] 源码如下 ---------------------------------------------------------- <HTML> <HEAD> <META NAME="keywords" CONTENT="Pi, machin, javascipt, numerical methods, multiple precision, series"> <TITLE>PI in JavaScript－www.51windows.Net</TITLE> <SCRIPT LANGUAGE="JavaScript"> <!-- // // arctangent series for Pi a la mode by Bohr // var vectorsize = 53; // number of elements in each of four arrays var nCells = 52; // number of array elements displayed
// constant var BASE = 10000.0; // The number base var TENTHOUSANDTH = 0.0001; // avoids floating point division var SQ_239E4 = 571210000.0; // = BASE x 239 squared var SQ_5E4 = 250000.0; // = BASE x 5 squared var REPAIR = 0.000005; // roundoff correction var RECIPR_25 = 0.04; // the reciprocal of 25 is 0.04 var RECIPR_239= 1.0 / 57121; // reciprocal of 239 squared
// arrays var term5 = null; // = (1/5)^(2n+1), integer array var term239 = null; // = (1/239)^(2n+1), integer array var sum = null; // = 16term5-4term239, integer array var series = null; // = PI
// floating or integer var TwoNplus1 = 1.0; // 2n+1 = 1,3,5,7,... var Basex2n_1 = BASE; // = 10000*(2n+1) var Currdigit = 1; // index of the series[] element being printed var sgn = 1; // = (-1)^n; takes the values -1 and 1
// strings var IntroString = " Pi = 3.";
function MakeArray(n) { this.length = n; for (var i = 1; i <= n; i++) { this[i] = 0; } return this; } function PiSetup() { term5 = new MakeArray(vectorsize); term239 = new MakeArray(vectorsize); sum = new MakeArray(vectorsize); series = new MakeArray(vectorsize);
term5[1] = 5; term239[1] = 239; } function DivideTerms() { var total5 = term5[1]; var total239 = term239[1];
// Divide the terms by 25 or 57121 for (var i = Currdigit; i <= nCells + 2; i++) { term5[i] = Math.floor( RECIPR_25 * total5 + REPAIR ); total5 = BASE * total5 - SQ_5E4 * term5[i] + term5[i+1]; term239[i] = Math.floor( total239 * RECIPR_239 + REPAIR ); total239 = BASE * total239 - SQ_239E4 * term239[i] + term239[i+1]; } } function SubtractTerms() { var carry = 0; var total = 0;
for (var i = nCells + 1; i > Currdigit; i--) { total = 16.0 * term5[i] - 4.0 * term239[i] + carry + 60000.0; carry = Math.floor( total * TENTHOUSANDTH + REPAIR ) - 6.0; sum[i] = Math.floor( total - BASE * carry - 60000.0 ); } } function DivideSum() { var total = sum[1]; var reciprocal = 1.0 / TwoNplus1;
for( var i = Currdigit; i <= nCells + 2; i++) { sum[i] = Math.floor( total * reciprocal + REPAIR ); total = BASE * total - Basex2n_1 * sum[i] + sum[i+1]; } } function AddShowDigits(nchars,lwidth) { var i = 0; var total = 0; var carry = 0; var ccount = nchars / 4; var strg = "x"; var old1 = series[Currdigit+1]; var old2 = series[Currdigit+2];
// Add the sum to the series for (i = nCells + 1; i >= Currdigit; i--) { total = series[i] + sgn * sum[i] + carry + 30000.0; carry = Math.floor( total * TENTHOUSANDTH + REPAIR ) - 3.0; series[i] = Math.floor( total - BASE * carry - 30000.0 ); }
<CENTER> <B><FONT SIZE=+1>Machin's Arctangent Series for Pi in JavaScript</FONT></B> <A HREF="../mathindex.html"> <IMG SRC=http://www.haosc.com/wytx/114/"p_btn3.jpg" BORDER="0" HSPACE=14 HEIGHT=40 WIDTH=110 ALIGN=ABSCENTER></A> </CENTER> <hr>
<hr> <CENTER>Click "View: Page Source" to see the Javascript source code</CENTER> <P><B>In 1706 John Machin</B> discovered the trigonometric identity</P>
<PRE> PI = 16 arctan(1/5) - 4 arctan(1/239)</PRE>
and used it to calculate the first one hundred digits of Pi. He combined his formula with the Taylor series expansion for the inverse tangent of x. (Brook Taylor was Machin's contemporary in Cambridge.) The arctangent series is
<PRE> 3 5 7 n 2n+1 arctan(x) = x - x /3 + x /5 - x /7 +...+ (-1) x /(2n+1) +..</PRE>
Machin's formula remained the primary tool of Pi-hunters for centuries. As late as 1973, Guilloud and Bouyer used a variation of it to compute one million digits of Pi on a CDC 7600.
<P>This program uses arrays of multiple precision digits for the two series terms, for their sum, and for the series. Each 'digit' is a modulo 10000 integer. The number base is 10,000. </P>
<P>In the end, John Machin was never cured. </P>
<HR SIZE=3 WIDTH="100%"> <B><FONT SIZE=-1>AUTHOR:</FONT></B> <TT> John Bohr </TT> <A HREF="mailto:jnbohr@netscape.net">email me</A> <BR><TT>Last updated: 1 October 1998</TT> <BR> </BODY> </HTML>
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