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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 : 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

`
Running
To build and run this program see cmake_check .

Input File: example/get_started/get_started.cpp