Third Order Reverse Mode: Example and Test

reverse_three.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 . Third Order Reverse Mode: Example and Test
Taylor Coefficients
@[@ \begin{array}{rcl} X(t) & = & x^{(0)} + x^{(1)} t + x^{(2)} t^2 \\ X^{(1)} (t) & = & x^{(1)} + 2 x^{(2)} t \\ X^{(2)} (t) & = & 2 x^{(2)} \end{array} @]@Thus, we need to be careful to properly account for the fact that @(@ X^{(2)} (0) = 2 x^{(2)} @)@ (and similarly @(@ Y^{(2)} (0) = 2 y^{(2)} @)@).

# include <cppad/cppad.hpp>
namespace { // ----------------------------------------------------------
// define the template function cases<Vector> in empty namespace
template <class Vector>
bool cases(void)
{   bool ok    = true;
    double eps = 10. * CppAD::numeric_limits<double>::epsilon();
    using CppAD::AD;
    using CppAD::NearEqual;

    // domain space vector
    size_t n = 2;
    CPPAD_TESTVECTOR(AD<double>) X(n);
    X[0] = 0.;
    X[1] = 1.;

    // declare independent variables and start recording
    CppAD::Independent(X);

    // range space vector
    size_t m = 1;
    CPPAD_TESTVECTOR(AD<double>) Y(m);
    Y[0] = X[0] * X[1];

    // create f : X -> Y and stop recording
    CppAD::ADFun<double> f(X, Y);

    // define x^0 and compute y^0 using user zero order forward
    Vector x0(n), y0(m);
    x0[0]    = 2.;
    x0[1]    = 3.;
    y0       = f.Forward(0, x0);

    // y^0 = F(x^0)
    double check;
    check    =  x0[0] * x0[1];
    ok      &= NearEqual(y0[0] , check, eps, eps);

    // define x^1 and compute y^1 using first order forward mode
    Vector x1(n), y1(m);
    x1[0] = 4.;
    x1[1] = 5.;
    y1    = f.Forward(1, x1);

    // Y^1 (x) = partial_t F( x^0 + x^1 * t )
    // y^1     = Y^1 (0)
    check = x1[0] * x0[1] + x0[0] * x1[1];
    ok   &= NearEqual(y1[0], check, eps, eps);

    // define x^2 and compute y^2 using second order forward mode
    Vector x2(n), y2(m);
    x2[0] = 6.;
    x2[1] = 7.;
    y2    = f.Forward(2, x2);

    // Y^2 (x) = partial_tt F( x^0 + x^1 * t + x^2 * t^2 )
    // y^2     = (1/2) *  Y^2 (0)
    check  = x2[0] * x0[1] + x1[0] * x1[1] + x0[0] * x2[1];
    ok    &= NearEqual(y2[0], check, eps, eps);

    // W(x)  = Y^0 (x) + 2 * Y^1 (x) + 3 * (1/2) * Y^2 (x)
    size_t p = 3;
    Vector dw(n*p), w(m*p);
    w[0] = 1.;
    w[1] = 2.;
    w[2] = 3.;
    dw   = f.Reverse(p, w);

    // check partial w.r.t x^0_0 of W(x)
    check = x0[1] + 2. * x1[1] + 3. * x2[1];
    ok   &= NearEqual(dw[0*p+0], check, eps, eps);

    // check partial w.r.t x^0_1 of W(x)
    check = x0[0] + 2. * x1[0] + 3. * x2[0];
    ok   &= NearEqual(dw[1*p+0], check, eps, eps);

    // check partial w.r.t x^1_0 of W(x)
    check = 2. * x0[1] + 3. * x1[1];
    ok   &= NearEqual(dw[0*p+1], check, eps, eps);

    // check partial w.r.t x^1_1 of W(x)
    check = 2. * x0[0] + 3. * x1[0];
    ok   &= NearEqual(dw[1*p+1], check, eps, eps);

    // check partial w.r.t x^2_0 of W(x)
    check = 3. * x0[1];
    ok   &= NearEqual(dw[0*p+2], check, eps, eps);

    // check partial w.r.t x^2_1 of W(x)
    check = 3. * x0[0];
    ok   &= NearEqual(dw[1*p+2], check, eps, eps);

    return ok;
}
} // End empty namespace
# include <vector>
# include <valarray>
bool reverse_three(void)
{   bool ok = true;
    ok &= cases< CppAD::vector  <double> >();
    ok &= cases< std::vector    <double> >();
    ok &= cases< std::valarray  <double> >();
    return ok;
}

Input File: example/general/reverse_three.cpp