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<HTML><HEAD><TITLE>About Polynomials</TITLE></HEAD><BODY bgcolor="#FFFFFF"> <a href="../index.html" target="_top"><img src="../icons/usins.gif" alt="Logo" align=right></a><b>Data Structures and Algorithms with Object-Oriented Design Patterns in Python</b><br><A NAME="tex2html1936" HREF="page64.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="../icons/next_motif.gif"></A> <A NAME="tex2html1934" HREF="page59.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="../icons/up_motif.gif"></A> <A NAME="tex2html1928" HREF="page62.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="../icons/previous_motif.gif"></A>  <A NAME="tex2html1938" HREF="page611.html"><IMG WIDTH=43 HEIGHT=24 ALIGN=BOTTOM ALT="index" SRC="../icons/index_motif.gif"></A> <BR><HR><H2><A NAME="SECTION003140000000000000000">About Polynomials</A></H2><A NAME="secasymptoticpolyi">&#160;</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">&#160;</A>Consider a polynomial<A NAME=1555>&#160;</A>in <I>n</I> of the form<P> <IMG WIDTH=500 HEIGHT=68 ALIGN=BOTTOM ALT="eqnarray1556" SRC="img334.gif"  ><P>where  <IMG WIDTH=50 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline58967" SRC="img335.gif"  >.Then  <IMG WIDTH=97 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline58969" SRC="img336.gif"  >.</BLOCKQUOTE><P>	extbfProofEach of the terms in the summation is of the form  <IMG WIDTH=28 HEIGHT=28 ALIGN=MIDDLE ALT="tex2html_wrap_inline58971" SRC="img337.gif"  >.Since <I>n</I> is non-negative,a particular term will be negative only if  <IMG WIDTH=43 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline58975" SRC="img338.gif"  >.Hence, for each term in the summation,  <IMG WIDTH=86 HEIGHT=28 ALIGN=MIDDLE ALT="tex2html_wrap_inline58977" SRC="img339.gif"  >.Recall too that we have stipulated that the coefficient of thelargest power of <I>n</I> is positive, i.e.,  <IMG WIDTH=50 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline58967" SRC="img335.gif"  >.<P><P> <IMG WIDTH=500 HEIGHT=150 ALIGN=BOTTOM ALT="eqnarray1564" SRC="img340.gif"  ><P><P>Note that for integers  <IMG WIDTH=38 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline58983" SRC="img341.gif"  >, <IMG WIDTH=92 HEIGHT=28 ALIGN=MIDDLE ALT="tex2html_wrap_inline58985" SRC="img342.gif"  > for  <IMG WIDTH=69 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline58987" SRC="img343.gif"  >. Thus<P><A NAME="eqnbigohpoly">&#160;</A> <IMG WIDTH=500 HEIGHT=71 ALIGN=BOTTOM ALT="equation1576" SRC="img344.gif"  ><P><P>From Equation&nbsp;<A HREF="page63.html#eqnbigohpoly"><IMG  ALIGN=BOTTOM ALT="gif" SRC="../icons/cross_ref_motif.gif"></A> we see thatwe have found the constants  <IMG WIDTH=45 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline58989" SRC="img345.gif"  > and  <IMG WIDTH=90 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline58991" SRC="img346.gif"  >,such that for all  <IMG WIDTH=47 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline58525" SRC="img239.gif"  >,  <IMG WIDTH=177 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline58995" SRC="img347.gif"  >.Thus,  <IMG WIDTH=97 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline58969" SRC="img336.gif"  >.<P>This property of the asymptotic behavior of polynomialsis used extensively.In fact, whenever we have a function,which is a polynomial in <I>n</I>, <IMG WIDTH=356 HEIGHT=28 ALIGN=MIDDLE ALT="tex2html_wrap_inline59001" SRC="img348.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=20 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline59007" SRC="img349.gif"  >,to write  <IMG WIDTH=97 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline58969" SRC="img336.gif"  >.<P><HR><A NAME="tex2html1936" HREF="page64.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="../icons/next_motif.gif"></A> <A NAME="tex2html1934" HREF="page59.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="../icons/up_motif.gif"></A> <A NAME="tex2html1928" HREF="page62.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="../icons/previous_motif.gif"></A>  <A NAME="tex2html1938" HREF="page611.html"><IMG WIDTH=43 HEIGHT=24 ALIGN=BOTTOM ALT="index" SRC="../icons/index_motif.gif"></A> <P><ADDRESS><img src="../icons/bruno.gif" alt="Bruno" align=right><a href="../copyright.html">Copyright &#169; 2003</a> by <a href="../signature.html">Bruno R. Preiss, P.Eng.</a>  All rights reserved.</ADDRESS></BODY></HTML>

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