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  25 <div class="section">
  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.sf_poly.chebyshev"></a><a class="link" href="chebyshev.html" title="Chebyshev Polynomials">Chebyshev Polynomials</a>
  28 </h3></div></div></div>
  29 <h5>
  30 <a name="math_toolkit.sf_poly.chebyshev.h0"></a>
  31         <span class="phrase"><a name="math_toolkit.sf_poly.chebyshev.synopsis"></a></span><a class="link" href="chebyshev.html#math_toolkit.sf_poly.chebyshev.synopsis">Synopsis</a>
  32       </h5>
  33 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">chebyshev</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  34 </pre>
  35 <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>
  36
  37 <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">,</span> <span class="keyword">class</span> <span class="identifier">Real3</span><span class="special">&gt;</span>
  38 <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_next</span><span class="special">(</span><span class="identifier">Real1</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">Real2</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">Tn</span><span class="special">,</span> <span class="identifier">Real3</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">Tn_1</span><span class="special">);</span>
  39
  40 <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>
  41 <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>
  42
  43 <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;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  44 <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;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  45
  46 <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>
  47 <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>
  48
  49 <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;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  50 <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;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  51
  52 <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>
  53 <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>
  54
  55 <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>
  56 <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>
  57
  58 <span class="special">}}</span> <span class="comment">// namespaces</span>
  59 </pre>
  60 <p>
  61         <span class="emphasis"><em>"Real analysts cannot do without Fourier, complex analysts
  62         cannot do without Laurent, and numerical analysts cannot do without Chebyshev"</em></span>
  63         --Lloyd N. Trefethen
  64       </p>
  65 <p>
  66         The Chebyshev polynomials of the first kind are defined by the recurrence
  67         <span class="emphasis"><em>T</em></span><sub>n+1</sub>(<span class="emphasis"><em>x</em></span>) := <span class="emphasis"><em>2xT</em></span><sub>n</sub>(<span class="emphasis"><em>x</em></span>)
  68         - <span class="emphasis"><em>T</em></span><sub>n-1</sub>(<span class="emphasis"><em>x</em></span>), <span class="emphasis"><em>n &gt; 0</em></span>,
  69         where <span class="emphasis"><em>T</em></span><sub>0</sub>(<span class="emphasis"><em>x</em></span>) := 1 and <span class="emphasis"><em>T</em></span><sub>1</sub>(<span class="emphasis"><em>x</em></span>)
  70         := <span class="emphasis"><em>x</em></span>. These can be calculated in Boost using the following
  71         simple code
  72       </p>
  73 <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>
  74 <span class="keyword">double</span> <span class="identifier">T12</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">chebyshev_t</span><span class="special">(</span><span class="number">12</span><span class="special">,</span> <span class="identifier">x</span><span class="special">);</span>
  75 </pre>
  76 <p>
  77         Calculation of derivatives is also straightforward:
  78       </p>
  79 <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">T12_prime</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">chebyshev_t_prime</span><span class="special">(</span><span class="number">12</span><span class="special">,</span> <span class="identifier">x</span><span class="special">);</span>
  80 </pre>
  81 <p>
  82         The complexity of evaluation of the <span class="emphasis"><em>n</em></span>-th Chebyshev polynomial
  83         by these functions is linear. So they are unsuitable for use in calculation
  84         of (say) a Chebyshev series, as a sum of linear scaling functions scales
  85         quadratically. Though there are very sophisticated algorithms for the evaluation
  86         of Chebyshev series, a linear time algorithm is presented below:
  87       </p>
  88 <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>
  89 <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>
  90 <span class="keyword">double</span> <span class="identifier">T0</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span>
  91 <span class="keyword">double</span> <span class="identifier">T1</span> <span class="special">=</span> <span class="identifier">x</span><span class="special">;</span>
  92 <span class="keyword">double</span> <span class="identifier">f</span> <span class="special">=</span> <span class="identifier">c</span><span class="special">[</span><span class="number">0</span><span class="special">]*</span><span class="identifier">T0</span><span class="special">/</span><span class="number">2</span><span class="special">;</span>
  93 <span class="keyword">unsigned</span> <span class="identifier">l</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span>
  94 <span class="keyword">while</span><span class="special">(</span><span class="identifier">l</span> <span class="special">&lt;</span> <span class="identifier">c</span><span class="special">.</span><span class="identifier">size</span><span class="special">())</span>
  95 <span class="special">{</span>
  96    <span class="identifier">f</span> <span class="special">+=</span> <span class="identifier">c</span><span class="special">[</span><span class="identifier">l</span><span class="special">]*</span><span class="identifier">T1</span><span class="special">;</span>
  97    <span class="identifier">std</span><span class="special">::</span><span class="identifier">swap</span><span class="special">(</span><span class="identifier">T0</span><span class="special">,</span> <span class="identifier">T1</span><span class="special">);</span>
  98    <span class="identifier">T1</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">chebyshev_next</span><span class="special">(</span><span class="identifier">x</span><span class="special">,</span> <span class="identifier">T0</span><span class="special">,</span> <span class="identifier">T1</span><span class="special">);</span>
  99    <span class="special">++</span><span class="identifier">l</span><span class="special">;</span>
 100 <span class="special">}</span>
 101 </pre>
 102 <p>
 103         This uses the <code class="computeroutput"><span class="identifier">chebyshev_next</span></code>
 104         function to evaluate each term of the Chebyshev series in constant time.
