Data Structures and Algorithms with Object-Oriented Design Patterns in C#
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About Polynomials


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 tex2html_wrap_inline58741. Then tex2html_wrap_inline58743.

extbfProof Each of the terms in the summation is of the form tex2html_wrap_inline58745. Since n is non-negative, a particular term will be negative only if tex2html_wrap_inline58749. Hence, for each term in the summation, tex2html_wrap_inline58751. Recall too that we have stipulated that the coefficient of the largest power of n is positive, i.e., tex2html_wrap_inline58741.


Note that for integers tex2html_wrap_inline58757, tex2html_wrap_inline58759 for tex2html_wrap_inline58761. Thus


From Equation gif we see that we have found the constants tex2html_wrap_inline58763 and tex2html_wrap_inline58765, such that for all tex2html_wrap_inline58299, tex2html_wrap_inline58769. Thus, tex2html_wrap_inline58743.

This property of the asymptotic behavior of polynomials is used extensively. In fact, whenever we have a function, which is a polynomial in n, tex2html_wrap_inline58775 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, tex2html_wrap_inline58781, to write tex2html_wrap_inline58743.

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Bruno Copyright © 2001 by Bruno R. Preiss, P.Eng. All rights reserved.