mul_level.cpp¶

Multiple Level of AD: Example and Test¶

Purpose¶

In this example, we use AD< AD<double> > (level two taping), the compute values of the function \(f : \B{R}^n \rightarrow \B{R}\) where

\[f(x) = \frac{1}{2} \left( x_0^2 + \cdots + x_{n-1}^2 \right)\]

We then use AD<double> (level one taping) to compute the directional derivative

\[f^{(1)} (x) * v = x_0 v_0 + \cdots + x_{n-1} v_{n-1}\]

where \(v \in \B{R}^n\). We then use double (no taping) to compute

\[\frac{d}{dx} \left[ f^{(1)} (x) * v \right] = v\]

This is only meant as an example of multiple levels of taping. The example hes_times_dir.cpp computes the same value more efficiently by using the identity:

\[\frac{d}{dx} \left[ f^{(1)} (x) * v \right] = f^{(2)} (x) * v\]

The example mul_level_adolc.cpp computes the same values using Adolc’s type adouble and CppAD’s type AD<adouble> .

Source¶

# include <cppad/cppad.hpp>

namespace {
   // f(x) = |x|^2 / 2 = .5 * ( x[0]^2 + ... + x[n-1]^2 )
   template <class Type>
   Type f(const CPPAD_TESTVECTOR(Type)& x)
   {  Type sum;

      sum  = 0.;
      size_t i = size_t(x.size());
      for(i = 0; i < size_t(x.size()); i++)
         sum += x[i] * x[i];

      return .5 * sum;
   }
}

bool mul_level(void)
{  bool ok = true;                          // initialize test result

   typedef CppAD::AD<double>   a1type;    // for one level of taping
   typedef CppAD::AD<a1type>    a2type;    // for two levels of taping
   size_t n = 5;                           // dimension for example
   size_t j;                               // a temporary index variable

   // 10 times machine epsilon
   double eps = 10. * std::numeric_limits<double>::epsilon();

   CPPAD_TESTVECTOR(double) x(n);
   CPPAD_TESTVECTOR(a1type)  a1x(n), a1v(n), a1dy(1) ;
   CPPAD_TESTVECTOR(a2type)  a2x(n), a2y(1);

   // Values for the independent variables while taping the function f(x)
   for(j = 0; j < n; j++)
      a2x[j] = a1x[j] = x[j] = double(j);
   // Declare the independent variable for taping f(x)
   CppAD::Independent(a2x);

   // Use AD< AD<double> > to tape the evaluation of f(x)
   a2y[0] = f(a2x);

   // Declare a1f as the corresponding ADFun< AD<double> >
   // (make sure we do not run zero order forward during constructor)
   CppAD::ADFun<a1type> a1f;
   a1f.Dependent(a2x, a2y);

   // Values for the independent variables while taping f'(x) * v
   // Declare the independent variable for taping f'(x) * v
   // (Note we did not have to tape the creationg of a1f.)
   CppAD::Independent(a1x);

   // set the argument value x for computing f'(x) * v
   a1f.Forward(0, a1x);
   // compute f'(x) * v
   for(j = 0; j < n; j++)
      a1v[j] = double(n - j);
   a1dy = a1f.Forward(1, a1v);

   // declare g as ADFun<double> function corresponding to f'(x) * v
   CppAD::ADFun<double> g;
   g.Dependent(a1x, a1dy);

   // optimize out operations not necessary for function f'(x) * v
   g.optimize();

   // Evaluate f'(x) * v
   g.Forward(0, x);

   // compute the d/dx of f'(x) * v = f''(x) * v = v
   CPPAD_TESTVECTOR(double) w(1);
   CPPAD_TESTVECTOR(double) dw(n);
   w[0] = 1.;
   dw   = g.Reverse(1, w);

   for(j = 0; j < n; j++)
      ok &= CppAD::NearEqual(dw[j], a1v[j], eps, eps);

   return ok;
}

mul_level.cpp¶

Multiple Level of AD: Example and Test¶

See Also¶

Purpose¶

Source¶