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    24. 

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    25. 

  25. <div class="section">
    26. 

  26. <div class="titlepage"><div><div><h2 class="title" style="clear: both">
    27. 

  27. <a name="math_toolkit.autodiff"></a><a class="link" href="autodiff.html" title="Automatic Differentiation">Automatic Differentiation</a>
    28. 

  28. </h2></div></div></div>
    29. 

  29. <h2>
    30. 

  30. <a name="math_toolkit.autodiff.h0"></a>
    31. 

  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. 

  32.     </h2>
    33. 

  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. 

  34. 

  35. <span class="keyword">namespace</span> <span class="identifier">boost</span> <span class="special">{</span>
    36. 

  36. <span class="keyword">namespace</span> <span class="identifier">math</span> <span class="special">{</span>
    37. 

  37. <span class="keyword">namespace</span> <span class="identifier">differentiation</span> <span class="special">{</span>
    38. 

  38. 

  39. <span class="comment">// Function returning a single variable of differentiation. Recommended: Use auto for type.</span>
    40. 

  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. 

  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. 

  42. 

  43. <span class="comment">// Function returning multiple independent variables of differentiation in a std::tuple.</span>
    44. 

  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. 

  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. 

  46. 

  47. <span class="comment">// Type of combined autodiff types. Recommended: Use auto for return type (C++14).</span>
    48. 

  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. 

  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. 

  50. 

  51. <span class="keyword">namespace</span> <span class="identifier">detail</span> <span class="special">{</span>
    52. 

  52. 

  53. <span class="comment">// Single autodiff variable. Use make_fvar() or make_ftuple() to instantiate.</span>
    54. 

  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. 

  55. <span class="keyword">class</span> <span class="identifier">fvar</span> <span class="special">{</span>
    56. 

  56.  <span class="keyword">public</span><span class="special">:</span>
    57. 

  57.   <span class="comment">// Query return value of function to get the derivatives.</span>
    58. 

  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. 

  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. 

  60. 

  61.   <span class="comment">// All of the arithmetic and comparison operators are overloaded.</span>
    62. 

  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. 

  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. 

  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. 

  66. 

  67.   <span class="comment">// ...</span>
    68. 

  68. <span class="special">};</span>
    69. 

  69. 

  70. <span class="comment">// Standard math functions are overloaded and called via argument-dependent lookup (ADL).</span>
    71. 

  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. 

  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. 

  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. 

  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. 

  76. 

  77. <span class="comment">// ...</span>
    78. 

  78. 

  79. <span class="special">}</span>  <span class="comment">// namespace detail</span>
    80. 

  80. 

  81. <span class="special">}</span>  <span class="comment">// namespace differentiation</span>
    82. 

  82. <span class="special">}</span>  <span class="comment">// namespace math</span>
    83. 

  83. <span class="special">}</span>  <span class="comment">// namespace boost</span>
    84. 

  84. </pre>
    85. 

  85. <h2>
    86. 

  86. <a name="math_toolkit.autodiff.h1"></a>
    87. 

  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. 

  88.     </h2>
    89. 

  89. <p>
    90. 

  90.       Autodiff is a header-only C++ library that facilitates the <a href="https://en.wikipedia.org/wiki/Automatic_differentiation" target="_top">automatic
    91. 

  91.       differentiation</a> (forward mode) of mathematical functions of single
    92. 

  92.       and multiple variables.
    93. 

  93.     </p>
    94. 

  94. <p>
    95. 

  95.       This implementation is based upon the <a href="https://en.wikipedia.org/wiki/Taylor_series" target="_top">Taylor
    96. 

  96.       series</a> expansion of an analytic function <span class="emphasis"><em>f</em></span> at
    97. 

  97.       the point <span class="emphasis"><em>x<sub>0</sub></em></span>:
    98. 

  98.     </p>
    99. 

  99. <div class="blockquote"><blockquote class="blockquote"><div class="blockquote"><blockquote class="blockquote"><p>
    100. 

  100.           <span class="inlinemediaobject"><img src="../../equations/autodiff/taylor_series.svg"></span>
    101. 

  101.         </p></blockquote></div></blockquote></div>
    102. 

  102. <p>
    103. 

  103.       The essential idea of autodiff is the substitution of numbers with polynomials
    104. 

  104.       in the evaluation of <span class="emphasis"><em>f(x<sub>0</sub>)</em></span>. By substituting the number
    105. 

  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. 

  106.       and using the same algorithm to compute <span class="emphasis"><em>f(x<sub>0</sub>+&#949;)</em></span>,
    107. 

  107.       the resulting polynomial in <span class="emphasis"><em>&#949;</em></span> contains the function's
    108. 

