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base_std_math.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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content=" base type requirements for standard math functions require purpose unary cppad_standard_math_unary Cppad_standard_math_unary erf sign pow isnan limits numeric Base numeric_limits epsilon "/> <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='_base_std_math_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="base_ordered.xml" target="_top">Prev</a> </td><td><a href="base_example.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>AD</option> <option>base_require</option> <option>base_std_math</option> </select> </td> <td> <select onchange='choose_down3(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_down2(this)'> <option>AD-></option> <option>ad_ctor</option> <option>ad_assign</option> <option>Convert</option> <option>ADValued</option> <option>BoolValued</option> <option>VecAD</option> <option>base_require</option> </select> </td> <td> <select onchange='choose_down1(this)'> <option>base_require-></option> <option>base_member</option> <option>base_cond_exp</option> <option>base_identical</option> <option>base_ordered</option> <option>base_std_math</option> <option>base_example</option> </select> </td> <td>base_std_math</td> <td> <select onchange='choose_current0(this)'> <option>Headings-></option> <option>Purpose</option> <option>Unary Standard Math</option> <option>CPPAD_STANDARD_MATH_UNARY</option> <option>erf</option> <option>sign</option> <option>pow</option> <option>isnan</option> <option>limits</option> <option>---..epsilon</option> </select> </td> </tr></table><br/> <center><b><big><big>Base Type Requirements for Standard Math Functions</big></big></b></center> <br/> <b><big><a name="Purpose" id="Purpose">Purpose</a></big></b> <br/> These definitions are required for the user's code to use the type <code><font color="blue"><span style='white-space: nowrap'>AD<</span></font><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i><font color="blue"><span style='white-space: nowrap'>></span></font></code> : <br/> <br/> <b><big><a name="Unary Standard Math" id="Unary Standard Math">Unary Standard Math</a></big></b> <br/> The type <code><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i></code> must support the following functions unary standard math functions (in the CppAD namespace): <table><tr><td align='left' valign='top'> <b>Syntax</b> </td><td align='left' valign='top'> <b>Result</b> </td></tr><tr><td align='left' valign='top'> <code><i><font color="black"><span style='white-space: nowrap'>y</span></font></i><font color="blue"><span style='white-space: nowrap'> = abs(</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>)</span></font></code> </td><td align='left' valign='top'> absolute value </td></tr><tr><td align='left' valign='top'> <code><i><font color="black"><span style='white-space: nowrap'>y</span></font></i><font color="blue"><span style='white-space: nowrap'> = acos(</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>)</span></font></code> </td><td align='left' valign='top'> inverse cosine </td></tr><tr><td align='left' valign='top'> <code><i><font color="black"><span style='white-space: nowrap'>y</span></font></i><font color="blue"><span style='white-space: nowrap'> = asin(</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>)</span></font></code> </td><td align='left' valign='top'> inverse sine </td></tr><tr><td align='left' valign='top'> <code><i><font color="black"><span style='white-space: nowrap'>y</span></font></i><font color="blue"><span style='white-space: nowrap'> = atan(</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>)</span></font></code> </td><td align='left' valign='top'> inverse tangent </td></tr><tr><td align='left' valign='top'> <code><i><font color="black"><span style='white-space: nowrap'>y</span></font></i><font color="blue"><span style='white-space: nowrap'> = cos(</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>)</span></font></code> </td><td align='left' valign='top'> cosine </td></tr><tr><td align='left' valign='top'> <code><i><font color="black"><span style='white-space: nowrap'>y</span></font></i><font color="blue"><span style='white-space: nowrap'> = cosh(</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>)</span></font></code> </td><td align='left' valign='top'> hyperbolic cosine </td></tr><tr><td align='left' valign='top'> <code><i><font color="black"><span style='white-space: nowrap'>y</span></font></i><font color="blue"><span style='white-space: nowrap'> = exp(</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>)</span></font></code> </td><td align='left' valign='top'> exponential </td></tr><tr><td align='left' valign='top'> <code><i><font color="black"><span style='white-space: nowrap'>y</span></font></i><font color="blue"><span style='white-space: nowrap'> = fabs(</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>)</span></font></code> </td><td align='left' valign='top'> absolute value </td></tr><tr><td align='left' valign='top'> <code><i><font color="black"><span style='white-space: nowrap'>y</span></font></i><font color="blue"><span style='white-space: nowrap'> = log(</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>)</span></font></code> </td><td align='left' valign='top'> natural logarithm </td></tr><tr><td align='left' valign='top'> <code><i><font color="black"><span style='white-space: nowrap'>y</span></font></i><font color="blue"><span style='white-space: nowrap'> = sin(</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>)</span></font></code> </td><td