| <html> |
| <head> |
| <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> |
| <title>Relative Error</title> |
| <link rel="stylesheet" href="../../../../../../../doc/src/boostbook.css" type="text/css"> |
| <meta name="generator" content="DocBook XSL Stylesheets V1.74.0"> |
| <link rel="home" href="../../index.html" title="Math Toolkit"> |
| <link rel="up" href="../backgrounders.html" title="Backgrounders"> |
| <link rel="prev" href="implementation.html" title="Additional Implementation Notes"> |
| <link rel="next" href="lanczos.html" title="The Lanczos Approximation"> |
| </head> |
| <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> |
| <table cellpadding="2" width="100%"><tr> |
| <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../boost.png"></td> |
| <td align="center"><a href="../../../../../../../index.html">Home</a></td> |
| <td align="center"><a href="../../../../../../../libs/libraries.htm">Libraries</a></td> |
| <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> |
| <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> |
| <td align="center"><a href="../../../../../../../more/index.htm">More</a></td> |
| </tr></table> |
| <hr> |
| <div class="spirit-nav"> |
| <a accesskey="p" href="implementation.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../backgrounders.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="lanczos.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> |
| </div> |
| <div class="section" lang="en"> |
| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="math_toolkit.backgrounders.relative_error"></a><a class="link" href="relative_error.html" title="Relative Error"> Relative |
| Error</a> |
| </h3></div></div></div> |
| <p> |
| Given an actual value <span class="emphasis"><em>a</em></span> and a found value <span class="emphasis"><em>v</em></span> |
| the relative error can be calculated from: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/error2.png"></span> |
| </p> |
| <p> |
| However the test programs in the library use the symmetrical form: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/error1.png"></span> |
| </p> |
| <p> |
| which measures <span class="emphasis"><em>relative difference</em></span> and happens to be |
| less error prone in use since we don't have to worry which value is the "true" |
| result, and which is the experimental one. It guarantees to return a value |
| at least as large as the relative error. |
| </p> |
| <p> |
| Special care needs to be taken when one value is zero: we could either take |
| the absolute error in this case (but that's cheating as the absolute error |
| is likely to be very small), or we could assign a value of either 1 or infinity |
| to the relative error in this special case. In the test cases for the special |
| functions in this library, everything below a threshold is regarded as "effectively |
| zero", otherwise the relative error is assigned the value of 1 if only |
| one of the terms is zero. The threshold is currently set at <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><>::</span><span class="identifier">min</span><span class="special">()</span></code>: in other words all denormalised numbers |
| are regarded as a zero. |
| </p> |
| <p> |
| All the test programs calculate <span class="emphasis"><em>quantized relative error</em></span>, |
| whereas the graphs in this manual are produced with the <span class="emphasis"><em>actual |
| error</em></span>. The difference is as follows: in the test programs, the |
| test data is rounded to the target real type under test when the program |
| is compiled, so the error observed will then be a whole number of <span class="emphasis"><em>units |
| in the last place</em></span> either rounded up from the actual error, or |
| rounded down (possibly to zero). In contrast the <span class="emphasis"><em>true error</em></span> |
| is obtained by extending the precision of the calculated value, and then |
| comparing to the actual value: in this case the calculated error may be some |
| fraction of <span class="emphasis"><em>units in the last place</em></span>. |
| </p> |
| <p> |
| Note that throughout this manual and the test programs the relative error |
| is usually quoted in units of epsilon. However, remember that <span class="emphasis"><em>units |
| in the last place</em></span> more accurately reflect the number of contaminated |
| digits, and that relative error can <span class="emphasis"><em>"wobble"</em></span> |
| by a factor of 2 compared to <span class="emphasis"><em>units in the last place</em></span>. |
| In other words: two implementations of the same function, whose maximum relative |
| errors differ by a factor of 2, can actually be accurate to the same number |
| of binary digits. You have been warned! |
| </p> |
| <a name="zero_error"></a><p> |
| </p> |
| <a name="math_toolkit.backgrounders.relative_error.the_impossibility_of_zero_error"></a><h5> |
| <a name="id1286953"></a> |
| <a class="link" href="relative_error.html#math_toolkit.backgrounders.relative_error.the_impossibility_of_zero_error">The |
| Impossibility of Zero Error</a> |
| </h5> |
| <p> |
| For many of the functions in this library, it is assumed that the error is |
| "effectively zero" if the computation can be done with a number |
| of guard digits. However it should be remembered that if the result is a |
| <span class="emphasis"><em>transcendental number</em></span> then as a point of principle we |
| can never be sure that the result is accurate to more than 1 ulp. This is |
| an example of <span class="emphasis"><em>the table makers dilemma</em></span>: consider what |
| happens if the first guard digit is a one, and the remaining guard digits |
| are all zero. Do we have a tie or not? Since the only thing we can tell about |
| a transcendental number is that its digits have no particular pattern, we |
| can never tell if we have a tie, no matter how many guard digits we have. |
| Therefore, we can never be completely sure that the result has been rounded |
| in the right direction. Of course, transcendental numbers that just happen |
| to be a tie - for however many guard digits we have - are extremely rare, |
| and get rarer the more guard digits we have, but even so.... |
| </p> |
| <p> |
| Refer to the classic text <a href="http://docs.sun.com/source/806-3568/ncg_goldberg.html" target="_top">What |
| Every Computer Scientist Should Know About Floating-Point Arithmetic</a> |
| for more information. |
| </p> |
| </div> |
| <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> |
| <td align="left"></td> |
| <td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009, 2010 John Maddock, Paul A. Bristow, |
| Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and |
| Thijs van den Berg<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) |
| </p> |
| </div></td> |
| </tr></table> |
| <hr> |
| <div class="spirit-nav"> |
| <a accesskey="p" href="implementation.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../backgrounders.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="lanczos.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> |
| </div> |
| </body> |
| </html> |
