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<option>Headings-&gt;</option>
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<option>Derivative</option>
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<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">&#x02192;</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">&#x022EF;</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">&#x022EF;</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 &lt;iostream&gt;      // standard input/output 
#include &lt;vector&gt;        // standard vector
#include &lt;cppad/cppad.hpp&gt; // the CppAD package http://www.coin-or.org/CppAD/

namespace { 
      // define y(x) = Poly(a, x) in the empty namespace
      template &lt;class Type&gt;
      Type Poly(const std::vector&lt;double&gt; &amp;a, const Type &amp;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 &lt; 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&lt;double&gt; a(k);       // vector of polynomial coefficients
      for(i = 0; i &lt; k; i++)
            a[i] = 1.;           // value of polynomial coefficients

      // domain space vector
      size_t n = 1;              // number of domain space variables
      vector&lt; <a href="ad.xml" target="_top">AD</a>&lt;double&gt; &gt; 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&lt; <a href="ad.xml" target="_top">AD</a>&lt;double&gt; &gt; Y(m); // vector of ranges space variables
      Y[0] = Poly(a, X[0]);      // value during recording of operations

      // store operation sequence in f: X -&gt; Y and stop recording
      CppAD::<a href="funconstruct.xml" target="_top">ADFun</a>&lt;double&gt; f(X, Y);

      // compute derivative using operation sequence stored in f
      vector&lt;double&gt; jac(m * n); // Jacobian of f (m by n matrix)
      vector&lt;double&gt; 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 &lt;&lt; &quot;f'(3) computed by CppAD = &quot; &lt;&lt; jac[0] &lt;&lt; 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

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