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  26 <div class="titlepage"><div><div><h2 class="title" style="clear: both">
  27 <a name="math_toolkit.autodiff"></a><a class="link" href="autodiff.html" title="Automatic Differentiation">Automatic Differentiation</a>
  28 </h2></div></div></div>
  29 <h2>
  30 <a name="math_toolkit.autodiff.h0"></a>
  31       <span class="phrase"><a name="math_toolkit.autodiff.synopsis"></a></span><a class="link" href="autodiff.html#math_toolkit.autodiff.synopsis">Synopsis</a>
  32     </h2>
  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">differentiation</span><span class="special">/</span><span class="identifier">autodiff</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  34
  35 <span class="keyword">namespace</span> <span class="identifier">boost</span> <span class="special">{</span>
  36 <span class="keyword">namespace</span> <span class="identifier">math</span> <span class="special">{</span>
  37 <span class="keyword">namespace</span> <span class="identifier">differentiation</span> <span class="special">{</span>
  38
  39 <span class="comment">// Function returning a single variable of differentiation. Recommended: Use auto for type.</span>
  40 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">size_t</span> <span class="identifier">Order</span><span class="special">,</span> <span class="identifier">size_t</span><span class="special">...</span> <span class="identifier">Orders</span><span class="special">&gt;</span>
  41 <span class="identifier">autodiff_fvar</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">Order</span><span class="special">,</span> <span class="identifier">Orders</span><span class="special">...&gt;</span> <span class="identifier">make_fvar</span><span class="special">(</span><span class="identifier">RealType</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">ca</span><span class="special">);</span>
  42
  43 <span class="comment">// Function returning multiple independent variables of differentiation in a std::tuple.</span>
  44 <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">size_t</span><span class="special">...</span> <span class="identifier">Orders</span><span class="special">,</span> <span class="keyword">typename</span><span class="special">...</span> <span class="identifier">RealTypes</span><span class="special">&gt;</span>
  45 <span class="keyword">auto</span> <span class="identifier">make_ftuple</span><span class="special">(</span><span class="identifier">RealTypes</span> <span class="keyword">const</span><span class="special">&amp;...</span> <span class="identifier">ca</span><span class="special">);</span>
  46
  47 <span class="comment">// Type of combined autodiff types. Recommended: Use auto for return type (C++14).</span>
  48 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">typename</span><span class="special">...</span> <span class="identifier">RealTypes</span><span class="special">&gt;</span>
  49 <span class="keyword">using</span> <span class="identifier">promote</span> <span class="special">=</span> <span class="keyword">typename</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">promote_args_n</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">RealTypes</span><span class="special">...&gt;::</span><span class="identifier">type</span><span class="special">;</span>
  50
  51 <span class="keyword">namespace</span> <span class="identifier">detail</span> <span class="special">{</span>
  52
  53 <span class="comment">// Single autodiff variable. Use make_fvar() or make_ftuple() to instantiate.</span>
  54 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">size_t</span> <span class="identifier">Order</span><span class="special">&gt;</span>
  55 <span class="keyword">class</span> <span class="identifier">fvar</span> <span class="special">{</span>
  56  <span class="keyword">public</span><span class="special">:</span>
  57   <span class="comment">// Query return value of function to get the derivatives.</span>
  58   <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span><span class="special">...</span> <span class="identifier">Orders</span><span class="special">&gt;</span>
  59   <span class="identifier">get_type_at</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">sizeof</span><span class="special">...(</span><span class="identifier">Orders</span><span class="special">)</span> <span class="special">-</span> <span class="number">1</span><span class="special">&gt;</span> <span class="identifier">derivative</span><span class="special">(</span><span class="identifier">Orders</span><span class="special">...</span> <span class="identifier">orders</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
  60
  61   <span class="comment">// All of the arithmetic and comparison operators are overloaded.</span>
  62   <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">RealType2</span><span class="special">,</span> <span class="identifier">size_t</span> <span class="identifier">Order2</span><span class="special">&gt;</span>
  63   <span class="identifier">fvar</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">+=(</span><span class="identifier">fvar</span><span class="special">&lt;</span><span class="identifier">RealType2</span><span class="special">,</span> <span class="identifier">Order2</span><span class="special">&gt;</span> <span class="keyword">const</span><span class="special">&amp;);</span>
  64
  65   <span class="identifier">fvar</span><span class="special">&amp;</span> <span class="keyword">operator</span><span class="special">+=(</span><span class="identifier">root_type</span> <span class="keyword">const</span><span class="special">&amp;);</span>
  66
  67   <span class="comment">// ...</span>
  68 <span class="special">};</span>
  69
  70 <span class="comment">// Standard math functions are overloaded and called via argument-dependent lookup (ADL).</span>
  71 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">size_t</span> <span class="identifier">Order</span><span class="special">&gt;</span>
  72 <span class="identifier">fvar</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">Order</span><span class="special">&gt;</span> <span class="identifier">floor</span><span class="special">(</span><span class="identifier">fvar</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">Order</span><span class="special">&gt;</span> <span class="keyword">const</span><span class="special">&amp;);</span>
  73
  74 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">size_t</span> <span class="identifier">Order</span><span class="special">&gt;</span>
  75 <span class="identifier">fvar</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">Order</span><span class="special">&gt;</span> <span class="identifier">exp</span><span class="special">(</span><span class="identifier">fvar</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">Order</span><span class="special">&gt;</span> <span class="keyword">const</span><span class="special">&amp;);</span>
  76
  77 <span class="comment">// ...</span>
  78
  79 <span class="special">}</span>  <span class="comment">// namespace detail</span>
  80
  81 <span class="special">}</span>  <span class="comment">// namespace differentiation</span>
  82 <span class="special">}</span>  <span class="comment">// namespace math</span>
  83 <span class="special">}</span>  <span class="comment">// namespace boost</span>
  84 </pre>
  85 <h2>
  86 <a name="math_toolkit.autodiff.h1"></a>
  87       <span class="phrase"><a name="math_toolkit.autodiff.description"></a></span><a class="link" href="autodiff.html#math_toolkit.autodiff.description">Description</a>
  88     </h2>
  89 <p>
  90       Autodiff is a header-only C++ library that facilitates the <a href="https://en.wikipedia.org/wiki/Automatic_differentiation" target="_top">automatic
  91       differentiation</a> (forward mode) of mathematical functions of single
  92       and multiple variables.
