Polynomials can be used to approximate complicated curves, for example, the shapes of letters in typography,^{} given a few points. A relevant application is the evaluation of the natural logarithm and trigonometric functions: pick a few known data points, create a lookup table, and interpolate between those data points. This results in significantly faster computations.^{} Polynomial interpolation also forms the basis for algorithms in numerical quadrature and numerical ordinary differential equations and Secure Multi Party Computation, Secret Sharing schemes.

Polynomial interpolation is also essential to perform sub-quadratic multiplication and squaring such as Karatsuba multiplication and Toom–Cook multiplication, where an interpolation through points on a polynomial which defines the product yields the product itself. For example, given *a* = *f*(*x*) = *a*_{0}*x*^{0} + *a*_{1}*x*^{1} + ... and *b* = *g*(*x*) = *b*_{0}*x*^{0} + *b*_{1}*x*^{1} + ..., the product *ab* is equivalent to *W*(*x*) = *f*(*x*)*g*(*x*). Finding points along *W*(*x*) by substituting *x* for small values in *f*(*x*) and *g*(*x*) yields points on the curve. Interpolation based on those points will yield the terms of *W*(*x*) and subsequently the product *ab*. In the case of Karatsuba multiplication this technique is substantially faster than quadratic multiplication, even for modest-sized inputs. This is especially true when implemented in parallel hardware.

Given a set of *n* + 1 data points (*x _{i}*,

The unisolvence theorem states that such a polynomial *p* exists and is unique, and can be proved by the Vandermonde matrix, as described below.

The theorem states that for *n* + 1 interpolation nodes (*x _{i}*), polynomial interpolation defines a linear bijection

This page was last edited on 15 January 2018, at 03:11.

Reference: https://en.wikipedia.org/wiki/Polynomial_interpolation under CC BY-SA license.

Reference: https://en.wikipedia.org/wiki/Polynomial_interpolation under CC BY-SA license.

- Numerical Analysis
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- Karatsuba Multiplication
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- Unisolvence
- Vandermonde Matrix
- Bijection

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