Editing Polynomial interpolation (section)

== Applications ==
The original use of interpolation polynomials was to approximate values of important [[transcendental function]]s such as [[natural logarithm]] and [[trigonometric function]]s. Starting with a few accurately computed data points, the corresponding interpolation polynomial will approximate the function at an arbitrary nearby point. Polynomial interpolation also forms the basis for algorithms in [[numerical quadrature]] ([[Simpson's rule]]) and [[numerical ordinary differential equations]] ([[multigrid method]]s).

In [[computer graphics]], polynomials can be used to approximate complicated plane curves given a few specified points, for example the shapes of letters in [[typography]]. This is usually done with [[Bézier curve]]s, which are a simple generalization of interpolation polynomials (having specified tangents as well as specified points).

In numerical analysis, polynomial interpolation is essential to perform sub-quadratic multiplication and squaring, such as [[Karatsuba multiplication]] and [[Toom–Cook multiplication]], where interpolation through points on a product polynomial yields the specific product required. For example, given ''a'' = ''f''(''x'') = ''a''<sub>0</sub>''x''<sup>0</sup> + ''a''<sub>1</sub>''x''<sup>1</sup> + ··· and ''b'' = ''g''(''x'') = ''b''<sub>0</sub>''x''<sup>0</sup> + ''b''<sub>1</sub>''x''<sup>1</sup> + ···, the product ''ab'' is a specific value of ''W''(''x'') = ''f''(''x'')''g''(''x''). One may easily find points along ''W''(''x'') at small values of ''x'', and interpolation based on those points will yield the terms of ''W''(''x'') and the specific product ''ab''. As fomulated in Karatsuba multiplication, this technique is substantially faster than quadratic multiplication, even for modest-sized inputs, especially on parallel hardware.

In [[computer science]], polynomial interpolation also leads to algorithms for [[Secure multi-party computation|secure multi party computation]] and [[secret sharing]].