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get_started.cpp.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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 | <?xml version='1.0'?> <html xmlns='http://www.w3.org/1999/xhtml' xmlns:math='http://www.w3.org/1998/Math/MathML' > <head> <title>Getting Started Using CppAD to Compute Derivatives</title> <meta http-equiv='Content-Type' content='text/html' charset='utf-8'/> <meta name="description" id="description" content="Getting Started Using CppAD to Compute Derivatives"/> <meta name="keywords" id="keywords" content=" getting started using cppad to compute derivatives simple example start Cppad purpose function derivative value poly exercises program output running "/> <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='_get_started.cpp_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="introduction.xml" target="_top">Prev</a> </td><td><a href="exp_2.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>get_started.cpp</option> </select> </td> <td> <select onchange='choose_down2(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_down1(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>get_started.cpp</td> <td> <select onchange='choose_current0(this)'> <option>Headings-></option> <option>Purpose</option> <option>Function</option> <option>Derivative</option> <option>Value</option> <option>Poly</option> <option>Exercises</option> <option>Program</option> <option>Output</option> <option>Running</option> </select> </td> </tr></table><br/> <center><b><big><big>Getting Started Using CppAD to Compute Derivatives</big></big></b></center> <br/> <b><big><a name="Purpose" id="Purpose">Purpose</a></big></b> <br/> Demonstrate the use of CppAD by computing the derivative of a simple example function. <br/> <br/> <b><big><a name="Function" id="Function">Function</a></big></b> <br/> The example function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <mi mathvariant='italic'>f</mi> <mo stretchy="false">:</mo> <mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi> </mstyle></mrow> <mo stretchy="false">→</mo> <mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi> </mstyle></mrow> </mrow></math> is defined by <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> <msub><mi mathvariant='italic'>a</mi> <mn>0</mn> </msub> <mo stretchy="false">+</mo> <msub><mi mathvariant='italic'>a</mi> <mn>1</mn> </msub> <mo stretchy="false">*</mo> <msup><mi mathvariant='italic'>x</mi> <mn>1</mn> </msup> <mo stretchy="false">+</mo> <mo stretchy="false">⋯</mo> <mo stretchy="false">+</mo> <msub><mi mathvariant='italic'>a</mi> <mrow><mi mathvariant='italic'>k</mi> <mn>-1</mn> </mrow> </msub> <mo stretchy="false">*</mo> <msup><mi mathvariant='italic'>x</mi> <mrow><mi mathvariant='italic'>k</mi> <mn>-1</mn> </mrow> </msup> </mrow></math> where <code><i><font color="black"><span style='white-space: nowrap'>a</span></font></i></code> is a fixed vector of length <code><i><font color="black"><span style='white-space: nowrap'>k</span></font></i></code> . <br/> <br/> <b><big><a name="Derivative" id="Derivative">Derivative</a></big></b> <br/> The derivative of <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> is given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow> <mi mathvariant='italic'>f</mi> <mo stretchy="false">'</mo> <mo stretchy="false">(</mo> <mi mathvariant='italic'>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">=</mo> <msub><mi mathvariant='italic'>a</mi> <mn>1</mn> </msub> <mo stretchy="false">+</mo> <mn>2</mn> <mo stretchy="false">*</mo> <msub><mi mathvariant='italic'>a</mi> <mn>2</mn> </msub> <mo stretchy="false">*</mo> <mi mathvariant='italic'>x</mi> <mo stretchy="false">+</mo> <mo stretchy="false">⋯</mo> <mo stretchy="false">+</mo> <mo stretchy="false">(</mo> <mi mathvariant='italic'>k</mi> <mn>-1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">*</mo> <msub><mi mathvariant='italic'>a</mi> <mrow><mi mathvariant='italic'>k</mi> <mn>-1</mn> </mrow> </msub> <mo stretchy="false">*</mo> <msup><mi mathvariant='italic'>x</mi> <mrow><mi mathvariant='italic'>k</mi> <mn>-2</mn> </mrow> </msup> </mrow></math> <br/> <b><big><a name="Value" id="Value">Value</a></big></b> <br/> For the particular case in this example, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <mi mathvariant='italic'>k</mi> </mrow></math> is equal to 5, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <mi mathvariant='italic'>a</mi> <mo stretchy="false">=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">,</mo> <mn>1</mn> <mo stretchy="false">,</mo> <mn>1</mn> <mo stretchy="false">,</mo> <mn>1</mn> <mo stretchy="false">,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow></math> , and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <mi mathvariant='italic'>x</mi> <mo stretchy="false">=</mo> <mn>3</mn> </mrow></math> . If follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow> <mi mathvariant='italic'>f</mi> <mo