BenderQuad: Example and Test

BenderQuad.cpp

Headings

BenderQuad: Example and Test Define

F : R \times R \to R

F (x, y) = \frac{1}{2} \sum_{i = 1}^{N} [y * \sin (x * t_{i}) - z_{i}]^{2}

where

z \in R^{N}

is a fixed vector. It follows that

\begin{matrix} \partial_{y} F (x, y) & = & \sum_{i = 1}^{N} [y * \sin (x * t_{i}) - z_{i}] \sin (x * t_{i}) \\ \partial_{y} \partial_{y} F (x, y) & = & \sum_{i = 1}^{N} \sin (x t_{i})^{2} \end{matrix}

Furthermore if we define

Y (x)

as the argmin of

F (x, y)

with respect to

y

\begin{matrix} Y (x) & = & y - [\partial_{y} \partial_{y} F (x, y)]^{-1} \partial_{y} F [x, y] \\ = & \sum_{i = 1}^{N} z_{i} \sin (x t_{i}) / \sum_{i = 1}^{N} z_{i} \sin (x * t_{i})^{2} \end{matrix}

 

# include <cppad/cppad.hpp>

namespace {
	using CppAD::AD;
	typedef CPPAD_TEST_VECTOR<double>         BAvector;
	typedef CPPAD_TEST_VECTOR< AD<double> >   ADvector;

	class Fun {
	private:
		BAvector t_; // measurement times
		BAvector z_; // measurement values
	public:
		// constructor
		Fun(const BAvector &t, const BAvector &z)
		: t_(t), z_(z)
		{ }
		// Fun.f(x, y) = F(x, y)
		ADvector f(const ADvector &x, const ADvector &y)
		{	size_t i;
			size_t N = z_.size();

			ADvector F(1);
			F[0] = 0.;

			AD<double> residual;
			for(i = 0; i < N; i++)
			{	residual = y[0] * sin( x[0] * t_[i] ) - z_[i];
				F[0]    += .5 * residual * residual;
			}
			return F;
		}
		// Fun.h(x, y) = H(x, y) = F_y (x, y)
		ADvector h(const ADvector &x, const BAvector &y)
		{	size_t i;
			size_t N = z_.size();

			ADvector fy(1);
			fy[0] = 0.;

			AD<double> residual;
			for(i = 0; i < N; i++)
			{	residual = y[0] * sin( x[0] * t_[i] ) - z_[i];
				fy[0]   += residual * sin( x[0] * t_[i] );
			}
			return fy;
		}
		// Fun.dy(x, y, h) = - H_y (x,y)^{-1} * h 
		//                 = - F_yy (x, y)^{-1} * h
		ADvector dy(
			const BAvector &x , 
			const BAvector &y , 
			const ADvector &H )
		{	size_t i;
			size_t N = z_.size();

			ADvector Dy(1);
			AD<double> fyy = 0.;

			for(i = 0; i < N; i++)
			{	fyy += sin( x[0] * t_[i] ) * sin( x[0] * t_[i] );
			}
			Dy[0] = - H[0] / fyy;

			return Dy;
		}
	};

	// Used to test calculation of Hessian of G
	AD<double> G(const ADvector& x, const BAvector& t, const BAvector& z)
	{	// compute Y(x)
		AD<double> numerator = 0.;
		AD<double> denominator = 0.;
		size_t k;
		for(k = 0; k < t.size(); k++)
		{	numerator   += sin( x[0] * t[k] ) * z[k];
			denominator += sin( x[0] * t[k] ) * sin( x[0] * t[k] ); 	
		}
		AD<double> y = numerator / denominator;

		// V(x) = F[x, Y(x)]
		AD<double> sum = 0;
		for(k = 0; k < t.size(); k++)
		{	AD<double> residual = y * sin( x[0] * t[k] ) - z[k];
			sum += .5 * residual * residual;
		}
		return sum;
	}
}

bool BenderQuad(void)
{	bool ok = true;
	using CppAD::AD;
	using CppAD::NearEqual;

	// temporary indices
	size_t i, j;

	// x space vector
	size_t n = 1;
	BAvector x(n);
	x[0] = 2. * 3.141592653;

	// y space vector
	size_t m = 1;
	BAvector y(m);
	y[0] = 1.;

	// t and z vectors
	size_t N = 10;
	BAvector t(N);
	BAvector z(N);
	for(i = 0; i < N; i++)
	{	t[i] = double(i) / double(N);       // time of measurement
		z[i] = y[0] * sin( x[0] * t[i] );   // data without noise
	}

	// construct the function object 
	Fun fun(t, z);

	// evaluate the G(x), G'(x) and G''(x)
	BAvector g(1), gx(n), gxx(n * n);
	CppAD::BenderQuad(x, y, fun, g, gx, gxx);


	// create ADFun object Gfun corresponding to G(x)
	ADvector a_x(n), a_g(1);
	for(j = 0; j < n; j++)
		a_x[j] = x[j];
	Independent(a_x);
	a_g[0] = G(a_x, t, z);
	CppAD::ADFun<double> Gfun(a_x, a_g);

	// accuracy for checks
	double eps = 10. * CppAD::epsilon<double>();

	// check Jacobian
	BAvector check_gx = Gfun.Jacobian(x);
	for(j = 0; j < n; j++)
		ok &= NearEqual(gx[j], check_gx[j], eps, eps);

	// check Hessian
	BAvector check_gxx = Gfun.Hessian(x, 0);
	for(j = 0; j < n*n; j++)
		ok &= NearEqual(gxx[j], check_gxx[j], eps, eps);

	return ok;
}

Input File: example/bender_quad.cpp