example of under-determined polynomial interpolation

Consider the following interpolationMathworldPlanetmath problem:

Given x1,y1,x2,y2R with x1x2 to determine all cubic polynomials


such that


This is a linear problem. Let 𝒫3 denote the vector spaceMathworldPlanetmath of cubic polynomials. The underlying linear mapping is the multi-evaluation mapping


given by


The interpolation problem in question is represented by the equation


where p𝒫3 is the unknown. One can recast the problem into the traditional form by taking standard bases of 𝒫3 and 2 and then seeking all possible a,b,c,d such that


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


The general solution of our interpolation problem is therefore given as p0+q, where q𝒫3 is a solution of the homogeneousPlanetmathPlanetmath problem


A basis of solutions for the latter is, evidently,


The general solution to our interpolation problem is therefore


with a,b 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