  93     </p>
  94 <p>
  95       This implementation is based upon the <a href="https://en.wikipedia.org/wiki/Taylor_series" target="_top">Taylor
  96       series</a> expansion of an analytic function <span class="emphasis"><em>f</em></span> at
  97       the point <span class="emphasis"><em>x<sub>0</sub></em></span>:
  98     </p>
  99 <div class="blockquote"><blockquote class="blockquote"><div class="blockquote"><blockquote class="blockquote"><p>
 100           <span class="inlinemediaobject"><img src="../../equations/autodiff/taylor_series.svg"></span>
 101         </p></blockquote></div></blockquote></div>
 102 <p>
 103       The essential idea of autodiff is the substitution of numbers with polynomials
 104       in the evaluation of <span class="emphasis"><em>f(x<sub>0</sub>)</em></span>. By substituting the number
 105       <span class="emphasis"><em>x<sub>0</sub></em></span> with the first-order polynomial <span class="emphasis"><em>x<sub>0</sub>+&#949;</em></span>,
 106       and using the same algorithm to compute <span class="emphasis"><em>f(x<sub>0</sub>+&#949;)</em></span>,
 107       the resulting polynomial in <span class="emphasis"><em>&#949;</em></span> contains the function's
 108       derivatives <span class="emphasis"><em>f'(x<sub>0</sub>)</em></span>, <span class="emphasis"><em>f''(x<sub>0</sub>)</em></span>, <span class="emphasis"><em>f'''(x<sub>0</sub>)</em></span>,
 109       ... within the coefficients. Each coefficient is equal to the derivative of
 110       its respective order, divided by the factorial of the order.
 111     </p>
 112 <p>
 113       In greater detail, assume one is interested in calculating the first <span class="emphasis"><em>N</em></span>
 114       derivatives of <span class="emphasis"><em>f</em></span> at <span class="emphasis"><em>x<sub>0</sub></em></span>. Without loss
 115       of precision to the calculation of the derivatives, all terms <span class="emphasis"><em>O(&#949;<sup>N+1</sup>)</em></span>
 116       that include powers of <span class="emphasis"><em>&#949;</em></span> greater than <span class="emphasis"><em>N</em></span>
 117       can be discarded. (This is due to the fact that each term in a polynomial depends
 118       only upon equal and lower-order terms under arithmetic operations.) Under these
 119       truncation rules, <span class="emphasis"><em>f</em></span> provides a polynomial-to-polynomial
 120       transformation:
 121     </p>
 122 <div class="blockquote"><blockquote class="blockquote"><div class="blockquote"><blockquote class="blockquote"><p>
 123           <span class="inlinemediaobject"><img src="../../equations/autodiff/polynomial_transform.svg"></span>
 124         </p></blockquote></div></blockquote></div>
 125 <p>
 126       C++'s ability to overload operators and functions allows for the creation of
 127       a class <code class="computeroutput"><span class="identifier">fvar</span></code> (<span class="underline">f</span>orward-mode
 128       autodiff <span class="underline">var</span>iable) that represents polynomials
 129       in <span class="emphasis"><em>&#949;</em></span>. Thus the same algorithm <span class="emphasis"><em>f</em></span>
 130       that calculates the numeric value of <span class="emphasis"><em>y<sub>0</sub>=f(x<sub>0</sub>)</em></span>, when written
 131       to accept and return variables of a generic (template) type, is also used to
 132       calculate the polynomial <span class="emphasis"><em>&#931;<sub>n</sub>y<sub>n</sub>&#949;<sup>n</sup>=f(x<sub>0</sub>+&#949;)</em></span>.
