mirror of
https://github.com/saitohirga/WSJT-X.git
synced 2025-10-25 01:50:30 -04:00
282 lines
23 KiB
HTML
282 lines
23 KiB
HTML
|
<html>
|
||
|
|
<head>
|
||
|
|
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||
|
|
<title>Continued Fraction Evaluation</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 2.5.1">
|
||
|
|
<link rel="up" href="../internals.html" title="Internal tools">
|
||
|
|
<link rel="prev" href="series_evaluation.html" title="Series Evaluation">
|
||
|
|
<link rel="next" href="tuples.html" title="Tuples">
|
||
|
|
</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="series_evaluation.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../internals.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="tuples.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||
|
|
</div>
|
||
|
|
<div class="section">
|
||
|
|
<div class="titlepage"><div><div><h3 class="title">
|
||
|
|
<a name="math_toolkit.internals.cf"></a><a class="link" href="cf.html" title="Continued Fraction Evaluation">Continued Fraction Evaluation</a>
|
||
|
|
</h3></div></div></div>
|
||
|
|
<h5>
|
||
|
|
<a name="math_toolkit.internals.cf.h0"></a>
|
||
|
|
<span class="phrase"><a name="math_toolkit.internals.cf.synopsis"></a></span><a class="link" href="cf.html#math_toolkit.internals.cf.synopsis">Synopsis</a>
|
||
|
|
</h5>
|
||
|
|
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">fraction</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
||
|
|
</pre>
|
||
|
|
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">tools</span><span class="special">{</span>
|
||
|
|
|
||
|
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Gen</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
|
||
|
|
<span class="keyword">typename</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">fraction_traits</span><span class="special"><</span><span class="identifier">Gen</span><span class="special">>::</span><span class="identifier">result_type</span>
|
||
|
|
<span class="identifier">continued_fraction_b</span><span class="special">(</span><span class="identifier">Gen</span><span class="special">&</span> <span class="identifier">g</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">tolerance</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_terms</span><span class="special">)</span>
|
||
|
|
|
||
|
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Gen</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
|
||
|
|
<span class="keyword">typename</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">fraction_traits</span><span class="special"><</span><span class="identifier">Gen</span><span class="special">>::</span><span class="identifier">result_type</span>
|
||
|
|
<span class="identifier">continued_fraction_b</span><span class="special">(</span><span class="identifier">Gen</span><span class="special">&</span> <span class="identifier">g</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">tolerance</span><span class="special">)</span>
|
||
|
|
|
||
|
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Gen</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
|
||
|
|
<span class="keyword">typename</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">fraction_traits</span><span class="special"><</span><span class="identifier">Gen</span><span class="special">>::</span><span class="identifier">result_type</span>
|
||
|
|
<span class="identifier">continued_fraction_a</span><span class="special">(</span><span class="identifier">Gen</span><span class="special">&</span> <span class="identifier">g</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">tolerance</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_terms</span><span class="special">)</span>
|
||
|
|
|
||
|
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Gen</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
|
||
|
|
<span class="keyword">typename</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">fraction_traits</span><span class="special"><</span><span class="identifier">Gen</span><span class="special">>::</span><span class="identifier">result_type</span>
|
||
|
|
<span class="identifier">continued_fraction_a</span><span class="special">(</span><span class="identifier">Gen</span><span class="special">&</span> <span class="identifier">g</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">tolerance</span><span class="special">)</span>
|
||
|
|
|
||
|
|
<span class="comment">//</span>
|
||
|
|
<span class="comment">// These interfaces are present for legacy reasons, and are now deprecated:</span>
|
||
|
|
<span class="comment">//</span>
|
||
|
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Gen</span><span class="special">></span>
|
||
|
|
