1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Automatic Differentiation</title> 5<link rel="stylesheet" href="../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../quadrature.html" title="Chapter 13. Quadrature and Differentiation"> 9<link rel="prev" href="diff.html" title="Numerical Differentiation"> 10<link rel="next" href="diff0.html" title="Lanczos Smoothing Derivatives"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="diff.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../quadrature.html"><img src="../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="diff0.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 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"><</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">></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"><</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">></span> 41<span class="identifier">autodiff_fvar</span><span class="special"><</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">...></span> <span class="identifier">make_fvar</span><span class="special">(</span><span class="identifier">RealType</span> <span class="keyword">const</span><span class="special">&</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"><</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">></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">&...</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"><</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">></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"><</span><span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">RealTypes</span><span class="special">...>::</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"><</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> 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"><</span><span class="keyword">typename</span><span class="special">...</span> <span class="identifier">Orders</span><span class="special">></span> 59 <span class="identifier">get_type_at</span><span class="special"><</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">></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"><</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">></span> 63 <span class="identifier">fvar</span><span class="special">&</span> <span class="keyword">operator</span><span class="special">+=(</span><span class="identifier">fvar</span><span class="special"><</span><span class="identifier">RealType2</span><span class="special">,</span> <span class="identifier">Order2</span><span class="special">></span> <span class="keyword">const</span><span class="special">&);</span> 64 65 <span class="identifier">fvar</span><span class="special">&</span> <span class="keyword">operator</span><span class="special">+=(</span><span class="identifier">root_type</span> <span class="keyword">const</span><span class="special">&);</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"><</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> 72<span class="identifier">fvar</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">Order</span><span class="special">></span> <span class="identifier">floor</span><span class="special">(</span><span class="identifier">fvar</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">Order</span><span class="special">></span> <span class="keyword">const</span><span class="special">&);</span> 73 74<span class="keyword">template</span> <span class="special"><</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> 75<span class="identifier">fvar</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">Order</span><span class="special">></span> <span class="identifier">exp</span><span class="special">(</span><span class="identifier">fvar</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">Order</span><span class="special">></span> <span class="keyword">const</span><span class="special">&);</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>+ε</em></span>, 106 and using the same algorithm to compute <span class="emphasis"><em>f(x<sub>0</sub>+ε)</em></span>, 107 the resulting polynomial in <span class="emphasis"><em>ε</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(ε<sup>N+1</sup>)</em></span> 116 that include powers of <span class="emphasis"><em>ε</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>ε</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>Σ<sub>n</sub>y<sub>n</sub>ε<sup>n</sup>=f(x<sub>0</sub>+ε)</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"><</span><span class="keyword">double</span><span class="special">,</span> 155 <span class="identifier">Order</span><span class="special">>(</span><span class="number">2.0</span><span class="special">)</span></code> instantiates 156 the polynomial 2+<span class="emphasis"><em>ε</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"><</span><span class="keyword">double</span><span class="special">,</span><span class="number">6</span><span class="special">></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"><</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">></span> 172<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iostream</span><span class="special">></span> 173 174<span class="keyword">template</span> <span class="special"><</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">></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">&</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"><</span><span class="keyword">double</span><span class="special">,</span> <span class="identifier">Order</span><span class="special">>(</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"><=</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"><<</span> <span class="string">"y.derivative("</span> <span class="special"><<</span> <span class="identifier">i</span> <span class="special"><<</span> <span class="string">") = "</span> <span class="special"><<</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"><<</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">/* 192Output: 193y.derivative(0) = 16 194y.derivative(1) = 32 195y.derivative(2) = 48 196y.derivative(3) = 48 197y.derivative(4) = 24 198y.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"><</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">>(</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"><</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">></span> 228<span class="preprocessor">#include</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</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> 229<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iostream</span><span class="special">></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"><</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">></span> 234<span class="identifier">promote</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> <span class="identifier">f</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">W</span><span class="special">&</span> <span class="identifier">w</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">X</span><span class="special">&</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Y</span><span class="special">&</span> <span class="identifier">y</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Z</span><span class="special">&</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"><</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">>(</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">&</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"><</span><span class="number">0</span><span class="special">>(</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">&</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"><</span><span class="number">1</span><span class="special">>(</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">&</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"><</span><span class="number">2</span><span class="special">>(</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">&</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"><</span><span class="number">3</span><span class="special">>(</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"><<</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"><</span><span class="identifier">float50</span><span class="special">>::</span><span class="identifier">digits10</span><span class="special">)</span> 256 <span class="special"><<</span> <span class="string">"mathematica : "</span> <span class="special"><<</span> <span class="identifier">answer</span> <span class="special"><<</span> <span class="char">'\n'</span> 257 <span class="special"><<</span> <span class="string">"autodiff : "</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="char">'\n'</span> 258 <span class="special"><<</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"><<</span> <span class="string">"relative error: "</span> <span class="special"><<</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"><<</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">/* 263Output: 264mathematica : 1976.3196007477977177798818752904187209081211892188 265autodiff : 1976.3196007477977177798818752904187209081211892188 266relative 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"><</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">></span> 288<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iostream</span><span class="special">></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"><</span><span class="keyword">typename</span> <span class="identifier">X</span><span class="special">></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">&</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"><</span><span class="identifier">X</span><span class="special">>()</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"><</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">></span> 306<span class="identifier">promote</span><span class="special"><</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">></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">&</span> <span class="identifier">S</span><span class="special">,</span> 309 <span class="identifier">Sigma</span> <span class="keyword">const</span><span class="special">&</span> <span class="identifier">sigma</span><span class="special">,</span> 310 <span class="identifier">Tau</span> <span class="keyword">const</span><span class="special">&</span> <span class="identifier">tau</span><span class="special">,</span> 311 <span class="identifier">Rate</span> <span class="keyword">const</span><span class="special">&</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"><</span><span class="keyword">double</span><span class="special">,</span> <span class="number">2</span><span class="special">>(</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"><<</span> <span class="string">"black-scholes call price = "</span> <span class="special"><<</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"><<</span> <span class="char">'\n'</span> 333 <span class="special"><<</span> <span class="string">"black-scholes put price = "</span> <span class="special"><<</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"><<</span> <span class="char">'\n'</span> 334 <span class="special"><<</span> <span class="string">"call delta = "</span> <span class="special"><<</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"><<</span> <span class="char">'\n'</span> 335 <span class="special"><<</span> <span class="string">"put delta = "</span> <span class="special"><<</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"><<</span> <span class="char">'\n'</span> 336 <span class="special"><<</span> <span class="string">"call gamma = "</span> <span class="special"><<</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"><<</span> <span class="char">'\n'</span> 337 <span class="special"><<</span> <span class="string">"put gamma = "</span> <span class="special"><<</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"><<</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">/* 341Output: 342black-scholes call price = 56.5136 343black-scholes put price = 51.4109 344call delta = 0.773818 345put delta = -0.226182 346call gamma = 0.00199852 347put 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>Δx</em></span> 386 free variable that must be carefully chosen for each application. If <span class="emphasis"><em>Δx</em></span> 387 is too small, then numerical errors become large. If <span class="emphasis"><em>Δx</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>Δx</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 © 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 Råde, 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> 414<hr> 415<div class="spirit-nav"> 416<a accesskey="p" href="diff.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../quadrature.html"><img src="../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="diff0.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a> 417</div> 418</body> 419</html> 420