 133       The derivatives <span class="emphasis"><em>f<sup>(n)</sup>(x<sub>0</sub>)</em></span> are then found from the product
 134       of the respective factorial <span class="emphasis"><em>n!</em></span> and coefficient <span class="emphasis"><em>y<sub>n</sub></em></span>:
 135     </p>
 136 <div class="blockquote"><blockquote class="blockquote"><div class="blockquote"><blockquote class="blockquote"><p>
 137           <span class="inlinemediaobject"><img src="../../equations/autodiff/derivative_formula.svg"></span>
 138         </p></blockquote></div></blockquote></div>
 139 <h2>
 140 <a name="math_toolkit.autodiff.h2"></a>
 141       <span class="phrase"><a name="math_toolkit.autodiff.examples"></a></span><a class="link" href="autodiff.html#math_toolkit.autodiff.examples">Examples</a>
 142     </h2>
 143 <h3>
 144 <a name="math_toolkit.autodiff.h3"></a>
 145       <span class="phrase"><a name="math_toolkit.autodiff.example-single-variable"></a></span><a class="link" href="autodiff.html#math_toolkit.autodiff.example-single-variable">Example
 146       1: Single-variable derivatives</a>
 147     </h3>
 148 <h4>
 149 <a name="math_toolkit.autodiff.h4"></a>
 150       <span class="phrase"><a name="math_toolkit.autodiff.calculate_derivatives_of_f_x_x_s"></a></span><a class="link" href="autodiff.html#math_toolkit.autodiff.calculate_derivatives_of_f_x_x_s">Calculate
 151       derivatives of <span class="emphasis"><em>f(x)=x<sup>4</sup></em></span> at <span class="emphasis"><em>x</em></span>=2.</a>
 152     </h4>
 153 <p>
 154       In this example, <code class="computeroutput"><span class="identifier">make_fvar</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span>
 155       <span class="identifier">Order</span><span class="special">&gt;(</span><span class="number">2.0</span><span class="special">)</span></code> instantiates
 156       the polynomial 2+<span class="emphasis"><em>&#949;</em></span>. The <code class="computeroutput"><span class="identifier">Order</span><span class="special">=</span><span class="number">5</span></code> means that
 157       enough space is allocated (on the stack) to hold a polynomial of up to degree
 158       5 during the proceeding computation.
 159     </p>
 160 <p>
 161       Internally, this is modeled by a <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span><span class="number">6</span><span class="special">&gt;</span></code> whose elements <code class="computeroutput"><span class="special">{</span><span class="number">2</span><span class="special">,</span> <span class="number">1</span><span class="special">,</span> <span class="number">0</span><span class="special">,</span>
 162       <span class="number">0</span><span class="special">,</span> <span class="number">0</span><span class="special">,</span> <span class="number">0</span><span class="special">}</span></code>
 163       correspond to the 6 coefficients of the polynomial upon initialization. Its
 164       fourth power, at the end of the computation, is a polynomial with coefficients
 165       <code class="computeroutput"><span class="identifier">y</span> <span class="special">=</span>
 166       <span class="special">{</span><span class="number">16</span><span class="special">,</span>
 167       <span class="number">32</span><span class="special">,</span> <span class="number">24</span><span class="special">,</span> <span class="number">8</span><span class="special">,</span> <span class="number">1</span><span class="special">,</span>
 168       <span class="number">0</span><span class="special">}</span></code>. The
 169       derivatives are obtained using the formula <span class="emphasis"><em>f<sup>(n)</sup>(2)=n!*y[n]</em></span>.
 170     </p>
 171 <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">differentiation</span><span class="special">/</span><span class="identifier">autodiff</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
 172 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
 173
 174 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">&gt;</span>
 175 <span class="identifier">T</span> <span class="identifier">fourth_power</span><span class="special">(</span><span class="identifier">T</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">{</span>
 176   <span class="identifier">T</span> <span class="identifier">x4</span> <span class="special">=</span> <span class="identifier">x</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">;</span>  <span class="comment">// retval in operator*() uses x4's memory via NRVO.</span>
 177   <span class="identifier">x4</span> <span class="special">*=</span> <span class="identifier">x4</span><span class="special">;</span>      <span class="comment">// No copies of x4 are made within operator*=() even when squaring.</span>
 178   <span class="keyword">return</span> <span class="identifier">x4</span><span class="special">;</span>     <span class="comment">// x4 uses y's memory in main() via NRVO.</span>
 179 <span class="special">}</span>
 180
 181 <span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span> <span class="special">{</span>
