Divided differences

2013-11-06T11:02:08Z

69.143.237.51:

In [[mathematics]] a '''monomial basis''' is a way to describe uniquely a [[polynomial]] using a [[linear combination]] of [[monomial]]s. This description, the '''monomial form''' of a polynomial, is often used because of the simple structure of the monomial basis.

Polynomials in monomial form can be evaluated efficiently using [[Horner's method]].

==Definition==

The '''monomial basis''' for the vector space <math>\Pi_n</math> of polynomials with degree ''n'' is the [[polynomial sequence]] of monomials

:<math>1,x,x^2,.\ldots,x^n</math>

The '''monomial form''' of a polynomial <math>p \in \Pi_n</math> is a linear combination of monomials

:<math>a_0 1 + a_1 x + a_2 x^2 + \ldots + a_n x^n</math>

alternatively the shorter [[sigma notation]] can be used

:<math>p=\sum_{\nu=0}^n a_{\nu}x^\nu</math>

==Notes==

A polynomial can always be converted into monomial form by calculating its [[Taylor expansion]] around 0.

==Examples==

A polynomial in <math>\Pi_4</math>

:<math>1+x+3x^4</math>

==See also==
*[[Horner's method]]
*[[Polynomial sequence]]
*[[Newton polynomial]]
*[[Lagrange polynomial]]
*[[Legendre polynomial]]
*[[Bernstein form]]
*[[Chebyshev form]]

[[Category:Algebra]]
[[Category:Polynomials]]

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Divided differences