Data Structures and Algorithms
with Object-Oriented Design Patterns in C#|
In this section we examine the asymptotic behavior of polynomials in n. In particular, we will see that as n gets large, the term involving the highest power of n will dominate all the others. Therefore, the asymptotic behavior is determined by that term.
Theorem Consider a polynomial in n of the form
where . Then .
extbfProof Each of the terms in the summation is of the form . Since n is non-negative, a particular term will be negative only if . Hence, for each term in the summation, . Recall too that we have stipulated that the coefficient of the largest power of n is positive, i.e., .
Note that for integers , for . Thus
From Equation we see that we have found the constants and , such that for all , . Thus, .
This property of the asymptotic behavior of polynomials is used extensively. In fact, whenever we have a function, which is a polynomial in n, we will immediately ``drop'' the less significant terms (i.e., terms involving powers of n which are less than m), as well as the leading coefficient, , to write .