<span class="keyword">typename</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">fraction_traits</span><span class="special"><</span><span class="identifier">Gen</span><span class="special">>::</span><span class="identifier">result_type</span>
|
||
|
|
<span class="identifier">continued_fraction_b</span><span class="special">(</span><span class="identifier">Gen</span><span class="special">&</span> <span class="identifier">g</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</span><span class="special">);</span>
|
||
|
|
|
||
|
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Gen</span><span class="special">></span>
|
||
|
|
<span class="keyword">typename</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">fraction_traits</span><span class="special"><</span><span class="identifier">Gen</span><span class="special">>::</span><span class="identifier">result_type</span>
|
||
|
|
<span class="identifier">continued_fraction_b</span><span class="special">(</span><span class="identifier">Gen</span><span class="special">&</span> <span class="identifier">g</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_terms</span><span class="special">);</span>
|
||
|
|
|
||
|
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Gen</span><span class="special">></span>
|
||
|
|
<span class="keyword">typename</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">fraction_traits</span><span class="special"><</span><span class="identifier">Gen</span><span class="special">>::</span><span class="identifier">result_type</span>
|
||
|
|
<span class="identifier">continued_fraction_a</span><span class="special">(</span><span class="identifier">Gen</span><span class="special">&</span> <span class="identifier">g</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</span><span class="special">);</span>
|
||
|
|
|
||
|
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Gen</span><span class="special">></span>
|
||
|
|
<span class="keyword">typename</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">fraction_traits</span><span class="special"><</span><span class="identifier">Gen</span><span class="special">>::</span><span class="identifier">result_type</span>
|
||
|
|
<span class="identifier">continued_fraction_a</span><span class="special">(</span><span class="identifier">Gen</span><span class="special">&</span> <span class="identifier">g</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_terms</span><span class="special">);</span>
|
||
|
|
|
||
|
|
<span class="special">}}}</span> <span class="comment">// namespaces</span>
|
||
|
|
</pre>
|
||
|
|
<h5>
|
||
|
|
<a name="math_toolkit.internals.cf.h1"></a>
|
||
|
|
<span class="phrase"><a name="math_toolkit.internals.cf.description"></a></span><a class="link" href="cf.html#math_toolkit.internals.cf.description">Description</a>
|
||
|
|
</h5>
|
||
|
|
<p>
|
||
|
|
<a href="http://en.wikipedia.org/wiki/Continued_fraction" target="_top">Continued fractions
|
||
|
|
are a common method of approximation. </a> These functions all evaluate
|
||
|
|
the continued fraction described by the <span class="emphasis"><em>generator</em></span> type
|
||
|
|
argument. The functions with an "_a" suffix evaluate the fraction:
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
<span class="inlinemediaobject"><img src="../../../equations/fraction2.svg"></span>
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
and those with a "_b" suffix evaluate the fraction:
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
<span class="inlinemediaobject"><img src="../../../equations/fraction1.svg"></span>
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
This latter form is somewhat more natural in that it corresponds with the
|
||
|
|
usual definition of a continued fraction, but note that the first <span class="emphasis"><em>a</em></span>
|
||
|
|
value returned by the generator is discarded. Further, often the first <span class="emphasis"><em>a</em></span>
|
||
|
|
and <span class="emphasis"><em>b</em></span> values in a continued fraction have different
|
||
|
|
defining equations to the remaining terms, which may make the "_a"
|
||
|
|
suffixed form more appropriate.
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
The generator type should be a function object which supports the following
|
||
|
|
operations:
|
||
|
|
</p>
|
||
|
|
<div class="informaltable"><table class="table">
|
||
|
|
<colgroup>
|
||
|
|
<col>
|
||
|
|
<col>
|
||
|
|
</colgroup>
|
||
|
|
<thead><tr>
|
||
|
|
<th>
|
||
|
|
<p>
|
||
|
|
Expression
|
||
|
|
</p>
|
||
|
|
</th>
|
||
|
|
<th>
|
||
|
|
<p>
|
||
|
|
Description
|
||
|
|
</p>
|
||
|
|
</th>
|
||
|
|
</tr></thead>
|
||
|
|
<tbody>
|
||
|
|
<tr>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
Gen::result_type
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
The type that is the result of invoking operator(). This can be
|
||
|
|
either an arithmetic type, or a std::pair<> of arithmetic
|
||
|
|
types.
