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- <div class="section">
- <div class="titlepage"><div><div><h3 class="title">
- <a name="math_toolkit.internals.recurrence"></a><a class="link" href="recurrence.html" title="Tools For 3-Term Recurrence Relations">Tools For 3-Term Recurrence
- Relations</a>
- </h3></div></div></div>
- <h5>
- <a name="math_toolkit.internals.recurrence.h0"></a>
- <span class="phrase"><a name="math_toolkit.internals.recurrence.synopsis"></a></span><a class="link" href="recurrence.html#math_toolkit.internals.recurrence.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">recurrence</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">Recurrence</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">function_ratio_from_backwards_recurrence</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Recurrence</span><span class="special">&</span> <span class="identifier">r</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">factor</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_iter</span><span class="special">);</span>
- <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Recurrence</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">function_ratio_from_forwards_recurrence</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Recurrence</span><span class="special">&</span> <span class="identifier">r</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">factor</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_iter</span><span class="special">);</span>
- <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">NextCoefs</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">apply_recurrence_relation_forward</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">NextCoefs</span><span class="special">&</span> <span class="identifier">get_coefs</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">number_of_steps</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">second</span><span class="special">,</span> <span class="keyword">int</span><span class="special">*</span> <span class="identifier">log_scaling</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">T</span><span class="special">*</span> <span class="identifier">previous</span> <span class="special">=</span> <span class="number">0</span><span class="special">);</span>
- <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">class</span> <span class="identifier">NextCoefs</span><span class="special">></span>
- <span class="identifier">T</span> <span class="identifier">apply_recurrence_relation_backward</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">NextCoefs</span><span class="special">&</span> <span class="identifier">get_coefs</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">number_of_steps</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">second</span><span class="special">,</span> <span class="keyword">int</span><span class="special">*</span> <span class="identifier">log_scaling</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">T</span><span class="special">*</span> <span class="identifier">previous</span> <span class="special">=</span> <span class="number">0</span><span class="special">);</span>
- <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Recurrence</span><span class="special">></span>
- <span class="keyword">struct</span> <span class="identifier">forward_recurrence_iterator</span><span class="special">;</span>
- <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Recurrence</span><span class="special">></span>
- <span class="keyword">struct</span> <span class="identifier">backward_recurrence_iterator</span><span class="special">;</span>
- <span class="special">}}}</span> <span class="comment">// namespaces</span>
- </pre>
- <h5>
- <a name="math_toolkit.internals.recurrence.h1"></a>
- <span class="phrase"><a name="math_toolkit.internals.recurrence.description"></a></span><a class="link" href="recurrence.html#math_toolkit.internals.recurrence.description">Description</a>
- </h5>
- <p>
- All of the tools in this header require a description of the recurrence relation:
- this takes the form of a functor that returns a tuple containing the 3 coefficients,
- specifically, given a recurrence relation:
- </p>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../equations/three_term_recurrence.svg"></span>
- </p></blockquote></div>
- <p>
- And a functor <code class="computeroutput"><span class="identifier">F</span></code> then the
- expression:
- </p>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="serif_italic">F(n);</span>
- </p></blockquote></div>
- <p>
- Returns a tuple containing <span class="serif_italic">{ a<sub>n</sub>, b<sub>n</sub>, c<sub>n</sub> }</span>.