 182   <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">differentiation</span><span class="special">;</span>
 183
 184   <span class="keyword">constexpr</span> <span class="keyword">unsigned</span> <span class="identifier">Order</span> <span class="special">=</span> <span class="number">5</span><span class="special">;</span>                  <span class="comment">// Highest order derivative to be calculated.</span>
 185   <span class="keyword">auto</span> <span class="keyword">const</span> <span class="identifier">x</span> <span class="special">=</span> <span class="identifier">make_fvar</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span> <span class="identifier">Order</span><span class="special">&gt;(</span><span class="number">2.0</span><span class="special">);</span>  <span class="comment">// Find derivatives at x=2.</span>
 186   <span class="keyword">auto</span> <span class="keyword">const</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">fourth_power</span><span class="special">(</span><span class="identifier">x</span><span class="special">);</span>
 187   <span class="keyword">for</span> <span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">i</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">i</span> <span class="special">&lt;=</span> <span class="identifier">Order</span><span class="special">;</span> <span class="special">++</span><span class="identifier">i</span><span class="special">)</span>
 188     <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"y.derivative("</span> <span class="special">&lt;&lt;</span> <span class="identifier">i</span> <span class="special">&lt;&lt;</span> <span class="string">") = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">y</span><span class="special">.</span><span class="identifier">derivative</span><span class="special">(</span><span class="identifier">i</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
 189   <span class="keyword">return</span> <span class="number">0</span><span class="special">;</span>
 190 <span class="special">}</span>
 191 <span class="comment">/*
 192 Output:
 193 y.derivative(0) = 16
 194 y.derivative(1) = 32
 195 y.derivative(2) = 48
 196 y.derivative(3) = 48
 197 y.derivative(4) = 24
 198 y.derivative(5) = 0
 199 */</span>
 200 </pre>
 201 <p>
 202       The above calculates
 203     </p>
 204 <div class="blockquote"><blockquote class="blockquote"><div class="blockquote"><blockquote class="blockquote"><p>
 205           <span class="inlinemediaobject"><img src="../../equations/autodiff/example1.svg"></span>
 206         </p></blockquote></div></blockquote></div>
 207 <h3>
 208 <a name="math_toolkit.autodiff.h5"></a>
 209       <span class="phrase"><a name="math_toolkit.autodiff.example-multiprecision"></a></span><a class="link" href="autodiff.html#math_toolkit.autodiff.example-multiprecision">Example
 210       2: Multi-variable mixed partial derivatives with multi-precision data type</a>
 211     </h3>
 212 <h4>
 213 <a name="math_toolkit.autodiff.h6"></a>
 214       <span class="phrase"><a name="math_toolkit.autodiff.calculate_autodiff_equation_mixe"></a></span><a class="link" href="autodiff.html#math_toolkit.autodiff.calculate_autodiff_equation_mixe">Calculate
 215       <span class="inlinemediaobject"><img src="../../equations/autodiff/mixed12.svg"></span> with a precision of about 50 decimal digits, where <span class="inlinemediaobject"><img src="../../equations/autodiff/example2f.svg"></span>.</a>
 216     </h4>
 217 <p>
 218       In this example, <code class="computeroutput"><span class="identifier">make_ftuple</span><span class="special">&lt;</span><span class="identifier">float50</span><span class="special">,</span> <span class="identifier">Nw</span><span class="special">,</span>
 219       <span class="identifier">Nx</span><span class="special">,</span> <span class="identifier">Ny</span><span class="special">,</span> <span class="identifier">Nz</span><span class="special">&gt;(</span><span class="number">11</span><span class="special">,</span>
 220       <span class="number">12</span><span class="special">,</span> <span class="number">13</span><span class="special">,</span> <span class="number">14</span><span class="special">)</span></code> returns a <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">tuple</span></code> of
 221       4 independent <code class="computeroutput"><span class="identifier">fvar</span></code> variables,
 222       with values of 11, 12, 13, and 14, for which the maximum order derivative to
 223       be calculated for each are 3, 2, 4, 3, respectively. The order of the variables
 224       is important, as it is the same order used when calling <code class="computeroutput"><span class="identifier">v</span><span class="special">.</span><span class="identifier">derivative</span><span class="special">(</span><span class="identifier">Nw</span><span class="special">,</span>
 225       <span class="identifier">Nx</span><span class="special">,</span> <span class="identifier">Ny</span><span class="special">,</span> <span class="identifier">Nz</span><span class="special">)</span></code> in the example below.
