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@(@\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 .
Taylor's Ode Solver: A Multi-Level Adolc Example and Test

See Also
taylor_ode.cpp , mul_level_ode.cpp

Purpose
This is a realistic example using two levels of AD; see mul_level . The first level uses Adolc's adouble type to tape the solution of an ordinary differential equation. This solution is then differentiated with respect to a parameter vector. The second level uses CppAD's type AD<adouble> to take derivatives during the solution of the differential equation. These derivatives are used in the application of Taylor's method to the solution of the ODE.

ODE
For this example the function @(@ y : \B{R} \times \B{R}^n \rightarrow \B{R}^n @)@ is defined by @(@ y(0, x) = 0 @)@ and @(@ \partial_t y(t, x) = g(y, x) @)@ where @(@ g : \B{R}^n \times \B{R}^n \rightarrow \B{R}^n @)@ is defined by @[@ g(y, x) = \left( \begin{array}{c} x_0 \\ x_1 y_0 \\ \vdots \\ x_{n-1} y_{n-2} \end{array} \right) @]@

ODE Solution
The solution for this example can be calculated by starting with the first row and then using the solution for the first row to solve the second and so on. Doing this we obtain @[@ y(t, x ) = \left( \begin{array}{c} x_0 t \\ x_1 x_0 t^2 / 2 \\ \vdots \\ x_{n-1} x_{n-2} \ldots x_0 t^n / n ! \end{array} \right) @]@

Derivative of ODE Solution
Differentiating the solution above, with respect to the parameter vector @(@ x @)@, we notice that @[@ \partial_x y(t, x ) = \left( \begin{array}{cccc} y_0 (t,x) / x_0 & 0 & \cdots & 0 \\ y_1 (t,x) / x_0 & y_1 (t,x) / x_1 & 0 & \vdots \\ \vdots & \vdots & \ddots & 0 \\ y_{n-1} (t,x) / x_0 & y_{n-1} (t,x) / x_1 & \cdots & y_{n-1} (t,x) / x_{n-1} \end{array} \right) @]@


We define the function @(@ z(t, x) @)@ by the equation @[@ z ( t , x ) = g[ y ( t , x ) ] = h [ x , y( t , x ) ] @]@ see taylor_ode for the method used to compute the Taylor coefficients w.r.t @(@ t @)@ of @(@ y(t, x) @)@.

base_adolc.hpp
The file base_adolc.hpp is implements the Base type requirements where Base is adolc.

Memory Management
Adolc uses raw memory arrays that depend on the number of dependent and independent variables. The thread_alloc memory management utilities create_array and delete_array are used to manage this memory allocation.

Configuration Requirement
This example will be compiled and tested provided include_ipopt is on the cmake command line.

Source 

// suppress conversion warnings before other includes
# include <cppad/wno_conversion.hpp>
//

# include <adolc/adouble.h>
# include <adolc/taping.h>
# include <adolc/drivers/drivers.h>

// definitions not in Adolc distribution and required to use CppAD::AD<adouble>
# include <cppad/example/base_adolc.hpp>

# include <cppad/cppad.hpp>
// ==========================================================================
namespace { // BEGIN empty namespace
// define types for each level
typedef adouble           a1type;
typedef CppAD::AD<a1type> a2type;

// -------------------------------------------------------------------------
// class definition for C++ function object that defines ODE
class Ode {
private:
    // copy of a that is set by constructor and used by g(y)
    CPPAD_TESTVECTOR(a1type) a1x_;
public:
    // constructor
    Ode(const CPPAD_TESTVECTOR(a1type)& a1x) : a1x_(a1x)
    { }
    // the function g(y) is evaluated with two levels of taping
    CPPAD_TESTVECTOR(a2type) operator()
    ( const CPPAD_TESTVECTOR(a2type)& a2y) const
    {   size_t n = a2y.size();
        CPPAD_TESTVECTOR(a2type) a2g(n);
        size_t i;
        a2g[0] = a1x_[0];
        for(i = 1; i < n; i++)
            a2g[i] = a1x_[i] * a2y[i-1];

        return a2g;
    }
};

// -------------------------------------------------------------------------
// Routine that uses Taylor's method to solve ordinary differential equaitons
// and allows for algorithmic differentiation of the solution.
CPPAD_TESTVECTOR(a1type) taylor_ode_adolc(
    Ode                            G       ,  // function that defines the ODE
    size_t                         order   ,  // order of Taylor's method used
    size_t                         nstep   ,  // number of steps to take
    const a1type                   &a1dt   ,  // Delta t for each step
    const CPPAD_TESTVECTOR(a1type) &a1y_ini)  // y(t) at the initial time
{
    // some temporary indices
    size_t i, k, ell;

    // number of variables in the ODE
    size_t n = a1y_ini.size();

    // copies of x and g(y) with two levels of taping
    CPPAD_TESTVECTOR(a2type)   a2y(n), Z(n);