- </p>
- <p>
- For example, the recurrence relation for the Bessel J and Y functions when
- written in this form is:
- </p>
- <p>
- <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/three_term_recurrence_bessel_jy.svg" width="250" height="38"></object></span>
- </p>
- <p>
- Therefore, given local variables <span class="emphasis"><em>x</em></span> and <span class="emphasis"><em>v</em></span>
- of type <code class="computeroutput"><span class="keyword">double</span></code> the recurrence
- relation for Bessel J and Y can be encoded in a lambda expression like this:
- </p>
- <pre class="programlisting"><span class="keyword">auto</span> <span class="identifier">recurrence_functor_jy</span> <span class="special">=</span> <span class="special">[&](</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">)</span> <span class="special">{</span> <span class="keyword">return</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">make_tuple</span><span class="special">(</span><span class="number">1.0</span><span class="special">,</span> <span class="special">-</span><span class="number">2</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">v</span> <span class="special">+</span> <span class="identifier">n</span><span class="special">)</span> <span class="special">/</span> <span class="identifier">x</span><span class="special">,</span> <span class="number">1.0</span><span class="special">);</span> <span class="special">};</span>
- </pre>
- <p>
- Similarly, the Bessel I and K recurrence relation differs just by the sign
- of the final term:
- </p>
- <p>
- <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/three_term_recurrence_bessel_ik.svg" width="244" height="38"></object></span>
- </p>
- <p>
- And this could be encoded as:
- </p>
- <pre class="programlisting"><span class="keyword">auto</span> <span class="identifier">recurrence_functor_ik</span> <span class="special">=</span> <span class="special">[&](</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">)</span> <span class="special">{</span> <span class="keyword">return</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">make_tuple</span><span class="special">(</span><span class="number">1.0</span><span class="special">,</span> <span class="special">-</span><span class="number">2</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">v</span> <span class="special">+</span> <span class="identifier">n</span><span class="special">)</span> <span class="special">/</span> <span class="identifier">x</span><span class="special">,</span> <span class="special">-</span><span class="number">1.0</span><span class="special">);</span> <span class="special">};</span>
- </pre>
- <p>
- The tools are then as follows:
- </p>
- <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Recurrence</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">function_ratio_from_backwards_recurrence</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Recurrence</span><span class="special">&</span> <span class="identifier">r</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">factor</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_iter</span><span class="special">);</span>
- </pre>
- <p>
- Given a functor <code class="computeroutput"><span class="identifier">r</span></code> which encodes
- the recurrence relation for function <code class="computeroutput"><span class="identifier">F</span></code>
- at some location <span class="emphasis"><em>n</em></span>, then returns the ratio:
- </p>
- <p>
- <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/three_term_recurrence_backwards_ratio.svg" width="63" height="20"></object></span>
- </p>
- <p>
- This calculation is stable only if recurrence is stable in the backwards
- direction. Further the ratio calculated is for the dominant solution (in
- the backwards direction) of the recurrence relation, if there are multiple
- solutions, then there is no guarantee that this will find the one you want
- or expect.
- </p>
- <p>
- Argument <span class="emphasis"><em>factor</em></span> is the tolerance required for convergence
- of the continued fraction associated with the recurrence relation, and should
- be no smaller than machine epsilon. Argument <span class="emphasis"><em>max_iter</em></span>
- sets the maximum number of permitted iterations in the associated continued
- fraction.
- </p>
- <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Recurrence</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">function_ratio_from_forwards_recurrence</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Recurrence</span><span class="special">&</span> <span class="identifier">r</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">factor</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_iter</span><span class="special">);</span>
- </pre>
- <p>
- Given a functor <code class="computeroutput"><span class="identifier">r</span></code> which encodes
- the recurrence relation for function F at some location <span class="emphasis"><em>n</em></span>,
- then returns the ratio:
- </p>
- <p>
- <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/three_term_recurrence_forwards_ratio.svg" width="63" height="20"></object></span>
- </p>
- <p>
- This calculation is stable only if recurrence is stable in the forwards direction.
- Further the ratio calculated is for the dominant solution (in the forwards
- direction) of the recurrence relation, if there are multiple solutions, then
- there is no guarantee that this will find the one you want or expect.
- </p>
- <p>
- Argument <span class="emphasis"><em>factor</em></span> is the tolerance required for convergence
- of the continued fraction associated with the recurrence relation, and should
- be no smaller than machine epsilon. Argument <span class="emphasis"><em>max_iter</em></span>
- sets the maximum number of permitted iterations in the associated continued
- fraction.
