Multiple Level of AD: Example and Test
Keywords: multiple level ad: example test purpose source

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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 : R n → R where f ( x ) = 1 2 (x 0 2 + ⋯ + x n -1 2 ) We then use `AD<double>` (level one taping) to compute the directional derivative f ( 1 ) ( x ) * v = x 0 v 0 + ⋯ + x n -1 v n -1 . where v ∈ R n . We then use `double` (no taping) to compute d dx [f ( 1 ) ( x ) * v ] = 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: d dx [f ( 1 ) ( x ) * v ] = 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;
    }

`

Input File: example/mul_level.cpp