# 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 $i$.

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 Interpolation 2013-03-22 14:20:05 2013-03-22 14:20:05 CWoo (3771) CWoo (3771) 13 CWoo (3771) Definition msc 41A05 msc 65D05 breakpoints interpolating function