Order of a polynomial
In mathematics, the order of a polynomial may have several meanings, depending on the context.
- Order has been used to denoted the degree of a polynomial. Nowadays, this terminology is rarely used.
- Order may refer to the order of the polynomial, viewed as a power series, that is the degree of its nonzero monomial of lowest degree.
- The meaning in this article.
In this article, the order of a polynomial, relative to a particular set of polynomial basis functions spanning the polynomial vector space in which a given polynomial is included, is the highest degree among those basis functions used to express the polynomial.
Consider the following polynomial:
where are the polynomial coefficients and the set of basis functions which span the polynomial vector space.
If the polynomial coefficients are:
under this polynomial vector space, is expressed as follows:
The degree of this polynomial would be 1. Yet, due to the set of basis functions which is used to define this polynomial, its order would be .
then, the polynomial coefficients would be:
The degree of this polynomial would still be 1, but as the highest degree of the Lagrangian basis functions is , then the order of this polynomial is 2.
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