\(\newcommand{\W}[1]{ \; #1 \; }\) \(\newcommand{\R}[1]{ {\rm #1} }\) \(\newcommand{\B}[1]{ {\bf #1} }\) \(\newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} }\) \(\newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} }\) \(\newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} }\) \(\newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }\)
atomic_three_forward.cpp¶
View page sourceAtomic Functions and Forward Mode: Example and Test¶
Purpose¶
This example demonstrates forward mode derivative calculation using an atomic_three function.
Function¶
For this example, the atomic function \(g : \B{R}^3 \rightarrow \B{R}^2\) is defined by
\[\begin{split}g(x) = \left( \begin{array}{c}
x_2 * x_2 \\
x_0 * x_1
\end{array} \right)\end{split}\]
Jacobian¶
The corresponding Jacobian is
\[\begin{split}g^{(1)} (x) = \left( \begin{array}{ccc}
0 & 0 & 2 x_2 \\
x_1 & x_0 & 0
\end{array} \right)\end{split}\]
Hessian¶
The Hessians of the component functions are
\[\begin{split}g_0^{(2)} ( x ) = \left( \begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 2
\end{array} \right)
\W{,}
g_1^{(2)} ( x ) = \left( \begin{array}{ccc}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 0
\end{array} \right)\end{split}\]
Start Class Definition¶
# include <cppad/cppad.hpp>
namespace { // begin empty namespace
using CppAD::vector; // abbreviate CppAD::vector using vector
//
class atomic_forward : public CppAD::atomic_three<double> {
Constructor¶
public:
atomic_forward(const std::string& name) :
CppAD::atomic_three<double>(name)
{ }
private:
for_type¶
// calculate type_y
bool for_type(
const vector<double>& parameter_x ,
const vector<CppAD::ad_type_enum>& type_x ,
vector<CppAD::ad_type_enum>& type_y ) override
{ assert( parameter_x.size() == type_x.size() );
bool ok = type_x.size() == 3; // n
ok &= type_y.size() == 2; // m
if( ! ok )
return false;
type_y[0] = type_x[2];
type_y[1] = std::max(type_x[0], type_x[1]);
return true;
}
forward¶
// forward mode routine called by CppAD
bool forward(
const vector<double>& parameter_x ,
const vector<CppAD::ad_type_enum>& type_x ,
size_t need_y ,
size_t order_low ,
size_t order_up ,
const vector<double>& taylor_x ,
vector<double>& taylor_y ) override
{
size_t q1 = order_up + 1;
# ifndef NDEBUG
size_t n = taylor_x.size() / q1;
size_t m = taylor_y.size() / q1;
# endif
assert( n == 3 );
assert( m == 2 );
assert( order_low <= order_up );
// this example only implements up to second order forward mode
bool ok = order_up <= 2;
if( ! ok )
return ok;
// ------------------------------------------------------------------
// Zero forward mode.
// This case must always be implemented
// g(x) = [ x_2 * x_2 ]
// [ x_0 * x_1 ]
// y^0 = f( x^0 )
if( order_low <= 0 )
{ // y_0^0 = x_2^0 * x_2^0
taylor_y[0*q1+0] = taylor_x[2*q1+0] * taylor_x[2*q1+0];
// y_1^0 = x_0^0 * x_1^0
taylor_y[1*q1+0] = taylor_x[0*q1+0] * taylor_x[1*q1+0];
}
if( order_up <= 0 )
return ok;
// ------------------------------------------------------------------
// First order forward mode.
// This case is needed if first order forward mode is used.
// g'(x) = [ 0, 0, 2 * x_2 ]
// [ x_1, x_0, 0 ]
// y^1 = f'(x^0) * x^1
if( order_low <= 1 )
{ // y_0^1 = 2 * x_2^0 * x_2^1
taylor_y[0*q1+1] = 2.0 * taylor_x[2*q1+0] * taylor_x[2*q1+1];
// y_1^1 = x_1^0 * x_0^1 + x_0^0 * x_1^1
taylor_y[1*q1+1] = taylor_x[1*q1+0] * taylor_x[0*q1+1];
taylor_y[1*q1+1] += taylor_x[0*q1+0] * taylor_x[1*q1+1];
}
if( order_up <= 1 )
return ok;
// ------------------------------------------------------------------
// Second order forward mode.