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
g()
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
Returns an object of type Gen::result_type.
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
Each time this operator is called then the next pair of <span class="emphasis"><em>a</em></span>
|
||
|
|
and <span class="emphasis"><em>b</em></span> values is returned. Or, if result_type
|
||
|
|
is an arithmetic type, then the next <span class="emphasis"><em>b</em></span> value
|
||
|
|
is returned and all the <span class="emphasis"><em>a</em></span> values are assumed
|
||
|
|
to 1.
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
</tbody>
|
||
|
|
</table></div>
|
||
|
|
<p>
|
||
|
|
In all the continued fraction evaluation functions the <span class="emphasis"><em>tolerance</em></span>
|
||
|
|
parameter is the precision desired in the result, evaluation of the fraction
|
||
|
|
will continue until the last term evaluated leaves the relative error in
|
||
|
|
the result less than <span class="emphasis"><em>tolerance</em></span>. The deprecated interfaces
|
||
|
|
take a number of digits precision here, internally they just convert this
|
||
|
|
to a tolerance and forward call.
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
If the optional <span class="emphasis"><em>max_terms</em></span> parameter is specified then
|
||
|
|
no more than <span class="emphasis"><em>max_terms</em></span> calls to the generator will be
|
||
|
|
made, and on output, <span class="emphasis"><em>max_terms</em></span> will be set to actual
|
||
|
|
number of calls made. This facility is particularly useful when profiling
|
||
|
|
a continued fraction for convergence.
|
||
|
|
</p>
|
||
|
|
<h5>
|
||
|
|
<a name="math_toolkit.internals.cf.h2"></a>
|
||
|
|
<span class="phrase"><a name="math_toolkit.internals.cf.implementation"></a></span><a class="link" href="cf.html#math_toolkit.internals.cf.implementation">Implementation</a>
|
||
|
|
</h5>
|
||
|
|
<p>
|
||
|
|
Internally these algorithms all use the modified Lentz algorithm: refer to
|
||
|
|
Numeric Recipes in C++, W. H. Press et all, chapter 5, (especially 5.2 Evaluation
|
||
|
|
of continued fractions, p 175 - 179) for more information, also Lentz, W.J.
|
||
|
|
1976, Applied Optics, vol. 15, pp. 668-671.
|
||
|
|
</p>
|
||
|
|
<h5>
|
||
|
|
<a name="math_toolkit.internals.cf.h3"></a>
|
||
|
|
<span class="phrase"><a name="math_toolkit.internals.cf.examples"></a></span><a class="link" href="cf.html#math_toolkit.internals.cf.examples">Examples</a>
|
||
|
|
</h5>
|
||
|
|
<p>
|
||
|
|
The <a href="http://en.wikipedia.org/wiki/Golden_ratio" target="_top">golden ratio phi
|
||
|
|
= 1.618033989...</a> can be computed from the simplest continued fraction
|
||
|
|
of all:
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
<span class="inlinemediaobject"><img src="../../../equations/fraction3.svg"></span>
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
We begin by defining a generator function:
|
||
|
|
</p>
|
||
|
|
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||
|
|
<span class="keyword">struct</span> <span class="identifier">golden_ratio_fraction</span>
|
||
|
|
<span class="special">{</span>
|
||
|
|
<span class="keyword">typedef</span> <span class="identifier">T</span> <span class="identifier">result_type</span><span class="special">;</span>
|
||
|
|
|
||
|
|
<span class="identifier">result_type</span> <span class="keyword">operator</span><span class="special">()</span>
|
||
|
|
<span class="special">{</span>
|
||
|
|
<span class="keyword">return</span> <span class="number">1</span><span class="special">;</span>
|
||
|
|
<span class="special">}</span>
|
||
|
|
<span class="special">};</span>
|
||
|
|
</pre>
|
||
|
|
<p>
|
||
|
|
The golden ratio can then be computed to double precision using:
|
||
|
|
</p>
|
||
|
|
<pre class="programlisting"><span class="identifier">continued_fraction_a</span><span class="special">(</span>
|
||
|
|
<span class="identifier">golden_ratio_fraction</span><span class="special"><</span><span class="keyword">double</span><span class="special">>(),</span>
|
||
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">());</span>
|
||
|
|
</pre>
|
||
|
|
<p>
|
||
|
|
