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@(@\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }@)@
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 @]@

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