\(\newcommand{\W}[1]{ \; #1 \; }\) \(\newcommand{\R}[1]{ {\rm #1} }\) \(\newcommand{\B}[1]{ {\bf #1} }\) \(\newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} }\) \(\newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} }\) \(\newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} }\) \(\newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }\)

atomic_four_lin_ode_forward.cpp

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Atomic Linear ODE Forward Mode: Example and Test

Purpose

This example demonstrates using forward mode with the atomic_four_lin_ode class.

f(u)

For this example, the function \(f(u) = z(r, u)\) where \(z(t, u)\) solves the following ODE

\[\begin{split}z_t (t, u) = \left( \begin{array}{cccc} 0 & 0 & 0 & 0 \\ u_4 & 0 & 0 & 0 \\ 0 & u_5 & 0 & 0 \\ 0 & 0 & u_6 & 0 \\ \end{array} \right) z(t, u) \W{,} z(0, u) = \left( \begin{array}{c} u_0 \\ u_1 \\ u_2 \\ u_3 \\ \end{array} \right)\end{split}\]

z(t, u)

The actual solution to this ODE is

\[\begin{split}z(t, u) = \left( \begin{array}{l} u_0 \\ u_1 + u_4 u_0 t \\ u_2 + u_5 u_1 t + u_5 u_4 u_0 t^2 / 2 \\ u_3 + u_6 u_2 t + u_6 u_5 u_1 t^2 / 2 + u_6 u_5 u_4 u_0 t^3 / 6 \end{array} \right)\end{split}\]

g(u)

We define \(g(u) = \partial_{u0} f(u)\). It follows that

\[\begin{split}g (u) = \left( \begin{array}{l} 1 \\ u_4 r \\ u_5 u_4 r^2 / 2 \\ u_6 u_5 u_4 r^3 / 6 \end{array} \right)\end{split}\]

Source

# include <cppad/cppad.hpp>
# include <cppad/example/atomic_four/lin_ode/lin_ode.hpp>

namespace { // BEGIN_EMPTY_NAMESPACE

template <class Scalar, class Vector>
Vector Z(Scalar t, const Vector& u)
{  size_t nz = 4;
   Vector z(nz);
   //
   z[0]  = u[0];
   z[1]  = u[1] + u[4]*u[0]*t;
   z[2]  = u[2] + u[5]*u[1]*t + u[5]*u[4]*u[0]*t*t/2.0;
   z[3]  = u[3] + u[6]*u[2]*t + u[6]*u[5]*u[1]*t*t/2.0
          + u[6]*u[5]*u[4]*u[0]*t*t*t/6.0;
   //
   return z;
}

template <class Scalar, class Vector>
Vector G(Scalar t, const Vector& u)
{  size_t ng = 4;
   Vector g(ng);
   //
   g[0]  = Scalar(1.0);
   g[1]  = u[4]*t;
   g[2]  = u[5]*u[4]*t*t/2.0;
   g[3]  = u[6]*u[5]*u[4]*t*t*t/6.0;
   //
   return g;
}