It's more usual though to have to define both the <span class="emphasis"><em>a</em></span>'s
|
||
|
|
and the <span class="emphasis"><em>b</em></span>'s when evaluating special functions by continued
|
||
|
|
fractions, for example the tan function is defined by:
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
<span class="inlinemediaobject"><img src="../../../equations/fraction4.svg"></span>
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
So its generator object would look like:
|
||
|
|
</p>
|
||
|
|
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||
|
|
<span class="keyword">struct</span> <span class="identifier">tan_fraction</span>
|
||
|
|
<span class="special">{</span>
|
||
|
|
<span class="keyword">private</span><span class="special">:</span>
|
||
|
|
<span class="identifier">T</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">;</span>
|
||
|
|
<span class="keyword">public</span><span class="special">:</span>
|
||
|
|
<span class="identifier">tan_fraction</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">v</span><span class="special">)</span>
|
||
|
|
<span class="special">:</span> <span class="identifier">a</span><span class="special">(-</span><span class="identifier">v</span><span class="special">*</span><span class="identifier">v</span><span class="special">),</span> <span class="identifier">b</span><span class="special">(-</span><span class="number">1</span><span class="special">)</span>
|
||
|
|
<span class="special">{}</span>
|
||
|
|
|
||
|
|
<span class="keyword">typedef</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">result_type</span><span class="special">;</span>
|
||
|
|
|
||
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span><span class="identifier">T</span><span class="special">></span> <span class="keyword">operator</span><span class="special">()()</span>
|
||
|
|
<span class="special">{</span>
|
||
|
|
<span class="identifier">b</span> <span class="special">+=</span> <span class="number">2</span><span class="special">;</span>
|
||
|
|
<span class="keyword">return</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">make_pair</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">);</span>
|
||
|
|
<span class="special">}</span>
|
||
|
|
<span class="special">};</span>
|
||
|
|
</pre>
|
||
|
|
<p>
|
||
|
|
Notice that if the continuant is subtracted from the <span class="emphasis"><em>b</em></span>
|
||
|
|
terms, as is the case here, then all the <span class="emphasis"><em>a</em></span> terms returned
|
||
|
|
by the generator will be negative. The tangent function can now be evaluated
|
||
|
|
using:
|
||
|
|
</p>
|
||
|
|
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||
|
|
<span class="identifier">T</span> <span class="identifier">tan</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">a</span><span class="special">)</span>
|
||
|
|
<span class="special">{</span>
|
||
|
|
<span class="identifier">tan_fraction</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">fract</span><span class="special">(</span><span class="identifier">a</span><span class="special">);</span>
|
||
|
|
<span class="keyword">return</span> <span class="identifier">a</span> <span class="special">/</span> <span class="identifier">continued_fraction_b</span><span class="special">(</span><span class="identifier">fract</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">());</span>
|
||
|
|
<span class="special">}</span>
|
||
|
|
</pre>
|
||
|
|
<p>
|
||
|
|
Notice that this time we're using the "_b" suffixed version to
|
||
|
|
evaluate the fraction: we're removing the leading <span class="emphasis"><em>a</em></span>
|
||
|
|
term during fraction evaluation as it's different from all the others.
|
||
|
|
</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 © 2006-2010, 2012-2014 Nikhar Agrawal,
|
||
|
|
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
|
||
|
|
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
|
||
|
|
Benjamin Sobotta, 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="series_evaluation.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../internals.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="tuples.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||
|
|
</div>
|
||
|
|
</body>
|
||
|
|
</html>
|