align='left' valign='top'> sine </td></tr><tr><td align='left' valign='top'> <code><i><font color="black"><span style='white-space: nowrap'>y</span></font></i><font color="blue"><span style='white-space: nowrap'> = sinh(</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>)</span></font></code> </td><td align='left' valign='top'> hyperbolic sine </td></tr><tr><td align='left' valign='top'> <code><i><font color="black"><span style='white-space: nowrap'>y</span></font></i><font color="blue"><span style='white-space: nowrap'> = sqrt(</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>)</span></font></code> </td><td align='left' valign='top'> square root </td></tr><tr><td align='left' valign='top'> <code><i><font color="black"><span style='white-space: nowrap'>y</span></font></i><font color="blue"><span style='white-space: nowrap'> = tan(</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>)</span></font></code> </td><td align='left' valign='top'> tangent </td></tr> </table> where the arguments and return value have the prototypes <code><font color="blue"><span style='white-space: nowrap'><br/>      const </span></font><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i><font color="blue"><span style='white-space: nowrap'>& </span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>      </span></font><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i><font color="blue"><span style='white-space: nowrap'>        </span></font><i><font color="black"><span style='white-space: nowrap'>y</span></font></i><font color="blue"><span style='white-space: nowrap'><br/> </span></font></code> For example, <a href="base_alloc.hpp.xml#Unary Standard Math" target="_top"><span style='white-space: nowrap'>base_alloc</span></a> , <br/> <br/> <b><big><a name="CPPAD_STANDARD_MATH_UNARY" id="CPPAD_STANDARD_MATH_UNARY">CPPAD_STANDARD_MATH_UNARY</a></big></b> <br/> The macro invocation, within the CppAD namespace, <code><font color="blue"><span style='white-space: nowrap'><br/>      CPPAD_STANDARD_MATH_UNARY(</span></font><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>Fun</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/> </span></font></code> defines the syntax <code><font color="blue"><span style='white-space: nowrap'><br/>      </span></font><i><font color="black"><span style='white-space: nowrap'>y</span></font></i><font color="blue"><span style='white-space: nowrap'> = CppAD::</span></font><i><font color="black"><span style='white-space: nowrap'>Fun</span></font></i><font color="blue"><span style='white-space: nowrap'>(</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/> </span></font></code> This macro uses the functions <code><font color="blue"><span style='white-space: nowrap'>std::</span></font><i><font color="black"><span style='white-space: nowrap'>Fun</span></font></i></code> which must be defined and have the same prototype as <code><font color="blue"><span style='white-space: nowrap'>CppAD::</span></font><i><font color="black"><span style='white-space: nowrap'>Fun</span></font></i></code> . For example, <a href="base_float.hpp.xml#Unary Standard Math" target="_top"><span style='white-space: nowrap'>float</span></a> . <br/> <br/> <b><big><a name="erf" id="erf">erf</a></big></b> <br/> If the error function is supported by the compiler, <a href="erf.xml#Method.CPPAD_COMPILER_HAS_ERF" target="_top"><span style='white-space: nowrap'>CPPAD_COMPILER_HAS_ERF</span></a> is one, the type <code><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i></code> must support the syntax <code><font color="blue"><span style='white-space: nowrap'><br/>      </span></font><i><font color="black"><span style='white-space: nowrap'>y</span></font></i><font color="blue"><span style='white-space: nowrap'> = CppAD::erf(</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/> </span></font></code> where <code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code> and <code><i><font color="black"><span style='white-space: nowrap'>y</span></font></i></code> have the same prototype as above. For example, see <a href="base_alloc.hpp.xml#erf" target="_top"><span style='white-space: nowrap'>base_alloc</span></a> . <br/> <br/> <b><big><a name="sign" id="sign">sign</a></big></b> <br/> The type <code><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i></code> must support the syntax <code><font color="blue"><span style='white-space: nowrap'><br/>      </span></font><i><font color="black"><span style='white-space: nowrap'>y</span></font></i><font color="blue"><span style='white-space: nowrap'> = CppAD::sign(</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/> </span></font></code> which computes <math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow> <mi mathvariant='italic'>y</mi> <mo stretchy="false">=</mo> <mrow><mo stretchy="true">{</mo><mrow><mtable rowalign="center" ><mtr><mtd columnalign="left" > <mo stretchy="false">+</mo> <mn>1</mn> </mtd><mtd columnalign="left" > <mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>if</mi> </mstyle></mrow> <mspace width='.3em'/> <mi mathvariant='italic'>x</mi> <mo stretchy="false">></mo> <mn>0</mn> </mtd></mtr><mtr><mtd columnalign="left" > <mn>0</mn> </mtd><mtd columnalign="left" > <mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>if</mi> </mstyle></mrow> <mspace width='.3em'/> <mi mathvariant='italic'>x</mi> <mo stretchy="false">=</mo> <mn>0</mn> </mtd></mtr><mtr><mtd columnalign="left" > <mn>-1</mn> </mtd><mtd columnalign="left" > <mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>if</mi> </mstyle></mrow> <mspace width='.3em'/> <mi mathvariant='italic'>x</mi> <mo