Prev Next Index-> contents reference index search external Up-> CppAD AD ADValued atomic atomic_base atomic_forward atomic_forward.cpp atomic-> checkpoint atomic_base atomic_base-> atomic_ctor atomic_option atomic_afun atomic_forward atomic_reverse atomic_for_sparse_jac atomic_rev_sparse_jac atomic_for_sparse_hes atomic_rev_sparse_hes atomic_base_clear atomic_get_started.cpp atomic_norm_sq.cpp atomic_reciprocal.cpp atomic_set_sparsity.cpp atomic_tangent.cpp atomic_eigen_mat_mul.cpp atomic_eigen_mat_inv.cpp atomic_eigen_cholesky.cpp atomic_mat_mul.cpp atomic_forward-> atomic_forward.cpp atomic_forward.cpp Headings-> Purpose function Start Class Definition Constructor forward Use Atomic Function

$\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }$
Atomic Forward: Example and Test

Purpose
This example demonstrates forward mode derivative calculation using an atomic operation.

function
For this example, the atomic function $f : \B{R}^3 \rightarrow \B{R}^2$ is defined by $$f(x) = \left( \begin{array}{c} x_2 * x_2 \\ x_0 * x_1 \end{array} \right)$$ The corresponding Jacobian is $$f^{(1)} (x) = \left( \begin{array}{ccc} 0 & 0 & 2 x_2 \\ x_1 & x_0 & 0 \end{array} \right)$$ The Hessians of the component functions are $$f_0^{(2)} ( x ) = \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 2 \end{array} \right) \W{,} f_1^{(2)} ( x ) = \left( \begin{array}{ccc} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right)$$

Start Class Definition
# include <cppad/cppad.hpp>
namespace {          // isolate items below to this file
using CppAD::vector; // abbreviate as vector
//
class atomic_forward : public CppAD::atomic_base<double> {

Constructor
public:
// constructor (could use const char* for name)
atomic_forward(const std::string& name) :
// this example does not use sparsity patterns
{ }
private:

forward
     // forward mode routine called by CppAD
virtual bool forward(
size_t                    p ,
size_t                    q ,
const vector<bool>&      vx ,
vector<bool>&            vy ,
const vector<double>&    tx ,
vector<double>&          ty
)
{
size_t q1 = q + 1;
# ifndef NDEBUG
size_t n = tx.size() / q1;
size_t m = ty.size() / q1;
# endif
assert( n == 3 );
assert( m == 2 );
assert( p <= q );

// this example only implements up to second order forward mode
bool ok = q <= 2;
if( ! ok )
return ok;

// check for defining variable information
// This case must always be implemented
if( vx.size() > 0 )
{     vy[0] = vx[2];
vy[1] = vx[0] || vx[1];
}
// ------------------------------------------------------------------
// Zero forward mode.
// This case must always be implemented
// f(x) = [ x_2 * x_2 ]
//        [ x_0 * x_1 ]
// y^0  = f( x^0 )
if( p <= 0 )
{     // y_0^0 = x_2^0 * x_2^0
ty[0 * q1 + 0] = tx[2 * q1 + 0] * tx[2 * q1 + 0];
// y_1^0 = x_0^0 * x_1^0
ty[1 * q1 + 0] = tx[0 * q1 + 0] * tx[1 * q1 + 0];
}
if( q <= 0 )
return ok;
// ------------------------------------------------------------------
// First order one forward mode.
// This case is needed if first order forward mode is used.
// f'(x) = [   0,   0, 2 * x_2 ]
//         [ x_1, x_0,       0 ]
// y^1 =  f'(x^0) * x^1
if( p <= 1 )
{     // y_0^1 = 2 * x_2^0 * x_2^1
ty[0 * q1 + 1] = 2.0 * tx[2 * q1 + 0] * tx[2 * q1 + 1];
// y_1^1 = x_1^0 * x_0^1 + x_0^0 * x_1^1
ty[1 * q1 + 1]  = tx[1 * q1 + 0] * tx[0 * q1 + 1];
ty[1 * q1 + 1] += tx[0 * q1 + 0] * tx[1 * q1 + 1];
}
if( q <= 1 )
return ok;
// ------------------------------------------------------------------
// Second order forward mode.
// This case is neede if second order forwrd mode is used.
// f'(x) = [   0,   0, 2 x_2 ]
//         [ x_1, x_0,     0 ]
//
//            [ 0 , 0 , 0 ]                  [ 0 , 1 , 0 ]
// f_0''(x) = [ 0 , 0 , 0 ]  f_1^{(2)} (x) = [ 1 , 0 , 0 ]
//            [ 0 , 0 , 2 ]                  [ 0 , 0 , 0 ]
//
//  y_0^2 = x^1 * f_0''( x^0 ) x^1 / 2! + f_0'( x^0 ) x^2
//        = ( x_2^1 * 2.0 * x_2^1 ) / 2!
//        + 2.0 * x_2^0 * x_2^2
ty[0 * q1 + 2]  = tx[2 * q1 + 1] * tx[2 * q1 + 1];
ty[0 * q1 + 2] += 2.0 * tx[2 * q + 0] * tx[2 * q1 + 2];
//
//  y_1^2 = x^1 * f_1''( x^0 ) x^1 / 2! + f_1'( x^0 ) x^2
//        = ( x_1^1 * x_0^1 + x_0^1 * x_1^1) / 2
//        + x_1^0 * x_0^2 + x_0^0 + x_1^2
ty[1 * q1 + 2]  = tx[1 * q1 + 1] * tx[0 * q1 + 1];
ty[1 * q1 + 2] += tx[1 * q1 + 0] * tx[0 * q1 + 2];
ty[1 * q1 + 2] += tx[0 * q1 + 0] * tx[1 * q1 + 2];
// ------------------------------------------------------------------
return ok;
}
};
}  // End empty namespace