 105         However, this naive algorithm has a catastrophic loss of precision as <span class="emphasis"><em>x</em></span>
 106         approaches 1. A method to mitigate this way given by <a href="http://www.ams.org/journals/mcom/1955-09-051/S0025-5718-1955-0071856-0/S0025-5718-1955-0071856-0.pdf" target="_top">Clenshaw</a>,
 107         and is implemented in boost as
 108       </p>
 109 <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>
 110 <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>
 111 <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>
 112 </pre>
 113 <p>
 114         N.B.: There is factor of <span class="emphasis"><em>2</em></span> difference in our definition
 115         of the first coefficient in the Chebyshev series from Clenshaw's original
 116         work. This is because two traditions exist in notation for the Chebyshev
 117         series expansion,
 118       </p>
 119 <div class="blockquote"><blockquote class="blockquote"><p>
 120           <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>)
 121         </p></blockquote></div>
 122 <p>
 123         and
 124       </p>
 125 <div class="blockquote"><blockquote class="blockquote"><p>
 126           <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
 127           + &#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>)
 128         </p></blockquote></div>
 129 <p>
 130         <span class="emphasis"><em><span class="bold"><strong>boost math always uses the second convention,
 131         with the factor of 1/2 on the first coefficient.</strong></span></em></span>
 132       </p>
 133 <p>
 134         Chebyshev polynomials of the second kind can be evaluated via <code class="computeroutput"><span class="identifier">chebyshev_u</span></code>:
 135       </p>
 136 <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="special">-</span><span class="number">0.23</span><span class="special">;</span>
 137 <span class="keyword">double</span> <span class="identifier">U1</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">chebyshev_u</span><span class="special">(</span><span class="number">1</span><span class="special">,</span> <span class="identifier">x</span><span class="special">);</span>
 138 </pre>
 139 <p>
 140         The evaluation of Chebyshev polynomials by a three-term recurrence is known
 141         to be <a href="https://link.springer.com/article/10.1007/s11075-014-9925-x" target="_top">mixed
 142         forward-backward stable</a> for <span class="emphasis"><em>x</em></span> &#8714; [-1,
 143         1]. However, the author does not know of a similar result for <span class="emphasis"><em>x</em></span>
 144         outside [-1, 1]. For this reason, evaluation of Chebyshev polynomials outside
 145         of [-1, 1] is strongly discouraged. That said, small rounding errors in the
 146         course of a computation will often lead to this situation, and termination
 147         of the computation due to these small problems is very discouraging. For
 148         this reason, <code class="computeroutput"><span class="identifier">chebyshev_t</span></code>
 149         and <code class="computeroutput"><span class="identifier">chebyshev_u</span></code> have code
 150         paths for <span class="emphasis"><em>x &gt; 1</em></span> and <span class="emphasis"><em>x &lt; -1</em></span>
 151         which do not use three-term recurrences. These code paths are <span class="emphasis"><em>much
 152         slower</em></span>, and should be avoided if at all possible.
 153       </p>
 154 <p>
 155         Evaluation of a Chebyshev series is relatively simple. The real challenge
 156         is <span class="emphasis"><em>generation</em></span> of the Chebyshev series. For this purpose,
 157         boost provides a <span class="emphasis"><em>Chebyshev transform</em></span>, a projection operator
 158         which projects a function onto a finite-dimensional span of Chebyshev polynomials.