 226     </p>
 227 <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">differentiation</span><span class="special">/</span><span class="identifier">autodiff</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
 228 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">multiprecision</span><span class="special">/</span><span class="identifier">cpp_bin_float</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
 229 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
 230
 231 <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">differentiation</span><span class="special">;</span>
 232
 233 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">W</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">X</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Y</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Z</span><span class="special">&gt;</span>
 234 <span class="identifier">promote</span><span class="special">&lt;</span><span class="identifier">W</span><span class="special">,</span> <span class="identifier">X</span><span class="special">,</span> <span class="identifier">Y</span><span class="special">,</span> <span class="identifier">Z</span><span class="special">&gt;</span> <span class="identifier">f</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">W</span><span class="special">&amp;</span> <span class="identifier">w</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">X</span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Y</span><span class="special">&amp;</span> <span class="identifier">y</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Z</span><span class="special">&amp;</span> <span class="identifier">z</span><span class="special">)</span> <span class="special">{</span>
 235   <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span>
 236   <span class="keyword">return</span> <span class="identifier">exp</span><span class="special">(</span><span class="identifier">w</span> <span class="special">*</span> <span class="identifier">sin</span><span class="special">(</span><span class="identifier">x</span> <span class="special">*</span> <span class="identifier">log</span><span class="special">(</span><span class="identifier">y</span><span class="special">)</span> <span class="special">/</span> <span class="identifier">z</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">w</span> <span class="special">*</span> <span class="identifier">z</span> <span class="special">/</span> <span class="special">(</span><span class="identifier">x</span> <span class="special">*</span> <span class="identifier">y</span><span class="special">)))</span> <span class="special">+</span> <span class="identifier">w</span> <span class="special">*</span> <span class="identifier">w</span> <span class="special">/</span> <span class="identifier">tan</span><span class="special">(</span><span class="identifier">z</span><span class="special">);</span>
 237 <span class="special">}</span>
 238
 239 <span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span> <span class="special">{</span>
 240   <span class="keyword">using</span> <span class="identifier">float50</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">cpp_bin_float_50</span><span class="special">;</span>
 241
 242   <span class="keyword">constexpr</span> <span class="keyword">unsigned</span> <span class="identifier">Nw</span> <span class="special">=</span> <span class="number">3</span><span class="special">;</span>  <span class="comment">// Max order of derivative to calculate for w</span>
 243   <span class="keyword">constexpr</span> <span class="keyword">unsigned</span> <span class="identifier">Nx</span> <span class="special">=</span> <span class="number">2</span><span class="special">;</span>  <span class="comment">// Max order of derivative to calculate for x</span>
 244   <span class="keyword">constexpr</span> <span class="keyword">unsigned</span> <span class="identifier">Ny</span> <span class="special">=</span> <span class="number">4</span><span class="special">;</span>  <span class="comment">// Max order of derivative to calculate for y</span>
 245   <span class="keyword">constexpr</span> <span class="keyword">unsigned</span> <span class="identifier">Nz</span> <span class="special">=</span> <span class="number">3</span><span class="special">;</span>  <span class="comment">// Max order of derivative to calculate for z</span>
 246   <span class="comment">// Declare 4 independent variables together into a std::tuple.</span>
 247   <span class="keyword">auto</span> <span class="keyword">const</span> <span class="identifier">variables</span> <span class="special">=</span> <span class="identifier">make_ftuple</span><span class="special">&lt;</span><span class="identifier">float50</span><span class="special">,</span> <span class="identifier">Nw</span><span class="special">,</span> <span class="identifier">Nx</span><span class="special">,</span> <span class="identifier">Ny</span><span class="special">,</span> <span class="identifier">Nz</span><span class="special">&gt;(</span><span class="number">11</span><span class="special">,</span> <span class="number">12</span><span class="special">,</span> <span class="number">13</span><span class="special">,</span> <span class="number">14</span><span class="special">);</span>
 248   <span class="keyword">auto</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">w</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">get</span><span class="special">&lt;</span><span class="number">0</span><span class="special">&gt;(</span><span class="identifier">variables</span><span class="special">);</span>  <span class="comment">// Up to Nw derivatives at w=11</span>
 249   <span class="keyword">auto</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">x</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">get</span><span class="special">&lt;</span><span class="number">1</span><span class="special">&gt;(</span><span class="identifier">variables</span><span class="special">);</span>  <span class="comment">// Up to Nx derivatives at x=12</span>
 250   <span class="keyword">auto</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">get</span><span class="special">&lt;</span><span class="number">2</span><span class="special">&gt;(</span><span class="identifier">variables</span><span class="special">);</span>  <span class="comment">// Up to Ny derivatives at y=13</span>
 251   <span class="keyword">auto</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">z</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">get</span><span class="special">&lt;</span><span class="number">3</span><span class="special">&gt;(</span><span class="identifier">variables</span><span class="special">);</span>  <span class="comment">// Up to Nz derivatives at z=14</span>
 252   <span class="keyword">auto</span> <span class="keyword">const</span> <span class="identifier">v</span> <span class="special">=</span> <span class="identifier">f</span><span class="special">(</span><span class="identifier">w</span><span class="special">,</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">z</span><span class="special">);</span>
