Atomic Linear ODE Reverse Dependency Analysis: Example and Test

atomic_four_lin_ode_rev_depend.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 Linear ODE Reverse Dependency Analysis: Example and Test
Purpose
This example demonstrates calculating reverse dependency with the atomic_four_lin_ode class; see atomic_four_lin_ode_rev_depend.hpp .

f(x)
For this example, the function @(@ f(x) = z_2 (r, u) @)@ where @(@ z(t, u) @)@ solves the following ODE @[@ z_t (t, x) = \left( \begin{array}{cccc} 0 & 0 & 0 & 0 \\ x_0 & 0 & 0 & 0 \\ 0 & x_1 & 0 & 0 \\ 0 & 0 & x_2 & 0 \\ \end{array} \right) z(t, u) \W{,} z(0, u) = \left( \begin{array}{c} x_3 \\ x_4 \\ x_5 \\ x_6 \\ \end{array} \right) @]@

Solution
The actual solution to this ODE is @[@ z(t, x) = \left( \begin{array}{l} x_3 \\ x_4 + x_0 x_3 t \\ x_5 + x_1 x_4 t + x_1 x_0 x_3 t^2 / 2 \\ x_6 + x_2 x_5 t + x_2 x_1 x_4 t^2 / 2 + x_2 x_1 x_0 x_3 t^3 / 6 \end{array} \right) @]@

Source


# include <cppad/cppad.hpp>
# include <cppad/example/atomic_four/lin_ode/lin_ode.hpp>

namespace { // BEGIN_EMPTY_NAMESPACE

template <class Scalar, class Vector>
Vector Y(Scalar t, const Vector& x)
{   size_t m = 4;
    Vector y(m);
    //
    y[0]  = x[3];
    y[1]  = x[4] + x[0]*x[3]*t;
    y[2]  = x[5] + x[1]*x[4]*t + x[1]*x[0]*x[3]*t*t/2.0;
    y[3]  = x[6] + x[2]*x[5]*t + x[2]*x[1]*x[4]*t*t/2.0
          + x[2]*x[1]*x[0]*x[3]*t*t*t/6.0;
    //
    return y;
}

} // END_EMPTY_NAMESPACE

bool rev_depend(void)
{   // ok, eps
    bool ok = true;
    //
    // sparse_rc, AD, eps99
    typedef CppAD::sparse_rc< CppAD::vector<size_t> > sparse_rc;
    using CppAD::AD;
    double eps99 = std::numeric_limits<double>::epsilon() * 99.0;
    // -----------------------------------------------------------------------
    // Record f
    // -----------------------------------------------------------------------
    //
    // afun
    CppAD::atomic_lin_ode<double> afun("atomic_lin_ode");
    //
    // m, r
    size_t m      = 4;
    double r      = 2.0;
    double step   = 1.0;
    //
    // pattern, transpose
    size_t nr  = m;
    size_t nc  = m;
    size_t nnz = 3;
    sparse_rc pattern(nr, nc, nnz);
    for(size_t k = 0; k < nnz; ++k)
    {   size_t i = k + 1;
        size_t j = k;
        pattern.set(k, i, j);
    }
    bool transpose = false;
    //
    // ax
    CPPAD_TESTVECTOR( AD<double> ) ax(nnz + m);
    for(size_t k = 0; k < nnz + m; ++k)
        ax[k] = double(k + 1);
    CppAD::Independent(ax);
    //
    // ay
    CPPAD_TESTVECTOR( AD<double> ) ay(m);
    size_t call_id = afun.set(r, step, pattern, transpose);
    afun(call_id, ax, ay);
    //
    // z_index
    size_t z_index = 1;
    //
    // az
    CPPAD_TESTVECTOR( AD<double> ) az(1);
    az[0] = ay[z_index];
    //
    // f
    // optimize uses rev_depend
    CppAD::ADFun<double> f(ax, az);
    f.optimize();
    // -----------------------------------------------------------------------
    // check_f
    // -----------------------------------------------------------------------
    CppAD::Independent(ax);
    AD<double> ar = r;
    ay    = Y(ar, ax);
    az[0] = ay[z_index];
    CppAD::ADFun<double> check_f(ax, az);
    // -----------------------------------------------------------------------
    // rev_depend
    // use test_rev_depend to call rev_depend directly
    // -----------------------------------------------------------------------
    //
    // depend_x
    CppAD::vector<bool> ident_zero_x(nnz + m), depend_x(nnz + m), depend_y(m);
    for(size_t i = 0; i < m; ++i)
    {   depend_y[i]     = i == z_index;
        ident_zero_x[i] = false;
    }
    afun.test_rev_depend(call_id, ident_zero_x, depend_x, depend_y);
    //
    // x
    CPPAD_TESTVECTOR(double) x(nnz + m);
    for(size_t j = 0; j < nnz + m; ++j)
        x[j] = double( j + 2 );
    //
    // dw
    check_f.Forward(0, x);
    CPPAD_TESTVECTOR(double) w(1), dw(nnz + m);
    w[0] = 1.0;
    dw = check_f.Reverse(1, w);
    //
    // ok
    // note that for this x, partial w.r.t x[j] is non-zero if and only if
    // y[z_index] depends on x[j]
    for(size_t j = 0; j < nnz + m; ++j)
        ok &= depend_x[j] == (dw[j] != 0.0);
    //
    // -----------------------------------------------------------------------
    // forward mode on f
    // Check that the optimized version of agrees with check_f.
    // -----------------------------------------------------------------------
    //
    // z
    // zero order forward mode computation of f(x)
    CPPAD_TESTVECTOR(double) z = f.Forward(0, x);
    //
    // ok
    CPPAD_TESTVECTOR(double) check_z = check_f.Forward(0, x);
    ok &= CppAD::NearEqual(z[0], check_z[0], eps99, eps99);
    //
    // du, ok
    CPPAD_TESTVECTOR(double) dx(nnz + m), dz(1), check_dz(1);
    for(size_t j = 0; j < nnz + m; ++j)
        dx[j] = 0.0;
    //
    for(size_t j = 0; j < nnz + m; ++j)
    {   dx[j]     = 1.0;
        dz        = f.Forward(1, dx);
        check_dz  = check_f.Forward(1, dx);
        ok       &= CppAD::NearEqual(dz[0], check_dz[0], eps99, eps99);
        dx[j]     = 0.0;
    }
    // -----------------------------------------------------------------------
    return ok;
}

Input File: example/atomic_four/lin_ode/rev_depend.cpp