abs_eval: Example and Test

abs_eval.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 . abs_eval: Example and Test
Purpose
The function @(@ f : \B{R}^3 \rightarrow \B{R} @)@ defined by @[@ f( x_0, x_1, x_2 ) = | x_0 + x_1 | + | x_1 + x_2 | @]@ is affine, except for its absolute value terms. For this case, the abs_normal approximation should be equal to the function itself.

Source


# include <cppad/cppad.hpp>
# include "abs_eval.hpp"

namespace {
    CPPAD_TESTVECTOR(double) join(
        const CPPAD_TESTVECTOR(double)& x ,
        const CPPAD_TESTVECTOR(double)& u )
    {   size_t n = x.size();
        size_t s = u.size();
        CPPAD_TESTVECTOR(double) xu(n + s);
        for(size_t j = 0; j < n; j++)
            xu[j] = x[j];
        for(size_t j = 0; j < s; j++)
            xu[n + j] = u[j];
        return xu;
    }
}
bool abs_eval(void)
{   bool ok = true;
    //
    using CppAD::AD;
    using CppAD::ADFun;
    //
    typedef CPPAD_TESTVECTOR(double)       d_vector;
    typedef CPPAD_TESTVECTOR( AD<double> ) ad_vector;
    //
    double eps99 = 99.0 * std::numeric_limits<double>::epsilon();
    //
    size_t n = 3; // size of x
    size_t m = 1; // size of y
    size_t s = 2; // number of absolute value terms
    //
    // record the function f(x)
    ad_vector ad_x(n), ad_y(m);
    for(size_t j = 0; j < n; j++)
        ad_x[j] = double(j + 1);
    Independent( ad_x );
    // for this example, we ensure first absolute value is | x_0 + x_1 |
    AD<double> ad_0 = abs( ad_x[0] + ad_x[1] );
    // and second absolute value is | x_1 + x_2 |
    AD<double> ad_1 = abs( ad_x[1] + ad_x[2] );
    ad_y[0]         = ad_0 + ad_1;
    ADFun<double> f(ad_x, ad_y);

    // create its abs_normal representation in g, a
    ADFun<double> g, a;
    f.abs_normal_fun(g, a);

    // check dimension of domain and range space for g
    ok &= g.Domain() == n + s;
    ok &= g.Range()  == m + s;

    // check dimension of domain and range space for a
    ok &= a.Domain() == n;
    ok &= a.Range()  == s;

    // --------------------------------------------------------------------
    // Choose a point x_hat
    d_vector x_hat(n);
    for(size_t j = 0; j < n; j++)
        x_hat[j] = double(j - 1);

    // value of a_hat = a(x_hat)
    d_vector a_hat = a.Forward(0, x_hat);

    // (x_hat, a_hat)
    d_vector xu_hat = join(x_hat, a_hat);

    // value of g[ x_hat, a_hat ]
    d_vector g_hat = g.Forward(0, xu_hat);

    // Jacobian of g[ x_hat, a_hat ]
    d_vector g_jac = g.Jacobian(xu_hat);

    // value of delta_x
    d_vector delta_x(n);
    delta_x[0] =  1.0;
    delta_x[1] = -2.0;
    delta_x[2] = +2.0;

    // value of x
    d_vector x(n);
    for(size_t j = 0; j < n; j++)
        x[j] = x_hat[j] + delta_x[j];

    // value of f(x)
    d_vector y = f.Forward(0, x);

    // value of g_tilde
    d_vector g_tilde = CppAD::abs_eval(n, m, s, g_hat, g_jac, delta_x);

    // should be equal because f is affine, except for abs terms
    ok &= CppAD::NearEqual(y[0], g_tilde[0], eps99, eps99);

    return ok;
}

Input File: example/abs_normal/abs_eval.cpp