 253   <span class="comment">// Calculated from Mathematica symbolic differentiation.</span>
 254   <span class="identifier">float50</span> <span class="keyword">const</span> <span class="identifier">answer</span><span class="special">(</span><span class="string">"1976.319600747797717779881875290418720908121189218755"</span><span class="special">);</span>
 255   <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</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">float50</span><span class="special">&gt;::</span><span class="identifier">digits10</span><span class="special">)</span>
 256             <span class="special">&lt;&lt;</span> <span class="string">"mathematica   : "</span> <span class="special">&lt;&lt;</span> <span class="identifier">answer</span> <span class="special">&lt;&lt;</span> <span class="char">'\n'</span>
 257             <span class="special">&lt;&lt;</span> <span class="string">"autodiff      : "</span> <span class="special">&lt;&lt;</span> <span class="identifier">v</span><span class="special">.</span><span class="identifier">derivative</span><span class="special">(</span><span class="identifier">Nw</span><span class="special">,</span> <span class="identifier">Nx</span><span class="special">,</span> <span class="identifier">Ny</span><span class="special">,</span> <span class="identifier">Nz</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="char">'\n'</span>
 258             <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span>
 259             <span class="special">&lt;&lt;</span> <span class="string">"relative error: "</span> <span class="special">&lt;&lt;</span> <span class="special">(</span><span class="identifier">v</span><span class="special">.</span><span class="identifier">derivative</span><span class="special">(</span><span class="identifier">Nw</span><span class="special">,</span> <span class="identifier">Nx</span><span class="special">,</span> <span class="identifier">Ny</span><span class="special">,</span> <span class="identifier">Nz</span><span class="special">)</span> <span class="special">/</span> <span class="identifier">answer</span> <span class="special">-</span> <span class="number">1</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="char">'\n'</span><span class="special">;</span>
 260   <span class="keyword">return</span> <span class="number">0</span><span class="special">;</span>
 261 <span class="special">}</span>
 262 <span class="comment">/*
 263 Output:
 264 mathematica   : 1976.3196007477977177798818752904187209081211892188
 265 autodiff      : 1976.3196007477977177798818752904187209081211892188
 266 relative error: 2.67e-50
 267 */</span>
 268 </pre>
 269 <h3>
 270 <a name="math_toolkit.autodiff.h7"></a>
 271       <span class="phrase"><a name="math_toolkit.autodiff.example-black_scholes"></a></span><a class="link" href="autodiff.html#math_toolkit.autodiff.example-black_scholes">Example
 272       3: Black-Scholes Option Pricing with Greeks Automatically Calculated</a>
 273     </h3>
 274 <h4>
 275 <a name="math_toolkit.autodiff.h8"></a>
 276       <span class="phrase"><a name="math_toolkit.autodiff.calculate_greeks_directly_from_t"></a></span><a class="link" href="autodiff.html#math_toolkit.autodiff.calculate_greeks_directly_from_t">Calculate
 277       greeks directly from the Black-Scholes pricing function.</a>
 278     </h4>
 279 <p>
 280       Below is the standard Black-Scholes pricing function written as a function
 281       template, where the price, volatility (sigma), time to expiration (tau) and
 282       interest rate are template parameters. This means that any greek based on these
 283       4 variables can be calculated using autodiff. The below example calculates
 284       delta and gamma where the variable of differentiation is only the price. For
 285       examples of more exotic greeks, see <code class="computeroutput"><span class="identifier">example</span><span class="special">/</span><span class="identifier">black_scholes</span><span class="special">.</span><span class="identifier">cpp</span></code>.
 286     </p>
 287 <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">differentiation</span><span class="special">/</span><span class="identifier">autodiff</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
 288 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
 289
 290 <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">constants</span><span class="special">;</span>
 291 <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">differentiation</span><span class="special">;</span>
 292
 293 <span class="comment">// Equations and function/variable names are from</span>
 294 <span class="comment">// https://en.wikipedia.org/wiki/Greeks_(finance)#Formulas_for_European_option_Greeks</span>
 295
 296 <span class="comment">// Standard normal cumulative distribution function</span>
 297 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">X</span><span class="special">&gt;</span>
 298 <span class="identifier">X</span> <span class="identifier">Phi</span><span class="special">(</span><span class="identifier">X</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">{</span>
 299   <span class="keyword">return</span> <span class="number">0.5</span> <span class="special">*</span> <span class="identifier">erfc</span><span class="special">(-</span><span class="identifier">one_div_root_two</span><span class="special">&lt;</span><span class="identifier">X</span><span class="special">&gt;()</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">);</span>
 300 <span class="special">}</span>
 301
 302 <span class="keyword">enum</span> <span class="keyword">class</span> <span class="identifier">CP</span> <span class="special">{</span> <span class="identifier">call</span><span class="special">,</span> <span class="identifier">put</span> <span class="special">};</span>
 303
 304 <span class="comment">// Assume zero annual dividend yield (q=0).</span>
 305 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">Price</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Sigma</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Tau</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Rate</span><span class="special">&gt;</span>
 306 <span class="identifier">promote</span><span class="special">&lt;</span><span class="identifier">Price</span><span class="special">,</span> <span class="identifier">Sigma</span><span class="special">,</span> <span class="identifier">Tau</span><span class="special">,</span> <span class="identifier">Rate</span><span class="special">&gt;</span> <span class="identifier">black_scholes_option_price</span><span class="special">(</span><span class="identifier">CP</span> <span class="identifier">cp</span><span class="special">,</span>
 307                                                             <span class="keyword">double</span> <span class="identifier">K</span><span class="special">,</span>