  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. 

  109.       ... within the coefficients. Each coefficient is equal to the derivative of
    110. 

  110.       its respective order, divided by the factorial of the order.
    111. 

  111.     </p>
    112. 

  112. <p>
    113. 

  113.       In greater detail, assume one is interested in calculating the first <span class="emphasis"><em>N</em></span>
    114. 

  114.       derivatives of <span class="emphasis"><em>f</em></span> at <span class="emphasis"><em>x<sub>0</sub></em></span>. Without loss
    115. 

  115.       of precision to the calculation of the derivatives, all terms <span class="emphasis"><em>O(&#949;<sup>N+1</sup>)</em></span>
    116. 

  116.       that include powers of <span class="emphasis"><em>&#949;</em></span> greater than <span class="emphasis"><em>N</em></span>
    117. 

  117.       can be discarded. (This is due to the fact that each term in a polynomial depends
    118. 

  118.       only upon equal and lower-order terms under arithmetic operations.) Under these
    119. 

  119.       truncation rules, <span class="emphasis"><em>f</em></span> provides a polynomial-to-polynomial
    120. 

  120.       transformation:
    121. 

  121.     </p>
    122. 

  122. <div class="blockquote"><blockquote class="blockquote"><div class="blockquote"><blockquote class="blockquote"><p>
    123. 

  123.           <span class="inlinemediaobject"><img src="../../equations/autodiff/polynomial_transform.svg"></span>
    124. 

  124.         </p></blockquote></div></blockquote></div>
    125. 

  125. <p>
    126. 

  126.       C++'s ability to overload operators and functions allows for the creation of
    127. 

  127.       a class <code class="computeroutput"><span class="identifier">fvar</span></code> (<span class="underline">f</span>orward-mode
    128. 

  128.       autodiff <span class="underline">var</span>iable) that represents polynomials
    129. 

  129.       in <span class="emphasis"><em>&#949;</em></span>. Thus the same algorithm <span class="emphasis"><em>f</em></span>
    130. 

  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. 

  131.       to accept and return variables of a generic (template) type, is also used to
    132. 

  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. 

  133.       The derivatives <span class="emphasis"><em>f<sup>(n)</sup>(x<sub>0</sub>)</em></span> are then found from the product
    134. 

  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. 

  135.     </p>
    136. 

  136. <div class="blockquote"><blockquote class="blockquote"><div class="blockquote"><blockquote class="blockquote"><p>
    137. 

  137.           <span class="inlinemediaobject"><img src="../../equations/autodiff/derivative_formula.svg"></span>
    138. 

  138.         </p></blockquote></div></blockquote></div>
    139. 

  139. <h2>
    140. 

  140. <a name="math_toolkit.autodiff.h2"></a>
    141. 

  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. 

  142.     </h2>
    143. 

  143. <h3>
    144. 

  144. <a name="math_toolkit.autodiff.h3"></a>
    145. 

  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. 

  146.       1: Single-variable derivatives</a>
    147. 

  147.     </h3>
    148. 

  148. <h4>
    149. 

  149. <a name="math_toolkit.autodiff.h4"></a>
    150. 

  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. 

  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. 

  152.     </h4>
    153. 

  153. <p>
    154. 

  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. 

  155.       <span class="identifier">Order</span><span class="special">&gt;(</span><span class="number">2.0</span><span class="special">)</span></code> instantiates
    156. 

  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. 

  157.       enough space is allocated (on the stack) to hold a polynomial of up to degree
    158. 

  158.       5 during the proceeding computation.
    159. 

  159.     </p>
    160. 

  160. <p>
    161. 

  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. 

  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. 

  163.       correspond to the 6 coefficients of the polynomial upon initialization. Its
    164. 

  164.       fourth power, at the end of the computation, is a polynomial with coefficients
    165. 

  165.       <code class="computeroutput"><span class="identifier">y</span> <span class="special">=</span>
    166. 

  166.       <span class="special">{</span><span class="number">16</span><span class="special">,</span>
    167. 

  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. 

  168.       <span class="number">0</span><span class="special">}</span></code>. The
    169. 

  169.       derivatives are obtained using the formula <span class="emphasis"><em>f<sup>(n)</sup>(2)=n!*y[n]</em></span>.
    170. 

  170.     </p>
    171. 

  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. 

  172. <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
    173. 

  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. 

  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. 

  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. 

  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. 

  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. 

  179. <span class="special">}</span>
    180. 

  180. 

  181. <span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span> <span class="special">{</span>
    182. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  189.   <span class="keyword">return</span> <span class="number">0</span><span class="special">;</span>
    190. 

