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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}} }@)@This is cppad-20221105 documentation. Here is a link to its current documentation .
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 @)@.
Input File: example/get_started/get_started.cpp