} // END_EMPTY_NAMESPACE

bool forward(void)
{  // ok
   bool ok = true;
   //
   // AD, NearEqual, eps99
   using CppAD::AD;
   using CppAD::NearEqual;
   double eps99 = std::numeric_limits<double>::epsilon() * 99.0;
   // -----------------------------------------------------------------------
   // Record f
   // -----------------------------------------------------------------------
   //
   // afun
   CppAD::atomic_lin_ode<double> afun("atomic_lin_ode");
   //
   // m, r
   size_t m      = 4;
   double r      = 2.0;
   double step   = 1.0;
   //
   // pattern, transpose
   size_t nr  = m;
   size_t nc  = m;
   size_t nnz = 3;
   CppAD::sparse_rc< CppAD::vector<size_t> > pattern(nr, nc, nnz);
   for(size_t k = 0; k < nnz; ++k)
   {  size_t i = k + 1;
      size_t j = k;
      pattern.set(k, i, j);
   }
   bool transpose = false;
   //
   // ny, ay
   size_t ny = m;
   CPPAD_TESTVECTOR( AD<double> ) ay(ny);
   //
   // nu, au
   size_t nu = nnz + m;
   CPPAD_TESTVECTOR( AD<double> ) au(nu);
   for(size_t j = 0; j < nu; ++j)
      au[j] = AD<double>(j + 1);
   CppAD::Independent(au);
   //
   // ax
   CPPAD_TESTVECTOR( AD<double> ) ax(nnz + m);
   for(size_t k = 0; k < nnz; ++k)
      ax[k] = au[m + k];
   for(size_t i = 0; i < m; ++i)
      ax[nnz + i] = au[i];
   //
   // ay
   size_t call_id = afun.set(r, step, pattern, transpose);
   afun(call_id, ax, ay);
   //
   // f
   CppAD::ADFun<double> f(au, ay);
   // -----------------------------------------------------------------------
   // ar, check_f
   CppAD::Independent(au);
   AD<double> ar = r;
   ay = Z(ar, au);
   CppAD::ADFun<double> check_f(au, ay);
   // -----------------------------------------------------------------------
   // forward mode on f
   // -----------------------------------------------------------------------
   //
   // u
   CPPAD_TESTVECTOR(double) u(nu);
   for(size_t j = 0; j < nu; ++j)
      u[j] = double( j + 2 );
   //
   // y
   // zero order forward mode computation of f(u)
   CPPAD_TESTVECTOR(double) y(ny);
   y = f.Forward(0, u);
   //
   // ok
   CPPAD_TESTVECTOR(double) z = check_f.Forward(0, u);
   for(size_t i = 0; i < ny; ++i)
      ok &= NearEqual(y[i], z[i], eps99, eps99);
   //
   // du, ok
   CPPAD_TESTVECTOR(double) du(nu), dy(ny), dz(ny);
   for(size_t j = 0; j < nu; ++j)
      du[j] = 0.0;
   for(size_t j = 0; j < nu; ++j)
   {  du[j] = 1.0;
      dy    = f.Forward(1, du);
      dz    = check_f.Forward(1, du);
      for(size_t i = 0; i < ny; ++i)
         ok &= NearEqual(dy[i], dz[i], eps99, eps99);
      du[j] = 0.0;
   }
   // -----------------------------------------------------------------------
   // Record g
   // -----------------------------------------------------------------------
   //
   // af
   CppAD::ADFun< AD<double>, double> af = f.base2ad();
   //
   // az
   CppAD::Independent(au);
   CPPAD_TESTVECTOR( AD<double> ) dau(nu), day(ny);
   af.Forward(0, au);
   for(size_t j = 0; j < nu; ++j)
      dau[j] = 0.0;
   dau[0] = 1.0;
   day    = af.Forward(1, dau);
   // g
   CppAD::ADFun<double> g(au, day);
   // -----------------------------------------------------------------------
   // check_g
   CppAD::Independent(au);
   ay = G(ar, au);
   CppAD::ADFun<double> check_g(au, ay);
   // -----------------------------------------------------------------------
   // forward mode on g
   // -----------------------------------------------------------------------
   //
   // y
   // zero order forward mode computation of g(u)
   dy = g.Forward(0, u);
   //
   // ok
   CPPAD_TESTVECTOR(double) v = check_g.Forward(0, u);
   for(size_t i = 0; i < ny; ++i)
      ok &= NearEqual(dy[i], v[i], eps99, eps99);
   //
   // du, ok
   CPPAD_TESTVECTOR(double) ddy(ny), ddz(ny);
   for(size_t j = 0; j < nu; ++j)
      du[j] = 0.0;
   for(size_t j = 0; j < nu; ++j)
   {  du[j] = 1.0;
      ddy    = g.Forward(1, du);
      ddz    = check_g.Forward(1, du);
      for(size_t i = 0; i < ny; ++i)
         ok &= NearEqual(ddy[i], ddz[i], eps99, eps99);
      du[j] = 0.0;
   }
   return ok;
}