)=\left(\begin{array}{c}\hfill {y}_{1}\hfill \\ \hfill {y}_{2}\hfill \end{array}\right)$$ |

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

$$\left(\begin{array}{cccc}\hfill {\left({x}_{1}\right)}^{3}\hfill & \hfill {\left({x}_{1}\right)}^{2}\hfill & \hfill {x}_{1}\hfill & \hfill 1\hfill \\ \hfill {\left({x}_{2}\right)}^{3}\hfill & \hfill {\left({x}_{2}\right)}^{2}\hfill & \hfill {x}_{2}\hfill & \hfill 1\hfill \end{array}\right)\left(\begin{array}{c}\hfill a\hfill \\ \hfill b\hfill \\ \hfill c\hfill \\ \hfill d\hfill \end{array}\right)=\left(\begin{array}{c}\hfill {y}_{1}\hfill \\ \hfill {y}_{2}\hfill \end{array}\right)$$ |

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

$${p}_{0}(x)=\frac{x-{x}_{1}}{{x}_{2}-{x}_{1}}{y}_{1}+\frac{x-{x}_{2}}{{x}_{1}-{x}_{2}}{y}_{2},x\in \mathbb{R}$$ |

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,

$${q}_{1}(x)=(x-{x}_{1})(x-{x}_{2}),{q}_{2}(x)=x{q}_{1}(x),x\in \mathbb{R}$$ |

The general solution to our interpolation problem is therefore

$$p(x)=\frac{x-{x}_{1}}{{x}_{2}-{x}_{1}}{y}_{1}+\frac{x-{x}_{2}}{{x}_{1}-{x}_{2}}{y}_{2}+(ax+b)(x-{x}_{1})(x-{x}_{2}),x\in \mathbb{R},$$ |

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 |