example of under-determined polynomial interpolation
Consider the following
interpolation problem:
Given with to determine all cubic polynomials
such that
This is a linear problem. Let denote the vector space 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 is the unknown. One can recast the problem into the traditional form by taking standard bases of and and then seeking all possible 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
, where is a solution of the homogeneous
problem
A basis of solutions for the latter is, evidently,
The general solution to our interpolation problem is therefore
with 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 |