    // y, y^{(k)} , z^{(k)}, and y^{(k+1)}
    CPPAD_TESTVECTOR(a1type)  a1y(n), a1y_k(n), a1z_k(n), a1y_kp(n);

    // initialize x
    for(i = 0; i < n; i++)
        a1y[i] = a1y_ini[i];

    // loop with respect to each step of Taylors method
    for(ell = 0; ell < nstep; ell++)
    {   // prepare to compute derivatives using a1type
        for(i = 0; i < n; i++)
            a2y[i] = a1y[i];
        CppAD::Independent(a2y);

        // evaluate ODE using a2type
        Z = G(a2y);

        // define differentiable version of g: X -> Y
        // that computes its derivatives using a1type
        CppAD::ADFun<a1type> a1g(a2y, Z);

        // Use Taylor's method to take a step
        a1y_k            = a1y;     // initialize y^{(k)}
        a1type dt_kp = a1dt;    // initialize dt^(k+1)
        for(k = 0; k <= order; k++)
        {   // evaluate k-th order Taylor coefficient of y
            a1z_k = a1g.Forward(k, a1y_k);

            for(i = 0; i < n; i++)
            {   // convert to (k+1)-Taylor coefficient for x
                a1y_kp[i] = a1z_k[i] / a1type(k + 1);

                // add term for to this Taylor coefficient
                // to solution for y(t, x)
                a1y[i]    += a1y_kp[i] * dt_kp;
            }
            // next power of t
            dt_kp *= a1dt;
            // next Taylor coefficient
            a1y_k   = a1y_kp;
        }
    }
    return a1y;
}
} // END empty namespace
// ==========================================================================
// Routine that tests algorithmic differentiation of solutions computed
// by the routine taylor_ode.
bool mul_level_adolc_ode(void)
{   bool ok = true;
    double eps = 100. * std::numeric_limits<double>::epsilon();

    // number of components in differential equation
    size_t n = 4;

    // some temporary indices
    size_t i, j;

    // set up for thread_alloc memory allocator
    using CppAD::thread_alloc; // the allocator
    size_t capacity;           // capacity of an allocation

    // the vector x with length n (or greater) in double
    double* x = thread_alloc::create_array<double>(n, capacity);

    // the vector x with length n in a1type
    CPPAD_TESTVECTOR(a1type) a1x(n);
    for(i = 0; i < n; i++)
        a1x[i] = x[i] = double(i + 1);

    // declare the parameters as the independent variable
    short tag = 0;                     // Adolc setup
    int keep = 1;
    trace_on(tag, keep);
    for(i = 0; i < n; i++)
        a1x[i] <<= double(i + 1);  // a1x is independent for adouble type

    // arguments to taylor_ode_adolc
    Ode G(a1x);                // function that defines the ODE
    size_t   order = n;      // order of Taylor's method used
    size_t   nstep = 2;      // number of steps to take
    a1type   a1dt  = 1.;     // Delta t for each step
    // value of y(t, x) at the initial time
    CPPAD_TESTVECTOR(a1type) a1y_ini(n);
    for(i = 0; i < n; i++)
        a1y_ini[i] = 0.;

    // integrate the differential equation
    CPPAD_TESTVECTOR(a1type) a1y_final(n);
    a1y_final = taylor_ode_adolc(G, order, nstep, a1dt, a1y_ini);

    // declare the differentiable fucntion f : x -> y_final
    // (corresponding to the tape of adouble operations)
    double* y_final = thread_alloc::create_array<double>(n, capacity);
    for(i = 0; i < n; i++)
        a1y_final[i] >>= y_final[i];
    trace_off();

    // check function values
    double check = 1.;
    double t     = nstep * a1dt.value();
    for(i = 0; i < n; i++)
    {   check *= x[i] * t / double(i + 1);
        ok &= CppAD::NearEqual(y_final[i], check, eps, eps);
    }

    // memory where Jacobian will be returned
    double* jac_ = thread_alloc::create_array<double>(n * n, capacity);
    double** jac = thread_alloc::create_array<double*>(n, capacity);
    for(i = 0; i < n; i++)
        jac[i] = jac_ + i * n;

    // evaluate Jacobian of h at a
    size_t m = n;              // # dependent variables
    jacobian(tag, int(m), int(n), x, jac);

    // check Jacobian
    for(i = 0; i < n; i++)
    {   for(j = 0; j < n; j++)
        {   if( i < j )
                check = 0.;
            else
                check = y_final[i] / x[j];
            ok &= CppAD::NearEqual(jac[i][j], check, eps, eps);
        }
    }

    // make memroy avaiable for other use by this thread
    thread_alloc::delete_array(x);
    thread_alloc::delete_array(y_final);
    thread_alloc::delete_array(jac_);
    thread_alloc::delete_array(jac);
    return ok;
}
Input File: example/general/mul_level_adolc_ode.cpp