$\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }$
Using Adolc with Multiple Levels of Taping: Example and Test

Purpose
In this example, we use AD< adouble> > (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 Adolc's adouble (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.cpp computes the same values using AD< AD<double> > and AD<double>.

Memory Management
Adolc uses raw memory arrays that depend on the number of dependent and independent variables. The memory management utility thread_alloc is used to manage this memory allocation.

Configuration Requirement
This example will be compiled and tested provided that the value adolc_prefix is specified on the cmake command line.

Source  // suppress conversion warnings before other includes # include <cppad/wno_conversion.hpp> // # include <adolc/adouble.h> # include <adolc/taping.h> # include <adolc/interfaces.h> // adouble definitions not in Adolc distribution and // required in order to use CppAD::AD<adouble> # include <cppad/example/base_adolc.hpp> # 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_adolc(void) { bool ok = true; // initialize test result using CppAD::thread_alloc; // The CppAD memory allocator typedef adouble a1type; // for first level of taping typedef CppAD::AD<a1type> a2type; // for second level of taping size_t n = 5; // number independent variables size_t j; // 10 times machine epsilon double eps = 10. * std::numeric_limits<double>::epsilon(); CPPAD_TESTVECTOR(double) x(n); CPPAD_TESTVECTOR(a1type) a1x(n); CPPAD_TESTVECTOR(a2type) a2x(n); // Values for the independent variables while taping the function f(x) for(j = 0; j < n; j++) a2x[j] = double(j); // Declare the independent variable for taping f(x) CppAD::Independent(a2x); // Use AD<adouble> to tape the evaluation of f(x) CPPAD_TESTVECTOR(a2type) a2y(1); a2y[0] = f(a2x); // Declare a1f as the corresponding ADFun<adouble> function f(x) // (make sure we do not run zero order forward during constructor) CppAD::ADFun<a1type> a1f; a1f.Dependent(a2x, a2y); // Value of the independent variables whitle taping f'(x) * v int tag = 0; int keep = 1; trace_on(tag, keep); for(j = 0; j < n; j++) a1x[j] <<= double(j); // set the argument value x for computing f'(x) * v a1f.Forward(0, a1x); // compute f'(x) * v CPPAD_TESTVECTOR(a1type) a1v(n); CPPAD_TESTVECTOR(a1type) a1df(1); for(j = 0; j < n; j++) a1v[j] = double(n - j); a1df = a1f.Forward(1, a1v); // declare Adolc function corresponding to f'(x) * v double df; a1df[0] >>= df; trace_off(); // compute the d/dx of f'(x) * v = f''(x) * v size_t m = 1; // # dependent in f'(x) * v // w = new double[capacity] where capacity >= m size_t capacity; double* w = thread_alloc::create_array<double>(m, capacity); // dw = new double[capacity] where capacity >= n double* dw = thread_alloc::create_array<double>(n, capacity); w[0] = 1.; fos_reverse(tag, int(m), int(n), w, dw); for(j = 0; j < n; j++) { double vj = a1v[j].value(); ok &= CppAD::NearEqual(dw[j], vj, eps, eps); } // make memory avaialble for other use by this thread thread_alloc::delete_array(w); thread_alloc::delete_array(dw); return ok; }