Codebase list cppad / upstream/2015.00.00.7 doc / exp_eps_cppad.xml
upstream/2015.00.00.7

Tree @upstream/2015.00.00.7 (Download .tar.gz)

exp_eps_cppad.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
<?xml version='1.0'?>
<html xmlns='http://www.w3.org/1999/xhtml'
      xmlns:math='http://www.w3.org/1998/Math/MathML'
>
<head>
<title>exp_eps: CppAD Forward and Reverse Sweeps</title>
<meta http-equiv='Content-Type' content='text/html' charset='utf-8'/>
<meta name="description" id="description" content="exp_eps: CppAD Forward and Reverse Sweeps"/>
<meta name="keywords" id="keywords" content=" exp_eps: cppad forward and reverse sweeps purpose 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_eps_cppad_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_eps_rev2.cpp.xml" target="_top">Prev</a>
</td><td><a href="exp_apx_main.cpp.xml" target="_top">Next</a>
</td><td>
<select onchange='choose_across0(this)'>
<option>Index-&gt;</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-&gt;</option>
<option>CppAD</option>
<option>Introduction</option>
<option>exp_eps</option>
<option>exp_eps_cppad</option>
</select>
</td>
<td>
<select onchange='choose_down3(this)'>
<option>CppAD-&gt;</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>Introduction-&gt;</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_down1(this)'>
<option>exp_eps-&gt;</option>
<option>exp_eps.hpp</option>
<option>exp_eps.cpp</option>
<option>exp_eps_for0</option>
<option>exp_eps_for1</option>
<option>exp_eps_rev1</option>
<option>exp_eps_for2</option>
<option>exp_eps_rev2</option>
<option>exp_eps_cppad</option>
</select>
</td>
<td>exp_eps_cppad</td>
<td>
<select onchange='choose_current0(this)'>
<option>Headings-&gt;</option>
<option>Purpose</option>
<option>Exercises</option>
</select>
</td>
</tr></table><br/>



<center><b><big><big>exp_eps: CppAD Forward and Reverse Sweeps</big></big></b></center>
.

<br/>
<br/>
<b><big><a name="Purpose" id="Purpose">Purpose</a></big></b>
<br/>
Use CppAD forward and reverse modes to compute the
partial derivative with respect to 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>x</mi>
</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>

 and 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='normal'>&#x003B5;</mi>
<mo stretchy="false">=</mo>
<mn>.2</mn>
</mrow></math>

,
of the function 

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;exp_eps(</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>epsilon</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>
as defined by the <a href="exp_eps.hpp.xml" target="_top"><span style='white-space: nowrap'>exp_eps.hpp</span></a>
 include file.

<br/>
<br/>
<b><big><a name="Exercises" id="Exercises">Exercises</a></big></b>

<ol type="1"><li>
Create and test a modified version of the routine below that computes
the same order derivatives with respect to 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>x</mi>
</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>.1</mn>
</mrow></math>

 and 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='normal'>&#x003B5;</mi>
<mo stretchy="false">=</mo>
<mn>.2</mn>
</mrow></math>

,
of the function 

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;exp_eps(</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>epsilon</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>
</li><li>
Create and test a modified version of the routine below that computes
partial derivative with respect to 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>x</mi>
</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>.1</mn>
</mrow></math>

 and 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='normal'>&#x003B5;</mi>
<mo stretchy="false">=</mo>
<mn>.2</mn>
</mrow></math>

,
of the function corresponding to the operation sequence 
for 
<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>

 and 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='normal'>&#x003B5;</mi>
<mo stretchy="false">=</mo>
<mn>.2</mn>
</mrow></math>

.
Hint: you could define a vector u with two components and use

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>f</span></font></i><font color="blue"><span style='white-space: nowrap'>.Forward(0,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>u</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>
to run zero order forward mode at a point different
form the point where the operation sequence corresponding to

