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@@ -1,10 +1,10 @@
<html>
<head>
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<title>Chebyshev Polynomials</title>
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@@ -40,27 +40,30 @@
<span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">chebyshev_t</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">Real</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">x</span><span class="special">);</span>
<span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">chebyshev_t</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">Real</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">chebyshev_t</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">Real</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">chebyshev_u</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">Real</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">x</span><span class="special">);</span>
<span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">chebyshev_u</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">Real</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">chebyshev_u</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">Real</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">chebyshev_t_prime</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">Real</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">x</span><span class="special">);</span>
<span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Real1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Real2</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">chebyshev_clenshaw_recurrence</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Real</span><span class="special">*</span> <span class="keyword">const</span> <span class="identifier">c</span><span class="special">,</span> <span class="identifier">size_t</span> <span class="identifier">length</span><span class="special">,</span> <span class="identifier">Real2</span> <span class="identifier">x</span><span class="special">);</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">chebyshev_clenshaw_recurrence</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Real1</span><span class="special">*</span> <span class="keyword">const</span> <span class="identifier">c</span><span class="special">,</span> <span class="identifier">size_t</span> <span class="identifier">length</span><span class="special">,</span> <span class="identifier">Real2</span> <span class="identifier">x</span><span class="special">);</span>
<span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">Real</span><span class="special">&gt;</span>
<span class="identifier">Real</span> <span class="identifier">chebyshev_clenshaw_recurrence</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Real</span><span class="special">*</span> <span class="keyword">const</span> <span class="identifier">c</span><span class="special">,</span> <span class="identifier">size_t</span> <span class="identifier">length</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">x</span><span class="special">);</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<p>
<span class="quote">&#8220;<span class="quote">Real analysts cannot do without Fourier, complex analysts cannot do
without Laurent, and numerical analysts cannot do without Chebyshev</span>&#8221;</span>--Lloyd
N. Trefethen
<span class="emphasis"><em>"Real analysts cannot do without Fourier, complex analysts
cannot do without Laurent, and numerical analysts cannot do without Chebyshev"</em></span>
--Lloyd N. Trefethen
</p>
<p>
The Chebyshev polynomials of the first kind are defined by the recurrence
@@ -108,7 +111,7 @@
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="number">0.5</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">c</span><span class="special">{</span><span class="number">14.2</span><span class="special">,</span> <span class="special">-</span><span class="number">13.7</span><span class="special">,</span> <span class="number">82.3</span><span class="special">,</span> <span class="number">96</span><span class="special">};</span>
<span class="keyword">double</span> <span class="identifier">f</span> <span class="special">=</span> <span class="identifier">chebyshev_clenshaw_recurrence</span><span class="special">(</span><span class="identifier">c</span><span class="special">.</span><span class="identifier">data</span><span class="special">(),</span> <span class="identifier">c</span><span class="special">.</span><span class="identifier">size</span><span class="special">(),</span> <span class="identifier">Real</span> <span class="identifier">x</span><span class="special">);</span>
<span class="keyword">double</span> <span class="identifier">f</span> <span class="special">=</span> <span class="identifier">chebyshev_clenshaw_recurrence</span><span class="special">(</span><span class="identifier">c</span><span class="special">.</span><span class="identifier">data</span><span class="special">(),</span> <span class="identifier">c</span><span class="special">.</span><span class="identifier">size</span><span class="special">(),</span> <span class="identifier">x</span><span class="special">);</span>
</pre>
<p>
N.B.: There is factor of <span class="emphasis"><em>2</em></span> difference in our definition
@@ -117,19 +120,33 @@
series expansion,
</p>
<p>
<span class="emphasis"><em>f</em></span>(<span class="emphasis"><em>x</em></span>) &#8776; &#8721;<sub>n=0</sub><sup>N-1</sup> <span class="emphasis"><em>a</em></span><sub>n</sub><span class="emphasis"><em>T</em></span><sub>n</sub>(<span class="emphasis"><em>x</em></span>)
<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../graphs/chebyshev_convention_1.svg" width="149" height="54"></object></span>
</p>
<p>
and
</p>
<p>
<span class="emphasis"><em>f</em></span>(<span class="emphasis"><em>x</em></span>) &#8776; <span class="emphasis"><em>c</em></span><sub>0</sub>/2
+ &#8721;<sub>n=1</sub><sup>N-1</sup> <span class="emphasis"><em>c</em></span><sub>n</sub><span class="emphasis"><em>T</em></span><sub>n</sub>(<span class="emphasis"><em>x</em></span>)
<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../graphs/chebyshev_convention_2.svg" width="193" height="54"></object></span>
</p>
<p>
<span class="emphasis"><em><span class="bold"><strong>boost math always uses the second convention,
with the factor of 1/2 on the first coefficient.</strong></span></em></span>
</p>
<p>
Another common use case is when the polynomial must be evaluated on some
interval [<span class="emphasis"><em>a</em></span>, <span class="emphasis"><em>b</em></span>]. The translation
to the interval [-1, 1] causes a few accuracy problems and also gives us
some opportunities. For this case, we use <a href="https://doi.org/10.1093/imamat/20.3.379" target="_top">Reinch's
modification</a> to the Clenshaw recurrence, which is also discussed
<a href="https://archive.siam.org/books/ot99/OT99SampleChapter.pdf" target="_top">here</a>.
