/* --------------------------------------------------------------------------
CppAD: C++ Algorithmic Differentiation: Copyright (C) 2003-19 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 rosen_34.cpp$$
$spell
    Rosen
$$

$section rosen_34: Example and Test$$


Define
$latex X : \B{R} \rightarrow \B{R}^n$$ by
$latex \[
    X_i (t) =  t^{i+1}
\] $$
for $latex i = 1 , \ldots , n-1$$.
It follows that
$latex \[
\begin{array}{rclr}
X_i(0)     & = & 0                             & {\rm for \; all \;} i \\
X_i ' (t)  & = & 1                             & {\rm if \;} i = 0      \\
X_i '(t)   & = & (i+1) t^i = (i+1) X_{i-1} (t) & {\rm if \;} i > 0
\end{array}
\] $$
The example tests Rosen34 using the relations above:

$srcfile%example/general/rosen_34.cpp%0%// BEGIN C++%// END C++%1%$$

$end
*/
// BEGIN C++

# include <cppad/cppad.hpp>        // For automatic differentiation

namespace {
    class Fun {
    public:
        // constructor
        Fun(bool use_x_) : use_x(use_x_)
        { }

        // compute f(t, x) both for double and AD<double>
        template <class Scalar>
        void Ode(
            const Scalar                    &t,
            const CPPAD_TESTVECTOR(Scalar) &x,
            CPPAD_TESTVECTOR(Scalar)       &f)
        {   size_t n  = x.size();
            Scalar ti(1);
            f[0]   = Scalar(1);
            size_t i;
            for(i = 1; i < n; i++)
            {   ti *= t;
                // convert int(size_t) to avoid warning
                // on _MSC_VER systems
                if( use_x )
                    f[i] = int(i+1) * x[i-1];
                else
                    f[i] = int(i+1) * ti;
            }
        }

        // compute partial of f(t, x) w.r.t. t using AD
        void Ode_ind(
            const double                    &t,
            const CPPAD_TESTVECTOR(double) &x,
            CPPAD_TESTVECTOR(double)       &f_t)
        {   using namespace CppAD;

            size_t n  = x.size();
            CPPAD_TESTVECTOR(AD<double>) T(1);
            CPPAD_TESTVECTOR(AD<double>) X(n);
            CPPAD_TESTVECTOR(AD<double>) F(n);

            // set argument values
            T[0] = t;
            size_t i;
            for(i = 0; i < n; i++)
                X[i] = x[i];

            // declare independent variables
            Independent(T);

            // compute f(t, x)
            this->Ode(T[0], X, F);

            // define AD function object
            ADFun<double> fun(T, F);

            // compute partial of f w.r.t t
            CPPAD_TESTVECTOR(double) dt(1);
            dt[0] = 1.;
            f_t = fun.Forward(1, dt);
        }

        // compute partial of f(t, x) w.r.t. x using AD
        void Ode_dep(
            const double                    &t,
            const CPPAD_TESTVECTOR(double) &x,
            CPPAD_TESTVECTOR(double)       &f_x)
        {   using namespace CppAD;

            size_t n  = x.size();
            CPPAD_TESTVECTOR(AD<double>) T(1);
            CPPAD_TESTVECTOR(AD<double>) X(n);
            CPPAD_TESTVECTOR(AD<double>) F(n);

            // set argument values
            T[0] = t;
            size_t i, j;
            for(i = 0; i < n; i++)
                X[i] = x[i];

            // declare independent variables
            Independent(X);

            // compute f(t, x)
            this->Ode(T[0], X, F);

            // define AD function object
            ADFun<double> fun(X, F);

            // compute partial of f w.r.t x
            CPPAD_TESTVECTOR(double) dx(n);
            CPPAD_TESTVECTOR(double) df(n);
            for(j = 0; j < n; j++)
                dx[j] = 0.;
            for(j = 0; j < n; j++)
            {   dx[j] = 1.;
                df = fun.Forward(1, dx);
                for(i = 0; i < n; i++)
                    f_x [i * n + j] = df[i];
                dx[j] = 0.;
            }
        }

    private:
        const bool use_x;

    };
}

bool rosen_34(void)
{   bool ok = true;     // initial return value
    size_t i;           // temporary indices

    using CppAD::NearEqual;
    double eps99 = 99.0 * std::numeric_limits<double>::epsilon();

    size_t  n = 4;      // number components in X(t) and order of method
    size_t  M = 2;      // number of Rosen34 steps in [ti, tf]
    double ti = 0.;     // initial time
    double tf = 2.;     // final time

    // xi = X(0)
    CPPAD_TESTVECTOR(double) xi(n);
    for(i = 0; i <n; i++)
        xi[i] = 0.;

    size_t use_x;
    for( use_x = 0; use_x < 2; use_x++)
    {   // function object depends on value of use_x
        Fun F(use_x > 0);

        // compute Rosen34 approximation for X(tf)
        CPPAD_TESTVECTOR(double) xf(n), e(n);
        xf = CppAD::Rosen34(F, M, ti, tf, xi, e);

        double check = tf;
        for(i = 0; i < n; i++)
        {   // check that error is always positive
            ok    &= (e[i] >= 0.);
            // 4th order method is exact for i < 4
            if( i < 4 ) ok &=
                NearEqual(xf[i], check, eps99, eps99);
            // 3rd order method is exact for i < 3
            if( i < 3 )
                ok &= (e[i] <= eps99);

            // check value for next i
            check *= tf;
        }
    }
    return ok;
}

// END C++