 308                                                             <span class="identifier">Price</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">S</span><span class="special">,</span>
 309                                                             <span class="identifier">Sigma</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">sigma</span><span class="special">,</span>
 310                                                             <span class="identifier">Tau</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">tau</span><span class="special">,</span>
 311                                                             <span class="identifier">Rate</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">r</span><span class="special">)</span> <span class="special">{</span>
 312   <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span>
 313   <span class="keyword">auto</span> <span class="keyword">const</span> <span class="identifier">d1</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">log</span><span class="special">(</span><span class="identifier">S</span> <span class="special">/</span> <span class="identifier">K</span><span class="special">)</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">r</span> <span class="special">+</span> <span class="identifier">sigma</span> <span class="special">*</span> <span class="identifier">sigma</span> <span class="special">/</span> <span class="number">2</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">tau</span><span class="special">)</span> <span class="special">/</span> <span class="special">(</span><span class="identifier">sigma</span> <span class="special">*</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">tau</span><span class="special">));</span>
 314   <span class="keyword">auto</span> <span class="keyword">const</span> <span class="identifier">d2</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">log</span><span class="special">(</span><span class="identifier">S</span> <span class="special">/</span> <span class="identifier">K</span><span class="special">)</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">r</span> <span class="special">-</span> <span class="identifier">sigma</span> <span class="special">*</span> <span class="identifier">sigma</span> <span class="special">/</span> <span class="number">2</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">tau</span><span class="special">)</span> <span class="special">/</span> <span class="special">(</span><span class="identifier">sigma</span> <span class="special">*</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">tau</span><span class="special">));</span>
 315   <span class="keyword">switch</span> <span class="special">(</span><span class="identifier">cp</span><span class="special">)</span> <span class="special">{</span>
 316     <span class="keyword">case</span> <span class="identifier">CP</span><span class="special">::</span><span class="identifier">call</span><span class="special">:</span>
 317       <span class="keyword">return</span> <span class="identifier">S</span> <span class="special">*</span> <span class="identifier">Phi</span><span class="special">(</span><span class="identifier">d1</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">exp</span><span class="special">(-</span><span class="identifier">r</span> <span class="special">*</span> <span class="identifier">tau</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">K</span> <span class="special">*</span> <span class="identifier">Phi</span><span class="special">(</span><span class="identifier">d2</span><span class="special">);</span>
 318     <span class="keyword">case</span> <span class="identifier">CP</span><span class="special">::</span><span class="identifier">put</span><span class="special">:</span>
 319       <span class="keyword">return</span> <span class="identifier">exp</span><span class="special">(-</span><span class="identifier">r</span> <span class="special">*</span> <span class="identifier">tau</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">K</span> <span class="special">*</span> <span class="identifier">Phi</span><span class="special">(-</span><span class="identifier">d2</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">S</span> <span class="special">*</span> <span class="identifier">Phi</span><span class="special">(-</span><span class="identifier">d1</span><span class="special">);</span>
 320   <span class="special">}</span>
 321 <span class="special">}</span>
 322
 323 <span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span> <span class="special">{</span>
 324   <span class="keyword">double</span> <span class="keyword">const</span> <span class="identifier">K</span> <span class="special">=</span> <span class="number">100.0</span><span class="special">;</span>                    <span class="comment">// Strike price.</span>
 325   <span class="keyword">auto</span> <span class="keyword">const</span> <span class="identifier">S</span> <span class="special">=</span> <span class="identifier">make_fvar</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">,</span> <span class="number">2</span><span class="special">&gt;(</span><span class="number">105</span><span class="special">);</span>  <span class="comment">// Stock price.</span>
 326   <span class="keyword">double</span> <span class="keyword">const</span> <span class="identifier">sigma</span> <span class="special">=</span> <span class="number">5</span><span class="special">;</span>                    <span class="comment">// Volatility.</span>
 327   <span class="keyword">double</span> <span class="keyword">const</span> <span class="identifier">tau</span> <span class="special">=</span> <span class="number">30.0</span> <span class="special">/</span> <span class="number">365</span><span class="special">;</span>             <span class="comment">// Time to expiration in years. (30 days).</span>
 328   <span class="keyword">double</span> <span class="keyword">const</span> <span class="identifier">r</span> <span class="special">=</span> <span class="number">1.25</span> <span class="special">/</span> <span class="number">100</span><span class="special">;</span>               <span class="comment">// Interest rate.</span>
 329   <span class="keyword">auto</span> <span class="keyword">const</span> <span class="identifier">call_price</span> <span class="special">=</span> <span class="identifier">black_scholes_option_price</span><span class="special">(</span><span class="identifier">CP</span><span class="special">::</span><span class="identifier">call</span><span class="special">,</span> <span class="identifier">K</span><span class="special">,</span> <span class="identifier">S</span><span class="special">,</span> <span class="identifier">sigma</span><span class="special">,</span> <span class="identifier">tau</span><span class="special">,</span> <span class="identifier">r</span><span class="special">);</span>