The usage is as follows:
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="number">9</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">a</span> <span class="special">=</span> <span class="number">7</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">b</span> <span class="special">=</span> <span class="number">12</span><span class="special">;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">c</span><span class="special">{</span><span class="number">14.2</span><span class="special">,</span> <span class="special">-</span><span class="number">13.7</span><span class="special">,</span> <span class="number">82.3</span><span class="special">,</span> <span class="number">96</span><span class="special">};</span>
<span class="keyword">double</span> <span class="identifier">f</span> <span class="special">=</span> <span class="identifier">chebyshev_clenshaw_recurrence</span><span class="special">(</span><span class="identifier">c</span><span class="special">.</span><span class="identifier">data</span><span class="special">(),</span> <span class="identifier">c</span><span class="special">.</span><span class="identifier">size</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="identifier">x</span><span class="special">);</span>
</pre>
<p>
Chebyshev polynomials of the second kind can be evaluated via <code class="computeroutput"><span class="identifier">chebyshev_u</span></code>:
</p>
@@ -139,7 +156,7 @@
<p>
The evaluation of Chebyshev polynomials by a three-term recurrence is known
to be <a href="https://link.springer.com/article/10.1007/s11075-014-9925-x" target="_top">mixed
forward-backward stable</a> for <span class="emphasis"><em>x</em></span> &#8714; [-1,
forward-backward stable</a> for <span class="emphasis"><em>x</em></span> [-1,
1]. However, the author does not know of a similar result for <span class="emphasis"><em>x</em></span>
outside [-1, 1]. For this reason, evaluation of Chebyshev polynomials outside
of [-1, 1] is strongly discouraged. That said, small rounding errors in the
@@ -161,7 +178,7 @@
</p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
We want a numerically stable way to evaluate the function's derivative
We want a numerically stable way to evaluate the function's derivative.
</li>
<li class="listitem">
Our function is expensive to evaluate, and we wish to find a less expensive
@@ -172,7 +189,7 @@
accelerate the computation of these functions.
</li>
<li class="listitem">
We wish to numerically integrate the function
We wish to numerically integrate the function.
</li>
</ul></div>
<p>
@@ -214,11 +231,12 @@
The Chebyshev transform works by creating a vector of values by evaluating
the input function at the Chebyshev points, and then performing a discrete
cosine transform on the resulting vector. In order to do this efficiently,
we have used FFTW3. So to compile, you must have <code class="computeroutput"><span class="identifier">FFTW3</span></code>
installed, and link with <code class="computeroutput"><span class="special">-</span><span class="identifier">lfftw3</span></code>
we have used <a href="http://www.fftw.org/" target="_top">FFTW3</a>. So to compile,
you must have <code class="computeroutput"><span class="identifier">FFTW3</span></code> installed,
and link with <code class="computeroutput"><span class="special">-</span><span class="identifier">lfftw3</span></code>
for double precision, <code class="computeroutput"><span class="special">-</span><span class="identifier">lfftw3f</span></code>
for float precision, <code class="computeroutput"><span class="special">-</span><span class="identifier">lfftw3l</span></code>
for long double precision, and -lfftwq for quad (<code class="computeroutput"><span class="identifier">__float128</span></code>)
for long double precision, and <code class="computeroutput"><span class="special">-</span><span class="identifier">lfftw3q</span></code> for quad (<code class="computeroutput"><span class="identifier">__float128</span></code>)
precision. After the coefficients of the Chebyshev series are known, the
routine goes back through them and filters out all the coefficients whose
absolute ratio to the largest coefficient are less than the tolerance requested
@@ -227,11 +245,11 @@
</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 &#169; 2006-2010, 2012-2014, 2017 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam
Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle Walker
and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
Lalande, John Maddock, Evan Miller, 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>