diff options
Diffstat (limited to 'old/dw/html/rounding-errors.html')
| -rw-r--r-- | old/dw/html/rounding-errors.html | 74 |
1 files changed, 74 insertions, 0 deletions
diff --git a/old/dw/html/rounding-errors.html b/old/dw/html/rounding-errors.html new file mode 100644 index 0000000..72814fa --- /dev/null +++ b/old/dw/html/rounding-errors.html @@ -0,0 +1,74 @@ +<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> +<html xmlns="http://www.w3.org/1999/xhtml"> +<head> +<meta http-equiv="Content-Type" content="text/xhtml;charset=UTF-8"/> +<meta http-equiv="X-UA-Compatible" content="IE=9"/> +<meta name="generator" content="Doxygen 1.8.8"/> +<title>Dillo: How to Avoid Rounding Errors</title> +<link href="tabs.css" rel="stylesheet" type="text/css"/> +<script type="text/javascript" src="https://www.dillo.org/dw/html/jquery.js"></script> +<script type="text/javascript" src="dynsections.js"></script> +<link href="doxygen.css" rel="stylesheet" type="text/css" /> +</head> +<body> +<div id="top"><!-- do not remove this div, it is closed by doxygen! --> +<div id="titlearea"> +<table cellspacing="0" cellpadding="0"> + <tbody> + <tr style="height: 56px;"> + <td style="padding-left: 0.5em;"> + <div id="projectname">Dillo + </div> + </td> + </tr> + </tbody> +</table> +</div> +<!-- end header part --> +<!-- Generated by Doxygen 1.8.8 --> + <div id="navrow1" class="tabs"> + <ul class="tablist"> + <li><a href="index.html"><span>Main Page</span></a></li> + <li class="current"><a href="pages.html"><span>Related Pages</span></a></li> + <li><a href="namespaces.html"><span>Namespaces</span></a></li> + <li><a href="annotated.html"><span>Classes</span></a></li> + <li><a href="files.html"><span>Files</span></a></li> + </ul> + </div> +</div><!-- top --> +<div class="header"> + <div class="headertitle"> +<div class="title">How to Avoid Rounding Errors </div> </div> +</div><!--header--> +<div class="contents"> +<div class="textblock"><p>(Probably, this is a standard algorithm, so if someone knows the name, drop me a note.)</p> +<p>If something like</p> +<p class="formulaDsp"> +<img class="formulaDsp" alt="\[y_i = {x_i a \over b}\]" src="form_49.png"/> +</p> +<p>is to be calculated, and all numbers are integers, a naive implementation would result in something, for which</p> +<p class="formulaDsp"> +<img class="formulaDsp" alt="\[\sum y_i \ne {(\sum x_i) a \over b}\]" src="form_50.png"/> +</p> +<p>because of rounding errors, due to the integer division. This can be avoided by transforming the formula into</p> +<p class="formulaDsp"> +<img class="formulaDsp" alt="\[y_i = {(\sum_{j=0}^{j=i} x_j) a \over b} - \sum_{j=0}^{j=i-1} y_j\]" src="form_51.png"/> +</p> +<p>Of corse, when all <img class="formulaInl" alt="$y_i$" src="form_8.png"/> are calculated in a sequence, <img class="formulaInl" alt="$\sum_{j=0}^{j=i} x_j$" src="form_52.png"/> and <img class="formulaInl" alt="$\sum_{j=0}^{j=i-1} y_j$" src="form_53.png"/> can be accumulated in the same loop. Regard this as sample:</p> +<div class="fragment"><div class="line"><span class="keywordtype">int</span> n, x[n], a, b; <span class="comment">// Should all be initialized.</span></div> +<div class="line"><span class="keywordtype">int</span> y[n], cumX = 0, cumY = 0;</div> +<div class="line"></div> +<div class="line"><span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i < n; i++) {</div> +<div class="line"> cumX += x[i]</div> +<div class="line"> y[i] = (cumX * a) / b - cumY;</div> +<div class="line"> cumY += y[i];</div> +<div class="line">}</div> +</div><!-- fragment --> </div></div><!-- contents --> +<!-- start footer part --> +<hr class="footer"/><address class="footer"><small> +Generated on Sat May 28 2016 11:47:43 for Dillo by  <a href="http://www.doxygen.org/index.html"> +<img class="footer" src="doxygen.png" alt="doxygen"/> +</a> 1.8.8 +</small></address> +</body> +</html> |