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// SPDX-License-Identifier: EPL-2.0 OR GPL-2.0-or-later
// SPDX-FileCopyrightText: Bradley M. Bell <bradbell@seanet.com>
// SPDX-FileContributor: 2003-22 Bradley M. Bell
// ----------------------------------------------------------------------------
/*
{xrst_begin runge_45.cpp}
Runge45: Example and Test
#########################
Define
:math:`X : \B{R} \times \B{R} \rightarrow \B{R}^n` by
.. math::
X_j (b, t) = b \left( \sum_{k=0}^j t^k / k ! \right)
for :math:`j = 0 , \ldots , n-1`.
It follows that
.. math::
:nowrap:
\begin{eqnarray}
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{eqnarray}
For a fixed :math:`t_f`,
we can use :ref:`Runge45-name` to define
:math:`f : \B{R} \rightarrow \B{R}^n` as an approximation for
:math:`f(b) = X(b, t_f )`.
We can then compute :math:`f^{(1)} (b)` which is an approximation for
.. math::
\partial_b X(b, t_f ) = \sum_{k=0}^j t_f^k / k !
{xrst_literal
// BEGIN C++
// END C++
}
{xrst_end runge_45.cpp}
*/
// 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(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);
}
// evaluate 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++
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