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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 $$f' ( 3 ) = 1 + 2 * 3 + 3 * 3^2 + 4 * 3^3 = 142$$

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
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

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 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

// 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

// 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