page68.html
来自「wqeqwvrw rkjqhwrjwq jkhrjqwhrwq jkhrwq」· HTML 代码 · 共 82 行
HTML
82 行
<HTML>
<HEAD>
<TITLE>About Polynomials Again</TITLE>
</HEAD>
<BODY bgcolor="#FFFFFF">
<img src="cover75.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cover75.gif" alt="Logo" align=right>
<b>Data Structures and Algorithms
with Object-Oriented Design Patterns in C++</b><br>
<A NAME="tex2html2735" HREF="page69.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page69.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="tex2html2733" HREF="page66.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page66.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="tex2html2729" HREF="page67.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page67.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="tex2html2737" 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="tex2html2738" 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="SECTION004220000000000000000">About Polynomials Again</A></H2>
<A NAME="secasymptoticpolyii"> </A>
<P>
In this section we reexamine the asymptotic behavior of polynomials in <I>n</I>.
In Section <A HREF="page61.html#secasymptoticpolyi" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page61.html#secasymptoticpolyi"><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 showed that <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" >.
That is, <I>f</I>(<I>n</I>) grows asymptotically no more quickly than <IMG WIDTH=20 HEIGHT=10 ALIGN=BOTTOM ALT="tex2html_wrap_inline60021" SRC="img418.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img418.gif" >.
This time we are interested in the asymptotic lower bound
rather than the asymptotic upper bound.
We will see that as <I>n</I> gets large,
the term involving <IMG WIDTH=20 HEIGHT=10 ALIGN=BOTTOM ALT="tex2html_wrap_inline60021" SRC="img418.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img418.gif" > also dominates the lower bound
in the sense that <I>f</I>(<I>n</I>) grows asymptotically <em>as quickly</em> as <IMG WIDTH=20 HEIGHT=10 ALIGN=BOTTOM ALT="tex2html_wrap_inline60021" SRC="img418.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img418.gif" >.
I.e., that <IMG WIDTH=96 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline60031" SRC="img419.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img419.gif" >.
<P>
<BLOCKQUOTE> <b>Theorem</b><A NAME="theoremviii"> </A>
Consider a polynomial<A NAME=1729> </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=96 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline60031" SRC="img419.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img419.gif" >.
</BLOCKQUOTE>
<P>
extbfProof
We begin by taking the term <IMG WIDTH=40 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline60039" SRC="img420.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img420.gif" > out of the summation:
<P> <IMG WIDTH=500 HEIGHT=98 ALIGN=BOTTOM ALT="eqnarray1738" SRC="img421.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img421.gif" ><P>
<P>
Since, <I>n</I> is a non-negative integer and <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" >,
the term <IMG WIDTH=40 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline60039" SRC="img420.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img420.gif" > is positive.
For each of the remaining terms in the summation, <IMG WIDTH=97 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline60047" SRC="img422.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img422.gif" >.
Hence
<P> <IMG WIDTH=500 HEIGHT=156 ALIGN=BOTTOM ALT="eqnarray1744" SRC="img423.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img423.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=118 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline60051" SRC="img424.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img424.gif" > for <IMG WIDTH=108 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline60053" SRC="img425.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img425.gif" >.
Thus
<P> <IMG WIDTH=500 HEIGHT=102 ALIGN=BOTTOM ALT="eqnarray1758" SRC="img426.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img426.gif" ><P>
<P>
Consider the term in parentheses on the right.
What we need to do is to find a positive constant <I>c</I>
and an integer <IMG WIDTH=16 HEIGHT=15 ALIGN=MIDDLE ALT="tex2html_wrap_inline59043" SRC="img238.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img238.gif" >
so that for all integers <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" >
this term is greater than or equal to <I>c</I>:
<P> <IMG WIDTH=372 HEIGHT=46 ALIGN=BOTTOM ALT="displaymath60013" SRC="img427.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img427.gif" ><P>
<P>
We choose the value <IMG WIDTH=16 HEIGHT=15 ALIGN=MIDDLE ALT="tex2html_wrap_inline59043" SRC="img238.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img238.gif" > for which the term is greater than zero:
<P> <IMG WIDTH=500 HEIGHT=101 ALIGN=BOTTOM ALT="eqnarray1775" SRC="img428.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img428.gif" ><P>
The value
<IMG WIDTH=176 HEIGHT=38 ALIGN=MIDDLE ALT="tex2html_wrap_inline60065" SRC="img429.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img429.gif" >
will suffice!
Thus
<P><A NAME="eqnomegapoly"> </A> <IMG WIDTH=500 HEIGHT=130 ALIGN=BOTTOM ALT="eqnarray1789" SRC="img430.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img430.gif" ><P>
<P>
From Equation <A HREF="page68.html#eqnomegapoly" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page68.html#eqnomegapoly"><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=16 HEIGHT=15 ALIGN=MIDDLE ALT="tex2html_wrap_inline59043" SRC="img238.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img238.gif" > and <I>c</I>,
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=176 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline60073" SRC="img431.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img431.gif" >.
Thus, <IMG WIDTH=96 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline60031" SRC="img419.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img419.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=96 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline60031" SRC="img419.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img419.gif" >.
<P>
<HR><A NAME="tex2html2735" HREF="page69.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page69.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="tex2html2733" HREF="page66.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page66.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="tex2html2729" HREF="page67.html" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page67.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="tex2html2737" 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="tex2html2738" 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> <P><ADDRESS>
<img src="bruno.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/bruno.gif" alt="Bruno" align=right>
<a href="javascript:if(confirm('http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/copyright.html \n\nThis file was not retrieved by Teleport Pro, because it is addressed on a domain or path outside the boundaries set for its Starting Address. \n\nDo you want to open it from the server?'))window.location='http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/copyright.html'" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/copyright.html">Copyright © 1997</a> by <a href="javascript:if(confirm('http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/signature.html \n\nThis file was not retrieved by Teleport Pro, because it is addressed on a domain or path outside the boundaries set for its Starting Address. \n\nDo you want to open it from the server?'))window.location='http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/signature.html'" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/signature.html">Bruno R. Preiss, P.Eng.</a> All rights reserved.
</ADDRESS>
</BODY>
</HTML>
⌨️ 快捷键说明
复制代码Ctrl + C
搜索代码Ctrl + F
全屏模式F11
增大字号Ctrl + =
减小字号Ctrl + -
显示快捷键?