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introduction.xml @upstream/2015.00.00.7 — raw · history · blame
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 | <?xml version='1.0'?> <html xmlns='http://www.w3.org/1999/xhtml' xmlns:math='http://www.w3.org/1998/Math/MathML' > <head> <title>An Introduction by Example to Algorithmic Differentiation</title> <meta http-equiv='Content-Type' content='text/html' charset='utf-8'/> <meta name="description" id="description" content="An Introduction by Example to Algorithmic Differentiation"/> <meta name="keywords" id="keywords" content=" introduction Ad Algorithmic Differentiation Automatic an by example to algorithmic differentiation purpose preface forward mode reverse operation count efficiency outline reference "/> <style type='text/css'> body { color : black } body { background-color : white } A:link { color : blue } A:visited { color : purple } A:active { color : purple } </style> <script type='text/javascript' language='JavaScript' src='_introduction_xml.js'> </script> </head> <body> <table><tr> <td> <a href="http://www.coin-or.org/CppAD/" target="_top"><img border="0" src="_image.gif"/></a> </td> <td><a href="installunix.xml" target="_top">Prev</a> </td><td><a href="get_started.cpp.xml" target="_top">Next</a> </td><td> <select onchange='choose_across0(this)'> <option>Index-></option> <option>contents</option> <option>reference</option> <option>index</option> <option>search</option> <option>external</option> </select> </td> <td> <select onchange='choose_up0(this)'> <option>Up-></option> <option>CppAD</option> <option>Introduction</option> </select> </td> <td> <select onchange='choose_down1(this)'> <option>CppAD-></option> <option>Install</option> <option>Introduction</option> <option>AD</option> <option>ADFun</option> <option>preprocessor</option> <option>multi_thread</option> <option>library</option> <option>ipopt_solve</option> <option>Example</option> <option>speed</option> <option>Appendix</option> </select> </td> <td> <select onchange='choose_down0(this)'> <option>Introduction-></option> <option>get_started.cpp</option> <option>exp_2</option> <option>exp_eps</option> <option>exp_apx_main.cpp</option> </select> </td> <td> <select onchange='choose_current0(this)'> <option>Headings-></option> <option>Purpose</option> <option>Preface</option> <option>---..Algorithmic Differentiation</option> <option>---..Forward Mode</option> <option>---..Reverse Mode</option> <option>---..Operation Count</option> <option>---..Efficiency</option> <option>Outline</option> <option>Reference</option> <option>Contents</option> </select> </td> </tr></table><br/> <center><b><big><big>An Introduction by Example to Algorithmic Differentiation</big></big></b></center> <br/> <b><big><a name="Purpose" id="Purpose">Purpose</a></big></b> <br/> This is an introduction by example to Algorithmic Differentiation. Its purpose is to aid in understand what AD calculates, how the calculations are preformed, and the amount of computation and memory required for a forward or reverse sweep. <br/> <br/> <b><big><a name="Preface" id="Preface">Preface</a></big></b> <br/> <br/> <b><a name="Preface.Algorithmic Differentiation" id="Preface.Algorithmic Differentiation">Algorithmic Differentiation</a></b> <br/> Algorithmic Differentiation (often referred to as Automatic Differentiation or just AD) uses the software representation of a function to obtain an efficient method for calculating its derivatives. These derivatives can be of arbitrary order and are analytic in nature (do not have any truncation error). <br/> <br/> <b><a name="Preface.Forward Mode" id="Preface.Forward Mode">Forward Mode</a></b> <br/> A forward mode sweep computes the partial derivative of all the dependent variables with respect to one independent variable (or independent variable direction). <br/> <br/> <b><a name="Preface.Reverse Mode" id="Preface.Reverse Mode">Reverse Mode</a></b> <br/> A reverse mode sweep computes the derivative of one dependent variable (or one dependent variable direction) with respect to all the independent variables. <br/> <br/> <b><a name="Preface.Operation Count" id="Preface.Operation Count">Operation Count</a></b> <br/> The number of floating point operations for either a forward or reverse mode sweep is a small multiple of the number required to