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exp_2_for0.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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 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528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 | <?xml version='1.0'?> <html xmlns='http://www.w3.org/1999/xhtml' xmlns:math='http://www.w3.org/1998/Math/MathML' > <head> <title>exp_2: Operation Sequence and Zero Order Forward Mode</title> <meta http-equiv='Content-Type' content='text/html' charset='utf-8'/> <meta name="description" id="description" content="exp_2: Operation Sequence and Zero Order Forward Mode"/> <meta name="keywords" id="keywords" content=" exp_2: operation sequence and zero order forward mode exp_2 example mathematical form expansion index code sweep return value verification exercises "/> <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='_exp_2_for0_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="exp_2.cpp.xml" target="_top">Prev</a> </td><td><a href="exp_2_for0.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> <option>exp_2</option> <option>exp_2_for0</option> </select> </td> <td> <select onchange='choose_down3(this)'> <option>CppAD-></option> <option>Install</option> <option>Introduction</option> 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<option>Zero Order Expansion</option> <option>Operation Sequence</option> <option>---..Index</option> <option>---..Code</option> <option>---..Operation</option> <option>---..Zero Order</option> <option>---..Sweep</option> <option>Return Value</option> <option>Verification</option> <option>Exercises</option> </select> </td> </tr></table><br/> <center><b><big><big>exp_2: Operation Sequence and Zero Order Forward Mode</big></big></b></center> <br/> <b><big><a name="Mathematical Form" id="Mathematical Form">Mathematical Form</a></big></b> <br/> The operation sequence (see below) corresponding to the algorithm <a href="exp_2.hpp.xml" target="_top"><span style='white-space: nowrap'>exp_2.hpp</span></a> is the same for all values of <code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code> . The mathematical form for the corresponding function is <math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow> <mi mathvariant='italic'>f</mi> <mo stretchy="false">(</mo> <mi mathvariant='italic'>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">=</mo> <mn>1</mn> <mo stretchy="false">+</mo> <mi mathvariant='italic'>x</mi> <mo stretchy="false">+</mo> <msup><mi mathvariant='italic'>x</mi> <mn>2</mn> </msup> <mo stretchy="false">/</mo> <mn>2</mn> </mrow></math> An algorithmic differentiation package does not operate on the mathematical function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <mi mathvariant='italic'>f</mi> <mo stretchy="false">(</mo> <mi mathvariant='italic'>x</mi> <mo stretchy="false">)</mo> </mrow></math> but rather on the particular algorithm used to compute the function (in this case <a href="exp_2.hpp.xml" target="_top"><span style='white-space: nowrap'>exp_2.hpp</span></a> ). <br/> <br/> <b><big><a name="Zero Order Expansion" id="Zero Order Expansion">Zero Order Expansion</a></big></b> <br/> In general, a zero order forward sweep is given a vector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <msup><mi mathvariant='italic'>x</mi> <mrow><mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">∈</mo> <msup><mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi> </mstyle></mrow> <mi mathvariant='italic'>n</mi> </msup> </mrow></math> and it returns the corresponding vector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <msup><mi mathvariant='italic'>y</mi> <mrow><mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">∈</mo> <msup><mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi> </mstyle></mrow> <mi mathvariant='italic'>m</mi> </msup> </mrow></math> given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow> <msup><mi mathvariant='italic'>y</mi> <mrow><mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">=</mo> <mi mathvariant='italic'>f</mi> <mo stretchy="false">(</mo> <msup><mi mathvariant='italic'>x</mi> <mrow><mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mrow></math> The superscript <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow></math> denotes zero order derivative; i.e., it is equal to the value of the corresponding variable. For the example we are considering here, both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <mi mathvariant='italic'>n</mi> </mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <mi mathvariant='italic'>m</mi> </mrow></math> are equal to one. <br/> <br/> <b><big><a name="Operation Sequence" id="Operation Sequence">Operation Sequence</a></big></b> <br/> An atomic <code><i><font