Prev Next Index-> contents reference index search external Up-> CppAD ADFun Reverse reverse_any reverse_three.cpp ADFun-> record_adfun drivers Forward Reverse sparsity_pattern sparse_derivative optimize abs_normal FunCheck check_for_nan Reverse-> reverse_one reverse_two reverse_any subgraph_reverse reverse_any-> reverse_three.cpp reverse_checkpoint.cpp reverse_three.cpp Headings-> Taylor Coefficients

$\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }$
Third Order Reverse Mode: Example and Test

Taylor Coefficients
$$\begin{array}{rcl} X(t) & = & x^{(0)} + x^{(1)} t + x^{(2)} t^2 \\ X^{(1)} (t) & = & x^{(1)} + 2 x^{(2)} t \\ X^{(2)} (t) & = & 2 x^{(2)} \end{array}$$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/general/reverse_three.cpp