  190. <span class="special">}</span>
    191. 

  191. <span class="comment">/*
    192. 

  192. Output:
    193. 

  193. y.derivative(0) = 16
    194. 

  194. y.derivative(1) = 32
    195. 

  195. y.derivative(2) = 48
    196. 

  196. y.derivative(3) = 48
    197. 

  197. y.derivative(4) = 24
    198. 

  198. y.derivative(5) = 0
    199. 

  199. */</span>
    200. 

  200. </pre>
    201. 

  201. <p>
    202. 

  202.       The above calculates
    203. 

  203.     </p>
    204. 

  204. <div class="blockquote"><blockquote class="blockquote"><div class="blockquote"><blockquote class="blockquote"><p>
    205. 

  205.           <span class="inlinemediaobject"><img src="../../equations/autodiff/example1.svg"></span>
    206. 

  206.         </p></blockquote></div></blockquote></div>
    207. 

  207. <h3>
    208. 

  208. <a name="math_toolkit.autodiff.h5"></a>
    209. 

  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. 

  210.       2: Multi-variable mixed partial derivatives with multi-precision data type</a>
    211. 

  211.     </h3>
    212. 

  212. <h4>
    213. 

  213. <a name="math_toolkit.autodiff.h6"></a>
    214. 

  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. 

  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. 

  216.     </h4>
    217. 

  217. <p>
    218. 

  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. 

  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. 

  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. 

  221.       4 independent <code class="computeroutput"><span class="identifier">fvar</span></code> variables,
    222. 

  222.       with values of 11, 12, 13, and 14, for which the maximum order derivative to
    223. 

  223.       be calculated for each are 3, 2, 4, 3, respectively. The order of the variables
    224. 

  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. 

  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. 

  226.     </p>
    227. 

  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. 

  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. 

  229. <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
    230. 

  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. 

  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. 

  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. 

  235.   <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span>
    236. 

  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. 

  237. <span class="special">}</span>
    238. 

  238. 

  239. <span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span> <span class="special">{</span>
    240. 

  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. 

  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. 

  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. 

  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. 

  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. 

  246.   <span class="comment">// Declare 4 independent variables together into a std::tuple.</span>
    247. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  253.   <span class="comment">// Calculated from Mathematica symbolic differentiation.</span>
    254. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  260.   <span class="keyword">return</span> <span class="number">0</span><span class="special">;</span>
    261. 

  261. <span class="special">}</span>
    262. 

  262. <span class="comment">/*
    263. 

  263. Output:
    264. 

  264. mathematica   : 1976.3196007477977177798818752904187209081211892188
    265. 

  265. autodiff      : 1976.3196007477977177798818752904187209081211892188
    266. 

  266. relative error: 2.67e-50
    267. 

  267. */</span>
    268. 

  268. </pre>
    269. 

  269. <h3>
    270. 

  270. <a name="math_toolkit.autodiff.h7"></a>
    271. 

  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. 

  272.       3: Black-Scholes Option Pricing with Greeks Automatically Calculated</a>
    273. 

  273.     </h3>
    274. 

  274. <h4>
    275. 

  275. <a name="math_toolkit.autodiff.h8"></a>
    276. 

  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. 

  277.       greeks directly from the Black-Scholes pricing function.</a>
    278. 

  278.     </h4>
    279. 

  279. <p>
    280. 

  280.       Below is the standard Black-Scholes pricing function written as a function
    281. 

  281.       template, where the price, volatility (sigma), time to expiration (tau) and
    282. 

  282.       interest rate are template parameters. This means that any greek based on these
    283. 

  283.       4 variables can be calculated using autodiff. The below example calculates
    284. 

  284.       delta and gamma where the variable of differentiation is only the price. For
    285. 

  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. 

  286.     </p>
    287. 

  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. 

  288. <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
    289. 

  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. 

  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. 

  292. 

  293. <span class="comment">// Equations and function/variable names are from</span>
    294. 

  294. <span class="comment">// https://en.wikipedia.org/wiki/Greeks_(finance)#Formulas_for_European_option_Greeks</span>
    295. 

  295. 

  296. <span class="comment">// Standard normal cumulative distribution function</span>
    297. 

  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. 

  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. 

  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. 

  300. <span class="special">}</span>
    301. 

  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. 

  303. 

  304. <span class="comment">// Assume zero annual dividend yield (q=0).</span>
    305. 

  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. 

  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. 

  307.                                                             <span class="keyword">double</span> <span class="identifier">K</span><span class="special">,</span>
    308. 

  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. 

  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. 

  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. 

  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. 

  312.   <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span>
    313. 

  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. 

  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. 

