127 lines
8.3 KiB
HTML
127 lines
8.3 KiB
HTML
<html>
|
|
<head>
|
|
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
|
<title>Relative Error</title>
|
|
<link rel="stylesheet" href="../math.css" type="text/css">
|
|
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
|
|
<link rel="home" href="../index.html" title="Math Toolkit 2.11.0">
|
|
<link rel="up" href="../backgrounders.html" title="Chapter 22. Backgrounders">
|
|
<link rel="prev" href="special_tut/special_tut_test.html" title="Testing">
|
|
<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="special_tut/special_tut_test.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">
|
|
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
|
|
<a name="math_toolkit.relative_error"></a><a class="link" href="relative_error.html" title="Relative Error">Relative Error</a>
|
|
</h2></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>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../equations/error2.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<p>
|
|
However the test programs in the library use the symmetrical form:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../equations/error1.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<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>
|
|
<h5>
|
|
<a name="math_toolkit.relative_error.h0"></a>
|
|
<span class="phrase"><a name="math_toolkit.relative_error.zero_error"></a></span><a class="link" href="relative_error.html#math_toolkit.relative_error.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 what <a href="http://en.wikipedia.org/wiki/William_Kahan" target="_top">http://en.wikipedia.org/wiki/William_Kahan</a>
|
|
called <a href="http://en.wikipedia.org/wiki/Rounding#The_table-maker.27s_dilemma" target="_top">http://en.wikipedia.org/wiki/Rounding#The_table-maker.27s_dilemma</a>:
|
|
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 © 2006-2019 Nikhar
|
|
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
|
|
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
|
|
Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
|
|
Daryle Walker and Xiaogang Zhang<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="special_tut/special_tut_test.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>
|