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<HTML><HEAD><TITLE>Canonical Matrix Multiplication</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="tex2html2313" HREF="page97.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="../icons/next_motif.gif"></A> <A NAME="tex2html2311" HREF="page89.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="../icons/up_motif.gif"></A> <A NAME="tex2html2307" HREF="page95.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="../icons/previous_motif.gif"></A> <A NAME="tex2html2315" HREF="page611.html"><IMG WIDTH=43 HEIGHT=24 ALIGN=BOTTOM ALT="index" SRC="../icons/index_motif.gif"></A> <BR><HR><H2><A NAME="SECTION004270000000000000000">Canonical Matrix Multiplication</A></H2><A NAME="secfdsmatmul"> </A><P>Given an <IMG WIDTH=42 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline60545" SRC="img618.gif" > matrix <I>A</I>and an <IMG WIDTH=37 HEIGHT=17 ALIGN=MIDDLE ALT="tex2html_wrap_inline60557" SRC="img620.gif" > matrix <I>B</I>,the product <I>C</I>=<I>AB</I> is an <IMG WIDTH=42 HEIGHT=17 ALIGN=MIDDLE ALT="tex2html_wrap_inline60563" SRC="img621.gif" > matrix.The elements of the result matrix are given by<P><A NAME="eqnmatmul"> </A> <IMG WIDTH=500 HEIGHT=47 ALIGN=BOTTOM ALT="equation3243" SRC="img622.gif" ><P>Accordingly, in order to compute the produce matrix, C,we need to compute <I>mp</I> summationseach of which is the sum of <I>n</I> product terms.An algorithm to compute the matrix productis given in Program <A HREF="page96.html#progdenseMatrixb"><IMG ALIGN=BOTTOM ALT="gif" SRC="../icons/cross_ref_motif.gif"></A>.The algorithm given is a direct implementation of Equation <A HREF="page96.html#eqnmatmul"><IMG ALIGN=BOTTOM ALT="gif" SRC="../icons/cross_ref_motif.gif"></A>.<P><P><A NAME="3312"> </A><A NAME="progdenseMatrixb"> </A> <IMG WIDTH=575 HEIGHT=294 ALIGN=BOTTOM ALT="program3253" SRC="img623.gif" ><BR><STRONG>Program:</STRONG> <tt>DenseMatrix</tt> class <tt>__times__</tt> method.<BR><P><P>The algorithm begins by checking to see that the matricesto be multiplied have compatible dimensions.That is, the number of columns of the first matrix must be equal tothe number of rows of the second one.This check takes <I>O</I>(1) time in the worst case.<P>Next a matrix in which the result will be formed is constructed (line 6).The running time for this is <I>O</I>(<I>mp</I>).For each value of <I>i</I> and <I>j</I>,the innermost loop (lines 10-11) does <I>n</I> iterations.Each iteration takes a constant amount of time.<P>The body of the middle loop (lines 9-12)takes time <I>O</I>(<I>n</I>) for each value of <I>i</I> and <I>j</I>.The middle loop is done for <I>p</I> iterations,giving the running time of <I>O</I>(<I>np</I>) for each value of <I>i</I>.Since, the outer loop (lines 7-12) does <I>m</I> iterations,its overall running time is <I>O</I>(<I>mnp</I>).Finally, the result matrix is returned on line 13.This takes a constant amount of time.<P>In summary, we have shown that lines 4-5 are <I>O</I>(1);line 6 is <I>O</I>(<I>mp</I>);lines 7-12 are <I>O</I>(<I>mnp</I>); andline 13 is <I>O</I>(1).Therefore, the running time of the canonical matrix multiplicationalgorithm is <I>O</I>(<I>mnp</I>).<P><HR><A NAME="tex2html2313" HREF="page97.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="../icons/next_motif.gif"></A> <A NAME="tex2html2311" HREF="page89.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="../icons/up_motif.gif"></A> <A NAME="tex2html2307" HREF="page95.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="../icons/previous_motif.gif"></A> <A NAME="tex2html2315" 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 © 2003</a> by <a href="../signature.html">Bruno R. Preiss, P.Eng.</a> All rights reserved.</ADDRESS></BODY></HTML>⌨️ 快捷键说明
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