color="black"><span style='white-space: nowrap'>Type</span></font></i></code> operation is an operation that has a <code><i><font color="black"><span style='white-space: nowrap'>Type</span></font></i></code> result and is not made up of other more basic operations. A sequence of atomic <code><i><font color="black"><span style='white-space: nowrap'>Type</span></font></i></code> operations is called a <code><i><font color="black"><span style='white-space: nowrap'>Type</span></font></i></code> operation sequence. Given an C++ algorithm and its inputs, there is a corresponding <code><i><font color="black"><span style='white-space: nowrap'>Type</span></font></i></code> operation sequence for each type. If <code><i><font color="black"><span style='white-space: nowrap'>Type</span></font></i></code> is clear from the context, we drop it and just refer to the operation sequence. <code><span style='white-space: nowrap'><br/> <br/> </span></code>We consider the case where <a href="exp_2.hpp.xml" target="_top"><span style='white-space: nowrap'>exp_2.hpp</span></a> is executed with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <msup><mi mathvariant='italic'>x</mi> <mrow><mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">=</mo> <mn>.5</mn> </mrow></math> . The table below contains the corresponding operation sequence and the results of a zero order sweep. <br/> <br/> <b><a name="Operation Sequence.Index" id="Operation Sequence.Index">Index</a></b> <br/> The Index column contains the index in the operation sequence of the corresponding atomic operation and variable. A Forward sweep starts with the first operation and ends with the last. <br/> <br/> <b><a name="Operation Sequence.Code" id="Operation Sequence.Code">Code</a></b> <br/> The Code column contains the C++ source code corresponding to the corresponding atomic operation in the sequence. <br/> <br/> <b><a name="Operation Sequence.Operation" id="Operation Sequence.Operation">Operation</a></b> <br/> The Operation column contains the mathematical function corresponding to each atomic operation in the sequence. <br/> <br/> <b><a name="Operation Sequence.Zero Order" id="Operation Sequence.Zero Order">Zero Order</a></b> <br/> The Zero Order column contains the zero order derivative for the corresponding variable in the operation sequence. Forward mode refers to the fact that these coefficients are computed in the same order as the original algorithm; i.e, in order of increasing index in the operation sequence. <br/> <br/> <b><a name="Operation Sequence.Sweep" id="Operation Sequence.Sweep">Sweep</a></b> <center> <table><tr><td align='left' valign='top'> <b>Index</b> </td><td align='left' valign='top'> <code><span style='white-space: nowrap'>  </span></code> </td><td align='left' valign='top'> <b>Code</b> </td><td align='left' valign='top'> <code><span style='white-space: nowrap'>  </span></code> </td><td align='left' valign='top'> <b>Operation</b> </td><td align='left' valign='top'> <code><span style='white-space: nowrap'>  </span></code> </td><td align='left' valign='top'> <b>Zero Order</b> </td></tr><tr><td align='left' valign='top'> 1 </td><td align='left' valign='top'> <code><span style='white-space: nowrap'>  </span></code> </td><td align='left' valign='top'> <code><font color="blue">Type v1 = x;</font></code> </td><td align='left' valign='top'> </td><td align='left' valign='top'> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <msub><mi mathvariant='italic'>v</mi> <mn>1</mn> </msub> <mo stretchy="false">=</mo> <mi mathvariant='italic'>x</mi> </mrow></math> </td><td align='left' valign='top'> </td><td align='left' valign='top'> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <msubsup><mi mathvariant='italic'>v</mi> <mn>1</mn> <mrow><mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">=</mo> <mn>0.5</mn> </mrow></math> </td></tr><tr><td align='left' valign='top'> 2 </td><td align='left' valign='top'> <code><span style='white-space: nowrap'>  </span></code> </td><td align='left' valign='top'> <code><font color="blue">Type v2 = Type(1) + v1;</font></code> </td><td align='left' valign='top'> </td><td align='left' valign='top'> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <msub><mi mathvariant='italic'>v</mi> <mn>2</mn> </msub> <mo stretchy="false">=</mo> <mn>1</mn> <mo stretchy="false">+</mo> <msub><mi mathvariant='italic'>v</mi> <mn>1</mn> </msub> </mrow></math> </td><td align='left' valign='top'> </td><td align='left' valign='top'> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <msubsup><mi mathvariant='italic'>v</mi> <mn>2</mn> <mrow><mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">=</mo> <mn>1.5</mn> </mrow></math> </td></tr><tr><td align='left' valign='top'> 3 </td><td align='left' valign='top'> <code><span