stretchy="false">'</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">=</mo> <mn>1</mn> <mo stretchy="false">+</mo> <mn>2</mn> <mo stretchy="false">*</mo> <mn>3</mn> <mo stretchy="false">+</mo> <mn>3</mn> <mo stretchy="false">*</mo> <msup><mn>3</mn> <mn>2</mn> </msup> <mo stretchy="false">+</mo> <mn>4</mn> <mo stretchy="false">*</mo> <msup><mn>3</mn> <mn>3</mn> </msup> <mo stretchy="false">=</mo> <mn>142</mn> </mrow></math> <br/> <b><big><a name="Poly" id="Poly">Poly</a></big></b> <br/> The routine <code><font color="blue">Poly</font></code> is defined below for this particular application. A general purpose polynomial evaluation routine is documented and distributed with CppAD (see <a href="poly.xml" target="_top"><span style='white-space: nowrap'>Poly</span></a> ). <br/> <br/> <b><big><a name="Exercises" id="Exercises">Exercises</a></big></b> <br/> Modify the program below to accomplish the following tasks using CppAD: <ol type="1"><li> Compute and print the derivative of <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> <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> <msup><mi mathvariant='italic'>x</mi> <mn>3</mn> </msup> <mo stretchy="false">+</mo> <msup><mi mathvariant='italic'>x</mi> <mn>4</mn> </msup> </mrow></math> at the point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <mi mathvariant='italic'>x</mi> <mo stretchy="false">=</mo> <mn>2</mn> </mrow></math> . </li><li> Compute and print the derivative of <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> <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> at the point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <mi mathvariant='italic'>x</mi> <mo stretchy="false">=</mo> <mn>.5</mn> </mrow></math> . </li><li> Compute and print the derivative of <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> <mo stretchy="false">=</mo> <mi>exp</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> at the point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <mi mathvariant='italic'>x</mi> <mo stretchy="false">=</mo> <mn>.5</mn> </mrow></math> . </li></ol> <br/> <br/> <b><big><a name="Program" id="Program">Program</a></big></b> <code><font color='blue'><pre style='display:inline'> #include <iostream> // standard input/output #include <vector> // standard vector #include <cppad/cppad.hpp> // the CppAD package http://www.coin-or.org/CppAD/ namespace { // define y(x) = Poly(a, x) in the empty namespace template <class Type> Type Poly(const std::vector<double> &a, const Type &x) { size_t k = a.size(); Type y = 0.; // initialize summation Type x_i = 1.; // initialize x^i size_t i; for(i = 0; i < k; i++) { y += a[i] * x_i; // y = y + a_i * x^i x_i *= x; // x_i = x_i * x } return y; } } // main program int main(void) { using CppAD::AD; // use AD as abbreviation for CppAD::AD using std::vector; // use vector as abbreviation for std::vector size_t i; // a temporary index // vector of polynomial coefficients size_t k = 5; // number of polynomial coefficients vector<double> a(k); // vector of polynomial coefficients for(i = 0; i < k; i++) a[i] = 1.; // value of polynomial coefficients // domain space vector size_t n = 1; // number of domain space variables vector< <a href="ad.xml" target="_top">AD</a><double> > X(n); // vector of domain space variables X[0] = 3.; // value corresponding to operation sequence // declare independent variables and start recording operation sequence CppAD::<a href="independent.xml" target="_top">Independent</a>(X); // range space vector size_t m = 1; // number of ranges space variables vector< <a href="ad.xml" target="_top">AD</a><double> > Y(m); // vector of ranges space variables Y[0] = Poly(a, X[0]); // value during recording of operations // store operation sequence in f: X -> Y and stop recording CppAD::<a href="funconstruct.xml" target="_top">ADFun</a><double> f(X, Y); // compute derivative using operation sequence stored in f vector<double> jac(m * n); // Jacobian of f (m by n matrix) vector<double> x(n); // domain space vector x[0] = 3.; // argument value for derivative jac = f.<a href="jacobian.xml" target="_top">Jacobian</a>(x); // Jacobian for operation sequence // print the results std::cout << "f'(3) computed by CppAD = " << jac[0] << std::endl; // check if the derivative is correct int error_code; if( jac[0] == 142. ) error_code = 0; // return code for correct case else error_code = 1; // return code for incorrect case return error_code; } </pre></font></code> <br/> <br/> <b><big><a name="Output" id="Output">Output</a></big></b> <br/> Executing the program above will generate the following output: <code><font color='blue'><pre style='display:inline'> f'(3) computed by CppAD = 142 </pre></font></code> <br/> <br/> <b><big><a name="Running" id="Running">Running</a></big></b> <br/> To build and run this program see <a href="cmake_check.xml" target="_top"><span style='white-space: nowrap'>cmake_check</span></a> . <hr/>Input File: introduction/get_started/get_started.cpp </body> </html> |