/* --------------------------------------------------------------------------
CppAD: C++ Algorithmic Differentiation: Copyright (C) 2003-17 Bradley M. Bell

CppAD is distributed under the terms of the
             Eclipse Public License Version 2.0.

This Source Code may also be made available under the following
Secondary License when the conditions for such availability set forth
in the Eclipse Public License, Version 2.0 are satisfied:
      GNU General Public License, Version 2.0 or later.
---------------------------------------------------------------------------- */

/*
$begin runge45_2.cpp$$
$spell
    Runge
$$

$section Runge45: Example and Test$$


Define
$latex X : \B{R} \times \B{R} \rightarrow \B{R}^n$$ by
$latex \[
    X_j (b, t) =  b \left( \sum_{k=0}^j t^k / k ! \right)
\] $$
for $latex j = 0 , \ldots , n-1$$.
It follows that
$latex \[
\begin{array}{rcl}
X_j  (b, 0)   & = & b                                                     \\
\partial_t X_j (b, t)   & = & b \left( \sum_{k=0}^{j-1} t^k / k ! \right) \\
\partial_t X_j (b, t)   & = & \left\{ \begin{array}{ll}
    0               & {\rm if} \; j = 0  \\
    X_{j-1} (b, t)  & {\rm otherwise}
\end{array} \right.
\end{array}
\] $$
For a fixed $latex t_f$$,
we can use $cref Runge45$$ to define
$latex f : \B{R} \rightarrow \B{R}^n$$ as an approximation for
$latex f(b) = X(b, t_f )$$.
We can then compute $latex f^{(1)} (b)$$ which is an approximation for
$latex \[
    \partial_b X(b, t_f ) =  \sum_{k=0}^j t_f^k / k !
\] $$

$srcfile%example/general/runge45_2.cpp%0%// BEGIN C++%// END C++%1%$$

$end
*/
// BEGIN C++

# include <cstddef>              // for size_t
# include <limits>               // for machine epsilon
# include <cppad/cppad.hpp>      // for all of CppAD

namespace {

    template <class Scalar>
    class Fun {
    public:
        // constructor
        Fun(void)
        { }

        // set return value to X'(t)
        void Ode(
            const Scalar                    &t,
            const CPPAD_TESTVECTOR(Scalar) &x,
            CPPAD_TESTVECTOR(Scalar)       &f)
        {   size_t n  = x.size();
            f[0]      = 0.;
            for(size_t k = 1; k < n; k++)
                f[k] = x[k-1];
        }
    };
}

bool runge_45_2(void)
{   typedef CppAD::AD<double> Scalar;
    using CppAD::NearEqual;

    bool ok = true;     // initial return value
    size_t j;           // temporary indices

    size_t     n = 5;   // number components in X(t) and order of method
    size_t     M = 2;   // number of Runge45 steps in [ti, tf]
    Scalar ad_ti = 0.;  // initial time
    Scalar ad_tf = 2.;  // final time

    // value of independent variable at which to record operations
    CPPAD_TESTVECTOR(Scalar) ad_b(1);
    ad_b[0] = 1.;

    // declare b to be the independent variable
    Independent(ad_b);

    // object to evaluate ODE
    Fun<Scalar> ad_F;

    // xi = X(0)
    CPPAD_TESTVECTOR(Scalar) ad_xi(n);
    for(j = 0; j < n; j++)
        ad_xi[j] = ad_b[0];

    // compute Runge45 approximation for X(tf)
    CPPAD_TESTVECTOR(Scalar) ad_xf(n), ad_e(n);
    ad_xf = CppAD::Runge45(ad_F, M, ad_ti, ad_tf, ad_xi, ad_e);

    // stop recording and use it to create f : b -> xf
    CppAD::ADFun<double> f(ad_b, ad_xf);

    // evaluate f(b)
    CPPAD_TESTVECTOR(double)  b(1);
    CPPAD_TESTVECTOR(double) xf(n);
    b[0] = 1.;
    xf   = f.Forward(0, b);

    // check that f(b) = X(b, tf)
    double tf    = Value(ad_tf);
    double term  = 1;
    double sum   = 0;
    double eps   = 10. * CppAD::numeric_limits<double>::epsilon();
    for(j = 0; j < n; j++)
    {   sum += term;
        ok &= NearEqual(xf[j], b[0] * sum, eps, eps);
        term *= tf;
        term /= double(j+1);
    }

    // evalute f'(b)
    CPPAD_TESTVECTOR(double) d_xf(n);
    d_xf = f.Jacobian(b);

    // check that f'(b) = partial of X(b, tf) w.r.t b
    term  = 1;
    sum   = 0;
    for(j = 0; j < n; j++)
    {   sum += term;
        ok &= NearEqual(d_xf[j], sum, eps, eps);
        term *= tf;
        term /= double(j+1);
    }

    return ok;
}

// END C++
