Definition of a Mathematica Module

<BODY LANG="EN"> <HR><center> <table border=1> <td><A NAME="tex2html33" HREF="node2.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A NAME="tex2html39" HREF="Lab5.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A NAME="tex2html41" HREF="node4.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node3.html"><img height=36 width=36 alt="CONTENTS" src="/UCESIcons/tools/toc.gif"></a></td> <td><a href="/hamlet-bin/qhandler?exit" target="results"><img height=36 width=36 alt="RESET" src="/UCESIcons/tools/stop.gif"></a></td> <td><a href="/UCES/hamlet/"><img height=36 width=36 alt="HAMLET" src="/UCESIcons/tools/hamlet.gif"></a></td> <td><a href="copyright.html"><img height=36 width=36 alt="COPYRIGHT" src="/UCESIcons/tools/copyright.gif"></a></td> <td><a href="mailto:uces_info@krellinst.org"><img height=36 width=36 alt="FEEDBACK" src="/UCESIcons/tools/feedback.gif"></a></td> <td><a href="/UCES/hamlet/usage.html"><img height=36 width=36 alt="ABOUT" src="/UCESIcons/tools/about.gif"></a></td> </table> </center> <HR><H1><A NAME="SECTION00030000000000000000">Definition of a Mathematica Module</A></H1> <P> To study systematically how good an estimate is obtained in an experiment with a given number of darts, <I>ndarts</I>, we can redo the experiment several times (in each experiment using <I>ndarts</I>) and check the consistency of the result. That is, we will carry out several experiments where pi is determined by throwing darts <I>ndarts</I> times. This set of different, independent estimates of pi is then analysed by calculating the spread or <I>variance</I> in the values obtained. This gives an estimate of the precision in an experiment with <I>ndarts</I> trials. This whole calculation and analysis can then be repeated with a different value of <I>ndarts</I> and the dependence of the precision on <I>ndarts</I> analysed (obviously the experiment becomes more precise the more darts we use, but the question is how fast does the precision improve). <P> To carry this out, it is necessary to make it easy to do repeated experiments. It is convenient to define a dart throw experiment as a callable Mathematica module. The variables listed in the beginning of the module are local, i.e. use of those variables outside the module does not affect their value inside the module and vice versa. You can easily modify the five lines of code you used before to make this dart throw experiment a self-contained module: <P> <table><td valign=top><a href="/hamlet-bin/qhandler/sende3.hamlet?execute" target="results"><img height=24 width=24 alt="EXECUTE" src="/UCESIcons/tools/execute.gif"></a></td><td valign=top><PRE>darts[ndarts_] := Module[ {xcoord,ycoord,distance2,nhits}, xcoord := Table[Random[]-0.5, {i,1,ndarts}]; ycoord := Table[Random[]-0.5, {i,1,ndarts}]; distance2 = xcoord^2 + ycoord^2 ; nhits = Count[distance2, x_ /; x <= 0.25]; piest = 4.0*nhits/ndarts ]</PRE></td></table> <P> Remember to include the last closing square bracket. It completes the module definition. An experiment with, for example, 100 darts can now be carried out by simply typing <B>darts[100]</B>. <I>Do a couple of experiments throwing 100 darts, for example. Do you get the same estimate of pi?</I> <P> <HR><center> <table border=1> <td><A NAME="tex2html33" HREF="node2.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A NAME="tex2html39" HREF="Lab5.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A NAME="tex2html41" HREF="node4.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node3.html"><img height=36 width=36 alt="CONTENTS" src="/UCESIcons/tools/toc.gif"></a></td> <td><a href="/hamlet-bin/qhandler?exit" target="results"><img height=36 width=36 alt="RESET" src="/UCESIcons/tools/stop.gif"></a></td> <td><a href="/UCES/hamlet/"><img height=36 width=36 alt="HAMLET" src="/UCESIcons/tools/hamlet.gif"></a></td> <td><a href="copyright.html"><img height=36 width=36 alt="COPYRIGHT" src="/UCESIcons/tools/copyright.gif"></a></td> <td><a href="mailto:uces_info@krellinst.org"><img height=36 width=36 alt="FEEDBACK" src="/UCESIcons/tools/feedback.gif"></a></td> <td><a href="/UCES/hamlet/usage.html"><img height=36 width=36 alt="ABOUT" src="/UCESIcons/tools/about.gif"></a></td> </table> </center> <HR><P><ADDRESS> <I>Hannes Jonsson<BR> Modified by Thomas L. Marchioro II<BR> and the <A NAME="tex2html1" HREF="/UCES/">Undergraduate Computational Engineering and Science project</A> </I> </ADDRESS> </BODY>