You are here
Homeinterpolation
Primary tabs
interpolation
Interpolation is a set of techniques in approximation where, given a set of paired data points
$(x_{1},y_{1}),(x_{2},y_{2}),\ldots,(x_{n},y_{n}),\ldots$ 
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 spline 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.
Mathematics Subject Classification
41A05 no label found65D05 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
 Corrections