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A Simple Program 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 : R → R is defined by f ( x ) = a 0 + a 1 * x 1 + ⋯ + 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 + ⋯ + ( 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
Poly
The routine Poly is defined below for this particular application. A general purpose polynomial evaluation routine is documented and distributed with CppAD (see Poly ).

Exercises
Modify the program below 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 .


Program 
 
#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< AD<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::Independent(X);

      // range space vector
      size_t m = 1;              // number of ranges space variables
      vector< AD<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::ADFun<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.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
Input File: introduction/get_started/get_started.cpp