Chemistry - Monomial basis

In mathematics a monomial basis is a way to uniquely describe a polynomial using a linear combination of monomials. 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 the Horner algorithm.

Contents

Definition

The monomial basis for the vector space $Π n$ of polynomials with degree n is the polynomial sequence of monomials

$1,x,x^2,.\ldots,x^n$

The monomial form of a polynomial $p \in \Pi_n$ is a linear combination of monomials

$a_0 1 + a_1 x + a_2 x^2 + \ldots + a_n x^n$

alternatively the shorter sigma notation can be used

$p=\sum_{\nu=0}^n a_{\nu}x^\nu$

Notes

A polynomial can always converted into monomial form by calculating Taylor expansion around 0.

Examples

A polynomial in $Π 4$

1 + x + 3 x 4