stretchy="false"><</mo> <mn>0</mn> </mtd></mtr></mtable> </mrow><mo stretchy="true"> </mo></mrow> </mrow></math> where <code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code> and <code><i><font color="black"><span style='white-space: nowrap'>y</span></font></i></code> have the same prototype as above. For example, see <a href="base_alloc.hpp.xml#sign" target="_top"><span style='white-space: nowrap'>base_alloc</span></a> . Note that, if ordered comparisons are not defined for the type <code><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i></code> , the <code><font color="blue">code sign</font></code> function should generate an assert if it is used; see <a href="base_complex.hpp.xml#Invalid Unary Math" target="_top"><span style='white-space: nowrap'>complex invalid unary math</span></a> . <br/> <br/> <b><big><a name="pow" id="pow">pow</a></big></b> <br/> The type <code><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i></code> must support the syntax <code><font color="blue"><span style='white-space: nowrap'><br/>      </span></font><i><font color="black"><span style='white-space: nowrap'>z</span></font></i><font color="blue"><span style='white-space: nowrap'> = CppAD::pow(</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>, </span></font><i><font color="black"><span style='white-space: nowrap'>y</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/> </span></font></code> which computes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <mi mathvariant='italic'>z</mi> <mo stretchy="false">=</mo> <msup><mi mathvariant='italic'>x</mi> <mi mathvariant='italic'>y</mi> </msup> </mrow></math> . The arguments <code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code> and <code><i><font color="black"><span style='white-space: nowrap'>y</span></font></i></code> have prototypes <code><font color="blue"><span style='white-space: nowrap'><br/>      const </span></font><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i><font color="blue"><span style='white-space: nowrap'>& </span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>      const </span></font><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i><font color="blue"><span style='white-space: nowrap'>& </span></font><i><font color="black"><span style='white-space: nowrap'>y</span></font></i><font color="blue"><span style='white-space: nowrap'><br/> </span></font></code> and the return value <code><i><font color="black"><span style='white-space: nowrap'>z</span></font></i></code> has prototype <code><font color="blue"><span style='white-space: nowrap'><br/>      </span></font><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i><font color="blue"><span style='white-space: nowrap'> </span></font><i><font color="black"><span style='white-space: nowrap'>z</span></font></i><font color="blue"><span style='white-space: nowrap'><br/> </span></font></code> For example, see <a href="base_alloc.hpp.xml#pow" target="_top"><span style='white-space: nowrap'>base_alloc</span></a> . <br/> <br/> <b><big><a name="isnan" id="isnan">isnan</a></big></b> <br/> If <code><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i></code> defines the <code><font color="blue">isnan</font></code> function, you may also have to provide a definition in the CppAD namespace (to avoid a function ambiguity). For example, see <a href="base_complex.hpp.xml#isnan" target="_top"><span style='white-space: nowrap'>base_complex</span></a> . <br/> <br/> <b><big><a name="limits" id="limits">limits</a></big></b> <br/> The <a href="limits.xml" target="_top"><span style='white-space: nowrap'>numeric_limits</span></a> functions <code><font color="blue"><span style='white-space: nowrap'><br/>      </span></font><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i><font color="blue"><span style='white-space: nowrap'> CppAD::numeric_limits<</span></font><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i><font color="blue"><span style='white-space: nowrap'>>::epsilon()<br/>      </span></font><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i><font color="blue"><span style='white-space: nowrap'> CppAD::numeric_limits<</span></font><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i><font color="blue"><span style='white-space: nowrap'>>::min()<br/>      </span></font><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i><font color="blue"><span style='white-space: nowrap'> CppAD::numeric_limits<</span></font><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i><font color="blue"><span style='white-space: nowrap'>>::max()<br/> </span></font></code> must return machine epsilon, minimum positive normalize value, and maximum finite value for the type <code><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i></code> . <br/> <br/> <b><a name="limits.epsilon" id="limits.epsilon">epsilon</a></b> <br/> The deprecated function <a href="epsilon.xml" target="_top"><span style='white-space: nowrap'>epsilon</span></a> function must also be included. It can be implemented using <code><font color="blue"><span style='white-space: nowrap'><br/> namespace CppAD {<br/>      template <> inline </span></font><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i><font color="blue"><span style='white-space: nowrap'> epsilon<</span></font><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i><font color="blue"><span style='white-space: nowrap'>>(void)<br/>      {    return numeric_limits<</span></font><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i><font color="blue"><span style='white-space: nowrap'>>::epsilon(); }<br/> }<br/> </span></font></code> <hr/>Input File: cppad/local/base_std_math.hpp </body> </html> |