evaluate the original function. Thus, using reverse mode, you can evaluate the derivative of a scalar valued function with respect to thousands of variables in a small multiple of the work to evaluate the original function. <br/> <br/> <b><a name="Preface.Efficiency" id="Preface.Efficiency">Efficiency</a></b> <br/> AD automatically takes advantage of the speed of your algorithmic representation of a function. For example, if you calculate a determinant using LU factorization, AD will use the LU representation for the derivative of the determinant (which is faster than using the definition of the determinant). <br/> <br/> <b><big><a name="Outline" id="Outline">Outline</a></big></b> <ol type="A"><li> Demonstrate the use of CppAD to calculate derivatives of a polynomial: <a href="get_started.cpp.xml" target="_top"><span style='white-space: nowrap'>get_started.cpp</span></a> . </li><li> Present two algorithms that approximate the exponential function. The first algorithm <a href="exp_2.hpp.xml" target="_top"><span style='white-space: nowrap'>exp_2.hpp</span></a> is simpler and does not include any logical variables or loops. The second algorithm <a href="exp_eps.hpp.xml" target="_top"><span style='white-space: nowrap'>exp_eps.hpp</span></a> includes logical operations and a <code><font color="blue">while</font></code> loop. For each of these algorithms, do the following: <ol type="1"><li> Define the mathematical function corresponding to the algorithm (<a href="exp_2.xml" target="_top"><span style='white-space: nowrap'>exp_2</span></a> and <a href="exp_eps.xml" target="_top"><span style='white-space: nowrap'>exp_eps</span></a> ). </li><li> Write out the floating point operation sequence, and corresponding values, that correspond to executing the algorithm for a specific input (<a href="exp_2_for0.xml" target="_top"><span style='white-space: nowrap'>exp_2_for0</span></a> and <a href="exp_eps_for0.xml" target="_top"><span style='white-space: nowrap'>exp_eps_for0</span></a> ). </li><li> Compute a forward sweep derivative of the operation sequence (<a href="exp_2_for1.xml" target="_top"><span style='white-space: nowrap'>exp_2_for1</span></a> and <a href="exp_eps_for1.xml" target="_top"><span style='white-space: nowrap'>exp_eps_for1</span></a> ). </li><li> Compute a reverse sweep derivative of the operation sequence (<a href="exp_2_rev1.xml" target="_top"><span style='white-space: nowrap'>exp_2_rev1</span></a> and <a href="exp_eps_rev1.xml" target="_top"><span style='white-space: nowrap'>exp_eps_rev1</span></a> ). </li><li> Use CppAD to compute both a forward and reverse sweep of the operation sequence (<a href="exp_2_cppad.xml" target="_top"><span style='white-space: nowrap'>exp_2_cppad</span></a> and <a href="exp_eps_cppad.xml" target="_top"><span style='white-space: nowrap'>exp_eps_cppad</span></a> ). </li></ol> </li><li> The program <a href="exp_apx_main.cpp.xml" target="_top"><span style='white-space: nowrap'>exp_apx_main.cpp</span></a> runs all of the test routines that validate the calculations in the <a href="exp_2.xml" target="_top"><span style='white-space: nowrap'>exp_2</span></a> and <a href="exp_eps.xml" target="_top"><span style='white-space: nowrap'>exp_eps</span></a> presentation. </li></ol> <br/> <br/> <b><big><a name="Reference" id="Reference">Reference</a></big></b> <br/> An in-depth review of AD theory and methods can be found in the book <code><i><font color="black"><span style='white-space: nowrap'><br/> Evaluating Derivatives:<br/> Principles and Techniques of Algorithmic Differentiation<br/> </span></font></i></code> , Andreas Griewank, SIAM Frontiers in Applied Mathematics, 2000. <br/> <br/> <b><big><a name="Contents" id="Contents">Contents</a></big></b> <br/> <table> <tr><td><a href="get_started.cpp.xml" target="_top">get_started.cpp</a></td><td>Getting Started Using CppAD to Compute Derivatives</td></tr><tr><td><a href="exp_2.xml" target="_top">exp_2</a></td><td>Second Order Exponential Approximation</td></tr><tr><td><a href="exp_eps.xml" target="_top">exp_eps</a></td><td>An Epsilon Accurate Exponential Approximation</td></tr><tr><td><a href="exp_apx_main.cpp.xml" target="_top">exp_apx_main.cpp</a></td><td>Correctness Tests For Exponential Approximation in Introduction</td></tr></table> <hr/>Input File: omh/introduction.omh </body> </html> |