Use Atomic Function
bool forward(void)
{     bool ok = true;
double eps = 10. * CppAD::numeric_limits<double>::epsilon();
//
// Create the atomic_forward object
atomic_forward afun("atomic_forward");
//
// Create the function f(u)
//
// domain space vector
size_t n  = 3;
double x_0 = 1.00;
double x_1 = 2.00;
double x_2 = 3.00;
au[0] = x_0;
au[1] = x_1;
au[2] = x_2;

// declare independent variables and start tape recording

// range space vector
size_t m = 2;

// call user function
vector< AD<double> > ax = au;
afun(ax, ay);

// create f: u -> y and stop tape recording
f.Dependent (au, ay);  // y = f(u)
//
// check function value
double check = x_2 * x_2;
ok &= NearEqual( Value(ay[0]) , check,  eps, eps);
check = x_0 * x_1;
ok &= NearEqual( Value(ay[1]) , check,  eps, eps);

// --------------------------------------------------------------------
// zero order forward
//
vector<double> x0(n), y0(m);
x0[0] = x_0;
x0[1] = x_1;
x0[2] = x_2;
y0   = f.Forward(0, x0);
check = x_2 * x_2;
ok &= NearEqual(y0[0] , check,  eps, eps);
check = x_0 * x_1;
ok &= NearEqual(y0[1] , check,  eps, eps);
// --------------------------------------------------------------------
// first order forward
//
// value of Jacobian of f
double check_jac[] = {
0.0, 0.0, 2.0 * x_2,
x_1, x_0,       0.0
};
vector<double> x1(n), y1(m);
// check first order forward mode
for(size_t j = 0; j < n; j++)
x1[j] = 0.0;
for(size_t j = 0; j < n; j++)
{     // compute partial in j-th component direction
x1[j] = 1.0;
y1    = f.Forward(1, x1);
x1[j] = 0.0;
// check this direction
for(size_t i = 0; i < m; i++)
ok &= NearEqual(y1[i], check_jac[i * n + j], eps, eps);
}
// --------------------------------------------------------------------
// second order forward
//
// value of Hessian of f_0
double check_hes_0[] = {
0.0, 0.0, 0.0,
0.0, 0.0, 0.0,
0.0, 0.0, 2.0
};
//
// value of Hessian of f_1
double check_hes_1[] = {
0.0, 1.0, 0.0,
1.0, 0.0, 0.0,
0.0, 0.0, 0.0
};
vector<double> x2(n), y2(m);
for(size_t j = 0; j < n; j++)
x2[j] = 0.0;
// compute diagonal elements of the Hessian
for(size_t j = 0; j < n; j++)
{     // first order forward in j-th direction
x1[j] = 1.0;
f.Forward(1, x1);
y2 = f.Forward(2, x2);
// check this element of Hessian diagonal
ok &= NearEqual(y2[0], check_hes_0[j * n + j] / 2.0, eps, eps);
ok &= NearEqual(y2[1], check_hes_1[j * n + j] / 2.0, eps, eps);
//
for(size_t k = 0; k < n; k++) if( k != j )
{     x1[k] = 1.0;
f.Forward(1, x1);
y2 = f.Forward(2, x2);
//
// y2 = (H_jj + H_kk + H_jk + H_kj) / 2.0
// y2 = (H_jj + H_kk) / 2.0 + H_jk
//
double H_jj = check_hes_0[j * n + j];
double H_kk = check_hes_0[k * n + k];
double H_jk = y2[0] - (H_kk + H_jj) / 2.0;
ok &= NearEqual(H_jk, check_hes_0[j * n + k], eps, eps);
//
H_jj = check_hes_1[j * n + j];
H_kk = check_hes_1[k * n + k];
H_jk = y2[1] - (H_kk + H_jj) / 2.0;
ok &= NearEqual(H_jk, check_hes_1[j * n + k], eps, eps);
//
x1[k] = 0.0;
}
x1[j] = 0.0;
}
// --------------------------------------------------------------------
return ok;
}

Input File: example/atomic/forward.cpp