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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: 2003-24 Bradley M. Bell
// ----------------------------------------------------------------------------
/*
{xrst_begin ode_gear.cpp}
{xrst_spell
rclr
}
OdeGear: Example and Test
#########################
Define
:math:`x : \B{R} \rightarrow \B{R}^n` by
.. math::
x_i (t) = t^{i+1}
for :math:`i = 1 , \ldots , n-1`.
It follows that
.. math::
\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 OdeGear using the relations above:
{xrst_literal
// BEGIN C++
// END C++
}
{xrst_end ode_gear.cpp}
*/
// BEGIN C++
# include <cppad/utility/ode_gear.hpp>
# 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;
}
}
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 OdeGear(void)
{ bool ok = true; // initial return value
size_t i, j; // temporary indices
double eps99 = 99.0 * std::numeric_limits<double>::epsilon();
size_t m = 4; // index of next value in X
size_t n = m; // number of components in x(t)
// vector of times
CPPAD_TESTVECTOR(double) T(m+1);
double step = .1;
T[0] = 0.;
for(j = 1; j <= m; j++)
{ T[j] = T[j-1] + step;
step = 2. * step;
}
// initial values for x( T[m-j] )
CPPAD_TESTVECTOR(double) X((m+1) * n);
for(j = 0; j < m; j++)
{ double ti = T[j];
for(i = 0; i < n; i++)
{ X[ j * n + i ] = ti;
ti *= T[j];
}
}
// error bound
CPPAD_TESTVECTOR(double) e(n);
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 OdeGear approximation for x( T[m] )
CppAD::OdeGear(F, m, n, T, X, e);
double check = T[m];
for(i = 0; i < n; i++)
{ // method is exact up to order m and x[i] = t^{i+1}
if( i + 1 <= m ) ok &= CppAD::NearEqual(
X[m * n + i], check, eps99, eps99
);
// error bound should be zero up to order m-1
if( i + 1 < m ) ok &= CppAD::NearEqual(
e[i], 0., eps99, eps99
);
// check value for next i
check *= T[m];
}
}
return ok;
}
// END C++
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