style='white-space: nowrap'>  </span></code> </td><td align='left' valign='top'> <code><font color="blue">Type v3 = v1 * v1;</font></code> </td><td align='left' valign='top'> </td><td align='left' valign='top'> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <msub><mi mathvariant='italic'>v</mi> <mn>3</mn> </msub> <mo stretchy="false">=</mo> <msub><mi mathvariant='italic'>v</mi> <mn>1</mn> </msub> <mo stretchy="false">*</mo> <msub><mi mathvariant='italic'>v</mi> <mn>1</mn> </msub> </mrow></math> </td><td align='left' valign='top'> </td><td align='left' valign='top'> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <msubsup><mi mathvariant='italic'>v</mi> <mn>3</mn> <mrow><mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">=</mo> <mn>0.25</mn> </mrow></math> </td></tr><tr><td align='left' valign='top'> 4 </td><td align='left' valign='top'> <code><span style='white-space: nowrap'>  </span></code> </td><td align='left' valign='top'> <code><font color="blue">Type v4 = v3 / Type(2);</font></code> </td><td align='left' valign='top'> </td><td align='left' valign='top'> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <msub><mi mathvariant='italic'>v</mi> <mn>4</mn> </msub> <mo stretchy="false">=</mo> <msub><mi mathvariant='italic'>v</mi> <mn>3</mn> </msub> <mo stretchy="false">/</mo> <mn>2</mn> </mrow></math> </td><td align='left' valign='top'> </td><td align='left' valign='top'> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <msubsup><mi mathvariant='italic'>v</mi> <mn>4</mn> <mrow><mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">=</mo> <mn>0.125</mn> </mrow></math> </td></tr><tr><td align='left' valign='top'> 5 </td><td align='left' valign='top'> <code><span style='white-space: nowrap'>  </span></code> </td><td align='left' valign='top'> <code><font color="blue"> Type v5 = v2 + v4;</font></code> </td><td align='left' valign='top'> </td><td align='left' valign='top'> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <msub><mi mathvariant='italic'>v</mi> <mn>5</mn> </msub> <mo stretchy="false">=</mo> <msub><mi mathvariant='italic'>v</mi> <mn>2</mn> </msub> <mo stretchy="false">+</mo> <msub><mi mathvariant='italic'>v</mi> <mn>4</mn> </msub> </mrow></math> </td><td align='left' valign='top'> </td><td align='left' valign='top'> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <msubsup><mi mathvariant='italic'>v</mi> <mn>5</mn> <mrow><mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">=</mo> <mn>1.625</mn> </mrow></math> </td></tr> </table> </center><b><big><a name="Return Value" id="Return Value">Return Value</a></big></b> <br/> The return value for this case is <math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow> <mn>1.625</mn> <mo stretchy="false">=</mo> <msubsup><mi mathvariant='italic'>v</mi> <mn>5</mn> <mrow><mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">=</mo> <mi mathvariant='italic'>f</mi> <mo stretchy="false">(</mo> <msup><mi mathvariant='italic'>x</mi> <mrow><mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mrow></math> <br/> <b><big><a name="Verification" id="Verification">Verification</a></big></b> <br/> The file <a href="exp_2_for0.cpp.xml" target="_top"><span style='white-space: nowrap'>exp_2_for0.cpp</span></a> contains a routine that verifies the values computed above. It returns true for success and false for failure. <br/> <br/> <b><big><a name="Exercises" id="Exercises">Exercises</a></big></b> <ol type="1"><li> Suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <msup><mi mathvariant='italic'>x</mi> <mrow><mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">=</mo> <mn>.2</mn> </mrow></math> , what is the result of a zero order forward sweep for the operation sequence above; i.e., what are the corresponding values for <math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow> <msubsup><mi mathvariant='italic'>v</mi> <mn>1</mn> <mrow><mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">,</mo> <msubsup><mi mathvariant='italic'>v</mi> <mn>2</mn> <mrow><mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">,</mo> <mo stretchy="false">⋯</mo> <mo stretchy="false">,</mo> <msubsup><mi mathvariant='italic'>v</mi> <mn>5</mn> <mrow><mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow></math> </li><li> Create a modified version of <a href="exp_2_for0.cpp.xml" target="_top"><span style='white-space: nowrap'>exp_2_for0.cpp</span></a> that verifies the values you obtained for the previous exercise. </li><li> Create and run a main program that reports the result of calling the modified version of <a href="exp_2_for0.cpp.xml" target="_top"><span style='white-space: nowrap'>exp_2_for0.cpp</span></a> in the previous exercise. </li></ol> <hr/>Input File: introduction/exp_apx/exp_2.omh </body> </html> |