Prev Next atomic_three_forward.cpp

@(@\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 .
Atomic 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 @[@ g(x) = \left( \begin{array}{c} x_2 * x_2 \\ x_0 * x_1 \end{array} \right) @]@

Jacobian
The corresponding Jacobian is @[@ g^{(1)} (x) = \left( \begin{array}{ccc} 0 & 0 & 2 x_2 \\ x_1 & x_0 & 0 \end{array} \right) @]@

Hessian
The Hessians of the component functions are @[@ 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) @]@

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;
}
Input File: example/atomic_three/forward.cpp