Hit and miss in 2-D to estimate Pi:

<BODY LANG="EN"> <HR><center> <table border=1> <td><A NAME="tex2html23" HREF="node1.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A NAME="tex2html29" HREF="Lab5.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A NAME="tex2html31" HREF="node3.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node2.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="SECTION00020000000000000000">Hit and miss in 2-D to estimate Pi:</A></H1> <P> A hit and miss Monte Carlo algorithm for estimating Pi can easily be typed into Mathematica. The principle here is that a ratio of two areas is estimated by counting how often pairs of random numbers <IMG WIDTH=39 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline88" SRC="img2.gif" > fall into each of the two areas (`darts' thrown at a dartboard inscribed in a square, so the ratio of areas is <IMG WIDTH=25 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline90" SRC="img3.gif" > ). <P> <table><td valign=top><a href="/hamlet-bin/qhandler/sende2.hamlet?execute" target="results"><img height=24 width=24 alt="EXECUTE" src="/UCESIcons/tools/execute.gif"></a></td><td valign=top><PRE>Clear[ndarts,xcoord,ycoord,distance2,nhits,piest]; ndarts=100; xcoord = Table[Random[]-0.5, {n,1,ndarts}]; ycoord = Table[Random[]-0.5, {n,1,ndarts}]; distance2 =xcoord^2 + ycoord^2; nhits = Count[distance2, x_ /; x <= .25]; piest = 4.0*nhits/ndarts</PRE></td></table> <P> Here the <B>Count </B> command finds how often a condition, <IMG WIDTH=59 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline92" SRC="img4.gif" > , is satisfied in the list <I>distance</I>2. This gives how often the darts land within the dartboard (the distance squared from the center is calculated and compared with the radius squared, which is <IMG WIDTH=25 HEIGHT=13 ALIGN=BOTTOM ALT="tex2html_wrap_inline96" SRC="img5.gif" > ). <P> The dartboard and the impact points can be displayed nicely by loading the graphics module located in the file <code>~hannes/Chem465/dartboard</code>, <P> <table><td valign=top><a href="/hamlet-bin/qhandler/send3.hamlet?execute" target="results"><img height=24 width=24 alt="EXECUTE" src="/UCESIcons/tools/execute.gif"></a></td><td valign=top><code><<dartboard</code></td></table> <P> <I>Run the dart calculation a few times varying the number of darts used to see roughly how good the estimate of pi is.</I> <P> <HR><center> <table border=1> <td><A NAME="tex2html23" HREF="node1.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A NAME="tex2html29" HREF="Lab5.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A NAME="tex2html31" HREF="node3.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node2.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>