Third Order Reverse Mode: Example and Test
Keywords: third order reverse mode: example test taylor coefficients

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Third Order Reverse Mode: Example and Test
Taylor Coefficients X ( t ) = x ( 0 ) + x ( 1 ) t + x ( 2 ) t 2 X ( 1 ) ( t ) = x ( 1 ) + 2 x ( 2 ) t X ( 2 ) ( t ) = 2 x ( 2 ) Thus, we need to be careful to properly account for the fact that X ( 2 ) ( 0 ) = 2 x ( 2 ) (and similarly Y ( 2 ) ( 0 ) = 2 y ( 2 ) ). `

    # include <cppad/cppad.hpp>
    namespace { // ----------------------------------------------------------
    // define the template function cases<Vector> in empty namespace
    template <typename Vector>
    bool cases(void)
    {	bool ok    = true;
    	double eps = 10. * CppAD::numeric_limits<double>::epsilon();
    	using CppAD::AD;
    	using CppAD::NearEqual;

    	// domain space vector
    	size_t n = 2;
    	CPPAD_TESTVECTOR(AD<double>) X(n);
    	X[0] = 0.;
    	X[1] = 1.;

    	// declare independent variables and start recording
    	CppAD::Independent(X);

    	// range space vector
    	size_t m = 1;
    	CPPAD_TESTVECTOR(AD<double>) Y(m);
    	Y[0] = X[0] * X[1];

    	// create f : X -> Y and stop recording
    	CppAD::ADFun<double> f(X, Y);

    	// define x^0 and compute y^0 using user zero order forward
    	Vector x0(n), y0(m);
    	x0[0]    = 2.;
    	x0[1]    = 3.;
    	y0       = f.Forward(0, x0);

    	// y^0 = F(x^0)
    	double check;
    	check    =  x0[0] * x0[1];
    	ok      &= NearEqual(y0[0] , check, eps, eps);

    	// define x^1 and compute y^1 using first order forward mode
    	Vector x1(n), y1(m);
    	x1[0] = 4.;
    	x1[1] = 5.;
    	y1    = f.Forward(1, x1);

    	// Y^1 (x) = partial_t F( x^0 + x^1 * t )
    	// y^1     = Y^1 (0)
    	check = x1[0] * x0[1] + x0[0] * x1[1];
    	ok   &= NearEqual(y1[0], check, eps, eps);

    	// define x^2 and compute y^2 using second order forward mode
    	Vector x2(n), y2(m);
    	x2[0] = 6.;
    	x2[1] = 7.;
    	y2    = f.Forward(2, x2);

    	// Y^2 (x) = partial_tt F( x^0 + x^1 * t + x^2 * t^2 )
    	// y^2     = (1/2) *  Y^2 (0)
    	check  = x2[0] * x0[1] + x1[0] * x1[1] + x0[0] * x2[1];
    	ok    &= NearEqual(y2[0], check, eps, eps);

    	// W(x)  = Y^0 (x) + 2 * Y^1 (x) + 3 * (1/2) * Y^2 (x)
    	size_t p = 3;
    	Vector dw(n*p), w(m*p);
    	w[0] = 1.;
    	w[1] = 2.;
    	w[2] = 3.;
    	dw   = f.Reverse(p, w);

    	// check partial w.r.t x^0_0 of W(x)
    	check = x0[1] + 2. * x1[1] + 3. * x2[1];
    	ok   &= NearEqual(dw[0*p+0], check, eps, eps);

    	// check partial w.r.t x^0_1 of W(x)
    	check = x0[0] + 2. * x1[0] + 3. * x2[0];
    	ok   &= NearEqual(dw[1*p+0], check, eps, eps);

    	// check partial w.r.t x^1_0 of W(x)
    	check = 2. * x0[1] + 3. * x1[1];
    	ok   &= NearEqual(dw[0*p+1], check, eps, eps);

    	// check partial w.r.t x^1_1 of W(x)
    	check = 2. * x0[0] + 3. * x1[0];
    	ok   &= NearEqual(dw[1*p+1], check, eps, eps);

    	// check partial w.r.t x^2_0 of W(x)
    	check = 3. * x0[1];
    	ok   &= NearEqual(dw[0*p+2], check, eps, eps);

    	// check partial w.r.t x^2_1 of W(x)
    	check = 3. * x0[0];
    	ok   &= NearEqual(dw[1*p+2], check, eps, eps);

    	return ok;
    }
    } // End empty namespace
    # include <vector>
    # include <valarray>
    bool reverse_three(void)
    {	bool ok = true;
    	ok &= cases< CppAD::vector  <double> >();
    	ok &= cases< std::vector    <double> >();
    	ok &= cases< std::valarray  <double> >();
    	return ok;
    }

`

Input File: example/reverse_three.cpp