 330   <span class="keyword">auto</span> <span class="keyword">const</span> <span class="identifier">put_price</span> <span class="special">=</span> <span class="identifier">black_scholes_option_price</span><span class="special">(</span><span class="identifier">CP</span><span class="special">::</span><span class="identifier">put</span><span class="special">,</span> <span class="identifier">K</span><span class="special">,</span> <span class="identifier">S</span><span class="special">,</span> <span class="identifier">sigma</span><span class="special">,</span> <span class="identifier">tau</span><span class="special">,</span> <span class="identifier">r</span><span class="special">);</span>
 331
 332   <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"black-scholes call price = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">call_price</span><span class="special">.</span><span class="identifier">derivative</span><span class="special">(</span><span class="number">0</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="char">'\n'</span>
 333             <span class="special">&lt;&lt;</span> <span class="string">"black-scholes put  price = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">put_price</span><span class="special">.</span><span class="identifier">derivative</span><span class="special">(</span><span class="number">0</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="char">'\n'</span>
 334             <span class="special">&lt;&lt;</span> <span class="string">"call delta = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">call_price</span><span class="special">.</span><span class="identifier">derivative</span><span class="special">(</span><span class="number">1</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="char">'\n'</span>
 335             <span class="special">&lt;&lt;</span> <span class="string">"put  delta = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">put_price</span><span class="special">.</span><span class="identifier">derivative</span><span class="special">(</span><span class="number">1</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="char">'\n'</span>
 336             <span class="special">&lt;&lt;</span> <span class="string">"call gamma = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">call_price</span><span class="special">.</span><span class="identifier">derivative</span><span class="special">(</span><span class="number">2</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="char">'\n'</span>
 337             <span class="special">&lt;&lt;</span> <span class="string">"put  gamma = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">put_price</span><span class="special">.</span><span class="identifier">derivative</span><span class="special">(</span><span class="number">2</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="char">'\n'</span><span class="special">;</span>
 338   <span class="keyword">return</span> <span class="number">0</span><span class="special">;</span>
 339 <span class="special">}</span>
 340 <span class="comment">/*
 341 Output:
 342 black-scholes call price = 56.5136
 343 black-scholes put  price = 51.4109
 344 call delta = 0.773818
 345 put  delta = -0.226182
 346 call gamma = 0.00199852
 347 put  gamma = 0.00199852
 348 */</span>
 349 </pre>
 350 <h2>
 351 <a name="math_toolkit.autodiff.h9"></a>
 352       <span class="phrase"><a name="math_toolkit.autodiff.advantages_of_automatic_differen"></a></span><a class="link" href="autodiff.html#math_toolkit.autodiff.advantages_of_automatic_differen">Advantages
 353       of Automatic Differentiation</a>
 354     </h2>
 355 <p>
 356       The above examples illustrate some of the advantages of using autodiff:
 357     </p>
 358 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 359 <li class="listitem">
 360           Elimination of code redundancy. The existence of <span class="emphasis"><em>N</em></span>
 361           separate functions to calculate derivatives is a form of code redundancy,
 362           with all the liabilities that come with it:
 363           <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: circle; ">
 364 <li class="listitem">
 365                 Changes to one function require <span class="emphasis"><em>N</em></span> additional
 366                 changes to other functions. In the 3rd example above, consider how
 367                 much larger and inter-dependent the above code base would be if a
 368                 separate function were written for <a href="https://en.wikipedia.org/wiki/Greeks_(finance)#Formulas_for_European_option_Greeks" target="_top">each
 369                 Greek</a> value.
 370               </li>
 371 <li class="listitem">
 372                 Dependencies upon a derivative function for a different purpose will
 373                 break when changes are made to the original function. What doesn't
 374                 need to exist cannot break.
 375               </li>
 376 <li class="listitem">
 377                 Code bloat, reducing conceptual integrity. Control over the evolution
 378                 of code is easier/safer when the code base is smaller and able to
 379                 be intuitively grasped.
 380               </li>
 381 </ul></div>
 382         </li>
 383 <li class="listitem">
 384           Accuracy of derivatives over finite difference methods. Single-iteration
 385           finite difference methods always include a <span class="emphasis"><em>&#916;x</em></span>
 386           free variable that must be carefully chosen for each application. If <span class="emphasis"><em>&#916;x</em></span>
 387           is too small, then numerical errors become large. If <span class="emphasis"><em>&#916;x</em></span>
 388           is too large, then mathematical errors become large. With autodiff, there
 389           are no free variables to set and the accuracy of the answer is generally
 390           superior to finite difference methods even with the best choice of <span class="emphasis"><em>&#916;x</em></span>.
 391         </li>
 392 </ul></div>
 393 <h2>
 394 <a name="math_toolkit.autodiff.h10"></a>
 395       <span class="phrase"><a name="math_toolkit.autodiff.manual"></a></span><a class="link" href="autodiff.html#math_toolkit.autodiff.manual">Manual</a>
 396     </h2>
 397 <p>
 398       Additional details are in the <a href="../../differentiation/autodiff.pdf" target="_top">autodiff
 399       manual</a>.
 400     </p>
 401 </div>
 402 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 403 <td align="left"></td>
 404 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 405       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 406       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 407       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 408       Daryle Walker and Xiaogang Zhang<p>
 409         Distributed under the Boost Software License, Version 1.0. (See accompanying
 410         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>)
 411       </p>
 412 </div></td>
 413 </tr></table>
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