<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
 was recorded.
</li></ol>

<code><font color='blue'><pre style='display:inline'> 
# include &lt;cppad/cppad.hpp&gt;  // http://www.coin-or.org/CppAD/ 
# include &quot;exp_eps.hpp&quot;      // our example exponential function approximation
bool exp_eps_cppad(void)
{	bool ok = true;
	using CppAD::AD;
	using CppAD::vector;    // can use any simple vector template class
	using CppAD::NearEqual; // checks if values are nearly equal

	// domain space vector
	size_t n = 2; // dimension of the domain space
	vector&lt; <a href="ad.xml" target="_top">AD</a>&lt;double&gt; &gt; U(n);
	U[0] = .5;    // value of x for this operation sequence
	U[1] = .2;    // value of e for this operation sequence

	// declare independent variables and start recording operation sequence
	CppAD::<a href="independent.xml" target="_top">Independent</a>(U);

	// evaluate our exponential approximation
	<a href="ad.xml" target="_top">AD</a>&lt;double&gt; x       = U[0];
	<a href="ad.xml" target="_top">AD</a>&lt;double&gt; epsilon = U[1];
	<a href="ad.xml" target="_top">AD</a>&lt;double&gt; apx = exp_eps(x, epsilon);  

	// range space vector
	size_t m = 1;  // dimension of the range space
	vector&lt; <a href="ad.xml" target="_top">AD</a>&lt;double&gt; &gt; Y(m);
	Y[0] = apx;    // variable that represents only range space component

	// Create f: U -&gt; Y corresponding to this operation sequence
	// and stop recording. This also executes a zero order forward 
	// mode sweep using values in U for x and e.
	CppAD::<a href="funconstruct.xml" target="_top">ADFun</a>&lt;double&gt; f(U, Y);

	// first order forward mode sweep that computes partial w.r.t x
	vector&lt;double&gt; du(n);      // differential in domain space
	vector&lt;double&gt; dy(m);      // differential in range space
	du[0] = 1.;                // x direction in domain space
	du[1] = 0.;
	dy    = f.<a href="forward.xml" target="_top">Forward</a>(1, du);  // partial w.r.t. x
	double check = 1.5;
	ok   &amp;= <a href="nearequal.xml" target="_top">NearEqual</a>(dy[0], check, 1e-10, 1e-10);

	// first order reverse mode sweep that computes the derivative
	vector&lt;double&gt;  w(m);     // weights for components of the range
	vector&lt;double&gt; dw(n);     // derivative of the weighted function
	w[0] = 1.;                // there is only one weight 
	dw   = f.<a href="reverse.xml" target="_top">Reverse</a>(1, w);   // derivative of w[0] * exp_eps(x, epsilon)
	check = 1.5;              // partial w.r.t. x
	ok   &amp;= <a href="nearequal.xml" target="_top">NearEqual</a>(dw[0], check, 1e-10, 1e-10);
	check = 0.;               // partial w.r.t. epsilon
	ok   &amp;= <a href="nearequal.xml" target="_top">NearEqual</a>(dw[1], check, 1e-10, 1e-10);

	// second order forward sweep that computes
	// second partial of exp_eps(x, epsilon) w.r.t. x
	vector&lt;double&gt; x2(n);     // second order Taylor coefficients
	vector&lt;double&gt; y2(m);
	x2[0] = 0.;               // evaluate partial w.r.t x
	x2[1] = 0.;
	y2    = f.<a href="forward.xml" target="_top">Forward</a>(2, x2);
	check = 0.5 * 1.;         // Taylor coef is 1/2 second derivative
	ok   &amp;= <a href="nearequal.xml" target="_top">NearEqual</a>(y2[0], check, 1e-10, 1e-10);

	// second order reverse sweep that computes
	// derivative of partial of exp_eps(x, epsilon) w.r.t. x
	dw.resize(2 * n);         // space for first and second derivative
	dw    = f.<a href="reverse.xml" target="_top">Reverse</a>(2, w);
	check = 1.;               // result should be second derivative
	ok   &amp;= <a href="nearequal.xml" target="_top">NearEqual</a>(dw[0*2+1], check, 1e-10, 1e-10);

	return ok;
}
</pre></font></code>


<hr/>Input File: introduction/exp_apx/exp_eps_cppad.cpp

</body>
</html>