  315.   <span class="keyword">switch</span> <span class="special">(</span><span class="identifier">cp</span><span class="special">)</span> <span class="special">{</span>
    316. 

  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. 

  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. 

  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. 

  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. 

  320.   <span class="special">}</span>
    321. 

  321. <span class="special">}</span>
    322. 

  322. 

  323. <span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span> <span class="special">{</span>
    324. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  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. 

  338.   <span class="keyword">return</span> <span class="number">0</span><span class="special">;</span>
    339. 

  339. <span class="special">}</span>
    340. 

  340. <span class="comment">/*
    341. 

  341. Output:
    342. 

  342. black-scholes call price = 56.5136
    343. 

  343. black-scholes put  price = 51.4109
    344. 

  344. call delta = 0.773818
    345. 

  345. put  delta = -0.226182
    346. 

  346. call gamma = 0.00199852
    347. 

  347. put  gamma = 0.00199852
    348. 

  348. */</span>
    349. 

  349. </pre>
    350. 

  350. <h2>
    351. 

  351. <a name="math_toolkit.autodiff.h9"></a>
    352. 

  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. 

  353.       of Automatic Differentiation</a>
    354. 

  354.     </h2>
    355. 

  355. <p>
    356. 

  356.       The above examples illustrate some of the advantages of using autodiff:
    357. 

  357.     </p>
    358. 

  358. <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
    359. 

  359. <li class="listitem">
    360. 

  360.           Elimination of code redundancy. The existence of <span class="emphasis"><em>N</em></span>
    361. 

  361.           separate functions to calculate derivatives is a form of code redundancy,
    362. 

  362.           with all the liabilities that come with it:
    363. 

  363.           <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: circle; ">
    364. 

  364. <li class="listitem">
    365. 

  365.                 Changes to one function require <span class="emphasis"><em>N</em></span> additional
    366. 

  366.                 changes to other functions. In the 3rd example above, consider how
    367. 

  367.                 much larger and inter-dependent the above code base would be if a
    368. 

  368.                 separate function were written for <a href="https://en.wikipedia.org/wiki/Greeks_(finance)#Formulas_for_European_option_Greeks" target="_top">each
    369. 

  369.                 Greek</a> value.
    370. 

  370.               </li>
    371. 

  371. <li class="listitem">
    372. 

  372.                 Dependencies upon a derivative function for a different purpose will
    373. 

  373.                 break when changes are made to the original function. What doesn't
    374. 

  374.                 need to exist cannot break.
    375. 

  375.               </li>
    376. 

  376. <li class="listitem">
    377. 

  377.                 Code bloat, reducing conceptual integrity. Control over the evolution
    378. 

  378.                 of code is easier/safer when the code base is smaller and able to
    379. 

  379.                 be intuitively grasped.
    380. 

  380.               </li>
    381. 

  381. </ul></div>
    382. 

  382.         </li>
    383. 

  383. <li class="listitem">
    384. 

  384.           Accuracy of derivatives over finite difference methods. Single-iteration
    385. 

  385.           finite difference methods always include a <span class="emphasis"><em>&#916;x</em></span>
    386. 

  386.           free variable that must be carefully chosen for each application. If <span class="emphasis"><em>&#916;x</em></span>
    387. 

  387.           is too small, then numerical errors become large. If <span class="emphasis"><em>&#916;x</em></span>
    388. 

  388.           is too large, then mathematical errors become large. With autodiff, there
    389. 

  389.           are no free variables to set and the accuracy of the answer is generally
    390. 

  390.           superior to finite difference methods even with the best choice of <span class="emphasis"><em>&#916;x</em></span>.
    391. 

  391.         </li>
    392. 

  392. </ul></div>
    393. 

  393. <h2>
    394. 

  394. <a name="math_toolkit.autodiff.h10"></a>
    395. 

  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. 

  396.     </h2>
    397. 

  397. <p>
    398. 

  398.       Additional details are in the <a href="../../differentiation/autodiff.pdf" target="_top">autodiff
    399. 

  399.       manual</a>.
    400. 

  400.     </p>
    401. 

  401. </div>
    402. 

  402. <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
    403. 

  403. <td align="left"></td>
    404. 

  404. <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
    405. 

  405.       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
    406. 

  406.       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
    407. 

  407.       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
    408. 

  408.       Daryle Walker and Xiaogang Zhang<p>
    409. 

  409.         Distributed under the Boost Software License, Version 1.0. (See accompanying
    410. 

  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. 

  411.       </p>
    412. 

  412. </div></td>
    413. 

  413. </tr></table>
    414. 

  414. <hr>
    415. 

  415. <div class="spirit-nav">
    416. 

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    420.