 159         But before we discuss the API, let's analyze why we might want to project
 160         a function onto a span of Chebyshev polynomials.
 161       </p>
 162 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 163 <li class="listitem">
 164             We want a numerically stable way to evaluate the function's derivative.
 165           </li>
 166 <li class="listitem">
 167             Our function is expensive to evaluate, and we wish to find a less expensive
 168             way to estimate its value. An example are the standard library transcendental
 169             functions: These functions are guaranteed to evaluate to within 1 ulp
 170             of the exact value, but often this accuracy is not needed. A projection
 171             onto the Chebyshev polynomials with a low accuracy requirement can vastly
 172             accelerate the computation of these functions.
 173           </li>
 174 <li class="listitem">
 175             We wish to numerically integrate the function.
 176           </li>
 177 </ul></div>
 178 <p>
 179         The API is given below.
 180       </p>
 181 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">chebyshev_transform</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
 182 </pre>
 183 <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>
 184
 185 <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>
 186 <span class="keyword">class</span> <span class="identifier">chebyshev_transform</span>
 187 <span class="special">{</span>
 188 <span class="keyword">public</span><span class="special">:</span>
 189     <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">&gt;</span>
 190     <span class="identifier">chebyshev_transform</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">F</span><span class="special">&amp;</span> <span class="identifier">f</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">tol</span><span class="special">=</span><span class="number">500</span><span class="special">*</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;::</span><span class="identifier">epsilon</span><span class="special">());</span>
 191
 192     <span class="identifier">Real</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span>
 193
 194     <span class="identifier">Real</span> <span class="identifier">integrate</span><span class="special">()</span> <span class="keyword">const</span>
 195
 196     <span class="keyword">const</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;&amp;</span> <span class="identifier">coefficients</span><span class="special">()</span> <span class="keyword">const</span>
 197
 198     <span class="identifier">Real</span> <span class="identifier">prime</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span>
 199 <span class="special">};</span>
 200
 201 <span class="special">}}//</span> <span class="identifier">end</span> <span class="identifier">namespaces</span>
 202 </pre>
 203 <p>
 204         The Chebyshev transform takes a function <span class="emphasis"><em>f</em></span> and returns
 205         a <span class="emphasis"><em>near-minimax</em></span> approximation to <span class="emphasis"><em>f</em></span>
 206         in terms of Chebyshev polynomials. By <span class="emphasis"><em>near-minimax</em></span>,
 207         we mean that the resulting Chebyshev polynomial is "very close"
 208         the polynomial <span class="emphasis"><em>p</em></span><sub>n</sub>  which minimizes the uniform norm of
 209         <span class="emphasis"><em>f</em></span> - <span class="emphasis"><em>p</em></span><sub>n</sub>. The notion of "very
 210         close" can be made rigorous; see Trefethen's "Approximation Theory
 211         and Approximation Practice" for details.
 212       </p>
 213 <p>
 214         The Chebyshev transform works by creating a vector of values by evaluating
 215         the input function at the Chebyshev points, and then performing a discrete
 216         cosine transform on the resulting vector. In order to do this efficiently,
 217         we have used <a href="http://www.fftw.org/" target="_top">FFTW3</a>. So to compile,
 218         you must have <code class="computeroutput"><span class="identifier">FFTW3</span></code> installed,
 219         and link with <code class="computeroutput"><span class="special">-</span><span class="identifier">lfftw3</span></code>
 220         for double precision, <code class="computeroutput"><span class="special">-</span><span class="identifier">lfftw3f</span></code>
 221         for float precision, <code class="computeroutput"><span class="special">-</span><span class="identifier">lfftw3l</span></code>
 222         for long double precision, and -lfftwq for quad (<code class="computeroutput"><span class="identifier">__float128</span></code>)
 223         precision. After the coefficients of the Chebyshev series are known, the
 224         routine goes back through them and filters out all the coefficients whose
 225         absolute ratio to the largest coefficient are less than the tolerance requested
 226         in the constructor.
 227       </p>
 228 </div>
 229 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 230 <td align="left"></td>
 231 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 232       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 233       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 234       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 235       Daryle Walker and Xiaogang Zhang<p>
 236         Distributed under the Boost Software License, Version 1.0. (See accompanying
 237         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>)
 238       </p>
 239 </div></td>
 240 </tr></table>
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