- </p>
- <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">NextCoefs</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">apply_recurrence_relation_forward</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">NextCoefs</span><span class="special">&</span> <span class="identifier">get_coefs</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">number_of_steps</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">second</span><span class="special">,</span> <span class="keyword">int</span><span class="special">*</span> <span class="identifier">log_scaling</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">T</span><span class="special">*</span> <span class="identifier">previous</span> <span class="special">=</span> <span class="number">0</span><span class="special">);</span>
- </pre>
- <p>
- Applies a recurrence relation in a stable forward direction, starting with
- the values F<sub>n-1</sub> and F<sub>n</sub>.
- </p>
- <div class="variablelist">
- <p class="title"><b></b></p>
- <dl class="variablelist">
- <dt><span class="term">get_coefs</span></dt>
- <dd><p>
- Functor that returns the corefficients of the recurrence relation.
- The coefficients should be centered on position <span class="emphasis"><em>second</em></span>.
- </p></dd>
- <dt><span class="term">number_of_steps</span></dt>
- <dd><p>
- The number of steps to apply the recurrence relation onwards from
- <span class="emphasis"><em>second</em></span>.
- </p></dd>
- <dt><span class="term">first</span></dt>
- <dd><p>
- The value of F<sub>n-1</sub>
- </p></dd>
- <dt><span class="term">second</span></dt>
- <dd><p>
- The value of F<sub>n</sub>
- </p></dd>
- <dt><span class="term">log_scaling</span></dt>
- <dd><p>
- When provided, the recurrence relations may be rescaled internally
- to avoid over/underflow issues. The result should be multiplied by
- <code class="computeroutput"><span class="identifier">exp</span><span class="special">(*</span><span class="identifier">log_scaling</span><span class="special">)</span></code>
- to get the true value of the result.
- </p></dd>
- <dt><span class="term">previous</span></dt>
- <dd><p>
- When provided, is set to the value of F<sub>n + number_of_steps - 1</sub>
- </p></dd>
- </dl>
- </div>
- <p>
- Returns F<sub>n + number_of_steps</sub>.
- </p>
- <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">NextCoefs</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">apply_recurrence_relation_backward</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">NextCoefs</span><span class="special">&</span> <span class="identifier">get_coefs</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">number_of_steps</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">second</span><span class="special">,</span> <span class="keyword">int</span><span class="special">*</span> <span class="identifier">log_scaling</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">T</span><span class="special">*</span> <span class="identifier">previous</span> <span class="special">=</span> <span class="number">0</span><span class="special">);</span>
- </pre>
- <p>
- Applies a recurrence relation in a stable backward direction, starting with
- the values F<sub>n+1</sub> and F<sub>n</sub>.
- </p>
- <div class="variablelist">
- <p class="title"><b></b></p>
- <dl class="variablelist">
- <dt><span class="term">get_coefs</span></dt>
- <dd><p>
- Functor that returns the corefficients of the recurrence relation.
- The coefficients should be centered on position <span class="emphasis"><em>second</em></span>.
- </p></dd>
- <dt><span class="term">number_of_steps</span></dt>
- <dd><p>
- The number of steps to apply the recurrence relation backwards from
- <span class="emphasis"><em>second</em></span>.
- </p></dd>
- <dt><span class="term">first</span></dt>
- <dd><p>
- The value of F<sub>n+1</sub>
- </p></dd>
- <dt><span class="term">second</span></dt>
- <dd><p>
- The value of F<sub>n</sub>
- </p></dd>
- <dt><span class="term">log_scaling</span></dt>
- <dd><p>
- When provided, the recurrence relations may be rescaled internally
- to avoid over/underflow issues. The result should be multiplied by
- <code class="computeroutput"><span class="identifier">exp</span><span class="special">(*</span><span class="identifier">log_scaling</span><span class="special">)</span></code>
- to get the true value of the result.
