get_started.cpp

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Getting Started Using CppAD to Compute Derivatives

Purpose

Demonstrate the use of CppAD by computing the derivative of a simple example function.

Function

The example function \(f : \B{R} \rightarrow \B{R}\) is defined by

\[f(x) = a_0 + a_1 * x^1 + \cdots + a_{k-1} * x^{k-1}\]

where a is a fixed vector of length k .

Derivative

The derivative of \(f(x)\) is given by

\[f' (x) = a_1 + 2 * a_2 * x + \cdots + (k-1) * a_{k-1} * x^{k-2}\]

Value

For the particular case in this example, \(k\) is equal to 5, \(a = (1, 1, 1, 1, 1)\), and \(x = 3\). If follows that

\[f' ( 3 ) = 1 + 2 * 3 + 3 * 3^2 + 4 * 3^3 = 142\]

Include File

The following command, in the program below, includes the CppAD package:

# include <cppad/cppad.hpp>

Poly

The routine Poly , defined below, evaluates a polynomial. A general purpose polynomial evaluation routine is documented and distributed with CppAD; see Poly .

CppAD Namespace

All of the functions and objects defined by CppAD are in the CppAD namespace. In the example below,

using CppAD::AD ;

enables one to abbreviate CppAD::AD using just AD .

CppAD Preprocessor Symbols

All the preprocessor symbols defined by CppAD begin with CPPAD_ (some deprecated symbols begin with CppAD_ ). The preprocessor symbol CPPAD_TESTVECTOR is used in the example below.

Program

# include <iostream>        // standard input/output
# include <vector>          // standard vector
# include <cppad/cppad.hpp> // the CppAD package

namespace { // begin the empty namespace
   // define the function Poly(a, x) = a[0] + a[1]*x[1] + ... + a[k-1]*x[k-1]
   template <class Type>
   Type Poly(const CPPAD_TESTVECTOR(double) &a, const Type &x)
   {  size_t k  = a.size();
      Type y   = 0.;  // initialize summation
      Type x_i = 1.;  // initialize x^i
      for(size_t 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

   // vector of polynomial coefficients
   size_t k = 5;                  // number of polynomial coefficients
   CPPAD_TESTVECTOR(double) a(k); // vector of polynomial coefficients
   for(size_t 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< AD<double> > ax(n); // vector of domain space variables
   ax[0] = 3.;                 // value at which function is recorded

   // declare independent variables and start recording operation sequence
   CppAD::Independent(ax);

   // range space vector
   size_t m = 1;               // number of ranges space variables
   vector< AD<double> > ay(m); // vector of ranges space variables
   ay[0] = Poly(a, ax[0]);     // record operations that compute ay[0]

   // store operation sequence in f: X -> Y and stop recording
   CppAD::ADFun<double> f(ax, ay);

   // 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 computing derivative
   jac  = f.Jacobian(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;
}

Output

Executing the program above will generate the following output:

f'(3) computed by CppAD = 142

Running

After you configure your system using the cmake command, you compile and run this example by executing the command

make check_example_get_started

in the build directory; i.e., the directory where the cmake command was executed.

Exercises

Modify the program above to accomplish the following tasks using CppAD:

  1. Compute and print the derivative of \(f(x) = 1 + x + x^2 + x^3 + x^4\) at the point \(x = 2\).

  2. Compute and print the derivative of \(f(x) = 1 + x + x^2 / 2\) at the point \(x = .5\).

  3. Compute and print the derivative of \(f(x) = \exp (x) - 1 - x - x^2 / 2\) at the point \(x = .5\).