interpolation
Interpolation^{} is a set of techniques in approximation where, given a set of paired data points
$$({x}_{1},{y}_{1}),({x}_{2},{y}_{2}),\mathrm{\dots},({x}_{n},{y}_{n}),\mathrm{\dots}$$ 
one is often interested in

•
finding a relation (usually in the form of a function $f$) that passes through (or is satisfied by) every one of these points, if the relation is unknown at the beginning,

•
finding a simplified relation to replace the original known relation that is very complicated and difficult to use,

•
finding other paired data points $({x}_{\alpha},{y}_{\alpha})$ in addition to the existing ones.
The data points $({x}_{i},{y}_{i})$ are called the breakpoints, and the function $f$ is the interpolating function such that $f({x}_{i})={y}_{i}$ for each $i$.
The choice of the interpolating function depends on what we wish to do with it. In some cases a polynomial is required, sometimes a piecewise linear function is prefered (linear interpolation), other times a http://planetmath.org/node/4339spline is of interest, when the interpolating function is required to not only to be continuous^{}, but differentiable^{}, or even smooth.
Even different strategies for finding the same interpolating function are of interest. The Lagrange interpolation formula is a direct way to calculate the interpolating polynomial. The Vandermonde interpolation formula is mainly of interest as a theoretical tool. Numerical implementation of Vandermonde interpolation involves solution of large ill conditioned linear systems, so numerical stability is questionable.
Title  interpolation 

Canonical name  Interpolation 
Date of creation  20130322 14:20:05 
Last modified on  20130322 14:20:05 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  13 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 41A05 
Classification  msc 65D05 
Defines  breakpoints 
Defines  interpolating function 