- </p></dd>
- <dt><span class="term">previous</span></dt>
- <dd><p>
- When provided, is set to the value of F<sub>n - number_of_steps + 1</sub>
- </p></dd>
- </dl>
- </div>
- <p>
- Returns F<sub>n - number_of_steps</sub>.
- </p>
- <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Recurrence</span><span class="special">></span>
- <span class="keyword">struct</span> <span class="identifier">forward_recurrence_iterator</span>
- <span class="special">{</span>
- <span class="keyword">typedef</span> <span class="keyword">typename</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">remove_reference</span><span class="special"><</span><span class="keyword">decltype</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">get</span><span class="special"><</span><span class="number">0</span><span class="special">>(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">declval</span><span class="special"><</span><span class="identifier">Recurrence</span><span class="special">&>()(</span><span class="number">0</span><span class="special">)))>::</span><span class="identifier">type</span> <span class="identifier">value_type</span><span class="special">;</span>
- <span class="identifier">forward_recurrence_iterator</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Recurrence</span><span class="special">&</span> <span class="identifier">r</span><span class="special">,</span> <span class="identifier">value_type</span> <span class="identifier">f_n_minus_1</span><span class="special">,</span> <span class="identifier">value_type</span> <span class="identifier">f_n</span><span class="special">);</span>
- <span class="identifier">forward_recurrence_iterator</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Recurrence</span><span class="special">&</span> <span class="identifier">r</span><span class="special">,</span> <span class="identifier">value_type</span> <span class="identifier">f_n</span><span class="special">);</span>
- <span class="comment">/* Operators omitted for clarity */</span>
- <span class="special">};</span>
- </pre>
- <p>
- Type <code class="computeroutput"><span class="identifier">forward_recurrence_iterator</span></code>
- defines a forward-iterator for a recurrence relation stable in the forward
- direction. The constructors take the recurrence relation, plus either one
- or two values: if only one value is provided, then the second is computed
- by using the recurrence relation to calculate the function ratio.
- </p>
- <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Recurrence</span><span class="special">></span>
- <span class="keyword">struct</span> <span class="identifier">backward_recurrence_iterator</span>
- <span class="special">{</span>
- <span class="keyword">typedef</span> <span class="keyword">typename</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">remove_reference</span><span class="special"><</span><span class="keyword">decltype</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">get</span><span class="special"><</span><span class="number">0</span><span class="special">>(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">declval</span><span class="special"><</span><span class="identifier">Recurrence</span><span class="special">&>()(</span><span class="number">0</span><span class="special">)))>::</span><span class="identifier">type</span> <span class="identifier">value_type</span><span class="special">;</span>
- <span class="identifier">backward_recurrence_iterator</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Recurrence</span><span class="special">&</span> <span class="identifier">r</span><span class="special">,</span> <span class="identifier">value_type</span> <span class="identifier">f_n_plus_1</span><span class="special">,</span> <span class="identifier">value_type</span> <span class="identifier">f_n</span><span class="special">);</span>
- <span class="identifier">backward_recurrence_iterator</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Recurrence</span><span class="special">&</span> <span class="identifier">r</span><span class="special">,</span> <span class="identifier">value_type</span> <span class="identifier">f_n</span><span class="special">);</span>
- <span class="comment">/* Operators omitted for clarity */</span>
- <span class="special">};</span>
- </pre>
- <p>
- Type <code class="computeroutput"><span class="identifier">backward_recurrence_iterator</span></code>
- defines a forward-iterator for a recurrence relation stable in the backward
- direction. The constructors take the recurrence relation, plus either one
- or two values: if only one value is provided, then the second is computed
- by using the recurrence relation to calculate the function ratio.
- </p>
- <p>
- Note that <span class="emphasis"><em>incrementing</em></span> this iterator moves the value
- returned successively to F<sub>n-1</sub>, F<sub>n-2</sub> etc.
- </p>
- </div>
- <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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- <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>)
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