// This case is neede if second order forwrd mode is used.
// g'(x) = [ 0, 0, 2 x_2 ]
// [ x_1, x_0, 0 ]
//
// [ 0 , 0 , 0 ] [ 0 , 1 , 0 ]
// g_0''(x) = [ 0 , 0 , 0 ] g_1^{(2)} (x) = [ 1 , 0 , 0 ]
// [ 0 , 0 , 2 ] [ 0 , 0 , 0 ]
//
// y_0^2 = x^1 * g_0''( x^0 ) x^1 / 2! + g_0'( x^0 ) x^2
// = ( x_2^1 * 2.0 * x_2^1 ) / 2!
// + 2.0 * x_2^0 * x_2^2
taylor_y[0*q1+2] = taylor_x[2*q1+1] * taylor_x[2*q1+1];
taylor_y[0*q1+2] += 2.0 * taylor_x[2*q1+0] * taylor_x[2*q1+2];
//
// y_1^2 = x^1 * g_1''( x^0 ) x^1 / 2! + g_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
taylor_y[1*q1+2] = taylor_x[1*q1+1] * taylor_x[0*q1+1];
taylor_y[1*q1+2] += taylor_x[1*q1+0] * taylor_x[0*q1+2];
taylor_y[1*q1+2] += taylor_x[0*q1+0] * taylor_x[1*q1+2];
// ------------------------------------------------------------------
return ok;
}
};
} // End empty namespace
Use Atomic Function¶
bool forward(void)
{ bool ok = true;
using CppAD::AD;
using CppAD::NearEqual;
double eps = 10. * CppAD::numeric_limits<double>::epsilon();
//
// Create the atomic_forward object corresponding to g(x)
atomic_forward afun("atomic_forward");
//
// Create the function f(u) = g(u) for this example.
//
// domain space vector
size_t n = 3;
double u_0 = 1.00;
double u_1 = 2.00;
double u_2 = 3.00;
vector< AD<double> > au(n);
au[0] = u_0;
au[1] = u_1;
au[2] = u_2;
// declare independent variables and start tape recording
CppAD::Independent(au);
// range space vector
size_t m = 2;
vector< AD<double> > ay(m);
// call atomic function
vector< AD<double> > ax = au;
afun(ax, ay);
// create f: u -> y and stop tape recording
CppAD::ADFun<double> f;
f.Dependent (au, ay); // y = f(u)
//
// check function value
double check = u_2 * u_2;
ok &= NearEqual( Value(ay[0]) , check, eps, eps);
check = u_0 * u_1;
ok &= NearEqual( Value(ay[1]) , check, eps, eps);
// --------------------------------------------------------------------
// zero order forward
//
vector<double> u0(n), y0(m);
u0[0] = u_0;
u0[1] = u_1;
u0[2] = u_2;
y0 = f.Forward(0, u0);
check = u_2 * u_2;
ok &= NearEqual(y0[0] , check, eps, eps);
check = u_0 * u_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 * u_2,
u_1, u_0, 0.0
};
vector<double> u1(n), y1(m);
// check first order forward mode
for(size_t j = 0; j < n; j++)
u1[j] = 0.0;
for(size_t j = 0; j < n; j++)
{ // compute partial in j-th component direction
u1[j] = 1.0;
y1 = f.Forward(1, u1);
u1[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 g_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 g_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> u2(n), y2(m);
for(size_t j = 0; j < n; j++)
u2[j] = 0.0;
// compute diagonal elements of the Hessian
for(size_t j = 0; j < n; j++)
{ // first order forward in j-th direction
u1[j] = 1.0;
f.Forward(1, u1);
y2 = f.Forward(2, u2);
// 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 )
{ u1[k] = 1.0;
f.Forward(1, u1);
y2 = f.Forward(2, u2);
//
// 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);
//
u1[k] = 0.0;
}
u1[j] = 0.0;
}
// --------------------------------------------------------------------
return ok;
}