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<TITLE>About Polynomials</TITLE>
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<b>Data Structures and Algorithms
with Object-Oriented Design Patterns in C++</b><br>
<A NAME="tex2html2653" HREF="page62.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page62.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="next_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/next_motif.gif"></A> <A NAME="tex2html2651" HREF="page57.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page57.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="up_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/up_motif.gif"></A> <A NAME="tex2html2645" HREF="page60.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page60.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="previous_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/previous_motif.gif"></A> <A NAME="tex2html2655" HREF="page9.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page9.html"><IMG WIDTH=65 HEIGHT=24 ALIGN=BOTTOM ALT="contents" SRC="contents_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/contents_motif.gif"></A> <A NAME="tex2html2656" HREF="page620.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page620.html"><IMG WIDTH=43 HEIGHT=24 ALIGN=BOTTOM ALT="index" SRC="index_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/index_motif.gif"></A> <BR><HR>
<H2><A NAME="SECTION004140000000000000000">About Polynomials</A></H2>
<A NAME="secasymptoticpolyi"> </A>
<P>
In this section we examine the asymptotic behavior of polynomials in <I>n</I>.
In particular, we will see that as <I>n</I> gets large,
the term involving the highest power of <I>n</I> will dominate all the others.
Therefore, the asymptotic behavior is determined by that term.
<P>
<BLOCKQUOTE> <b>Theorem</b><A NAME="theoremvi"> </A>
Consider a polynomial<A NAME=1489> </A>
in <I>n</I> of the form
<P> <IMG WIDTH=500 HEIGHT=67 ALIGN=BOTTOM ALT="eqnarray1490" SRC="img337.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img337.gif" ><P>
where <IMG WIDTH=49 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline59517" SRC="img338.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img338.gif" >.
Then <IMG WIDTH=98 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline59519" SRC="img339.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img339.gif" >.
</BLOCKQUOTE>
<P>
extbfProof
Each of the terms in the summation is of the form <IMG WIDTH=26 HEIGHT=28 ALIGN=MIDDLE ALT="tex2html_wrap_inline59521" SRC="img340.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img340.gif" >.
Since <I>n</I> is non-negative,
a particular term will be negative only if <IMG WIDTH=42 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline59525" SRC="img341.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img341.gif" >.
Hence, for each term in the summation, <IMG WIDTH=85 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline59527" SRC="img342.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img342.gif" >.
Recall too that we have stipulated that the coefficient of the
largest power of <I>n</I> is positive, i.e., <IMG WIDTH=49 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline59517" SRC="img338.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img338.gif" >.
<P>
<P> <IMG WIDTH=500 HEIGHT=149 ALIGN=BOTTOM ALT="eqnarray1498" SRC="img343.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img343.gif" ><P>
<P>
Note that for integers <IMG WIDTH=38 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline59533" SRC="img344.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img344.gif" >,
<IMG WIDTH=92 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline59535" SRC="img345.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img345.gif" > for <IMG WIDTH=69 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline59537" SRC="img346.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img346.gif" >. Thus
<P><A NAME="eqnbigohpoly"> </A> <IMG WIDTH=500 HEIGHT=70 ALIGN=BOTTOM ALT="equation1510" SRC="img347.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img347.gif" ><P>
<P>
From Equation <A HREF="page61.html#eqnbigohpoly" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page61.html#eqnbigohpoly"><IMG ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A> we see that
we have found the constants <IMG WIDTH=45 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline59539" SRC="img348.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img348.gif" > and <IMG WIDTH=91 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline59541" SRC="img349.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img349.gif" >,
such that for all <IMG WIDTH=46 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline59075" SRC="img242.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img242.gif" >, <IMG WIDTH=175 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline59545" SRC="img350.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img350.gif" >.
Thus, <IMG WIDTH=98 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline59519" SRC="img339.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img339.gif" >.
<P>
This property of the asymptotic behavior of polynomials
is used extensively.
In fact, whenever we have a function,
which is a polynomial in <I>n</I>,
<IMG WIDTH=357 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline59551" SRC="img351.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img351.gif" >
we will immediately ``drop'' the less significant terms
(i.e., terms involving powers of <I>n</I> which are less than <I>m</I>),
as well as the leading coefficient, <IMG WIDTH=18 HEIGHT=15 ALIGN=MIDDLE ALT="tex2html_wrap_inline59557" SRC="img352.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img352.gif" >,
to write <IMG WIDTH=98 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline59519" SRC="img339.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img339.gif" >.
<P>
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