exp_eps_cppad¶

exp_eps: CppAD Forward and Reverse Sweeps¶

Purpose¶

Use CppAD forward and reverse modes to compute the partial derivative with respect to \(x\), at the point \(x = .5\) and \(\varepsilon = .2\), of the function

exp_eps ( x , epsilon )

as defined by the exp_eps.hpp include file.

Exercises¶

Create and test a modified version of the routine below that computes the same order derivatives with respect to \(x\), at the point \(x = .1\) and \(\varepsilon = .2\), of the function

exp_eps ( x , epsilon )
Create and test a modified version of the routine below that computes partial derivative with respect to \(x\), at the point \(x = .1\) and \(\varepsilon = .2\), of the function corresponding to the operation sequence for \(x = .5\) and \(\varepsilon = .2\). Hint: you could define a vector u with two components and use

f . Forward (0, u )

to run zero order forward mode at a point different form the point where the operation sequence corresponding to f was recorded.

# include <cppad/cppad.hpp>  // https://www.coin-or.org/CppAD/
# include "exp_eps.hpp"      // our example exponential function approximation
bool exp_eps_cppad(void)
{  bool ok = true;
    using CppAD::AD;
    using CppAD::vector;    // can use any simple vector template class
    using CppAD::NearEqual; // checks if values are nearly equal

    // domain space vector
    size_t n = 2; // dimension of the domain space
    vector< AD<double> > U(n);
    U[0] = .5;    // value of x for this operation sequence
    U[1] = .2;    // value of e for this operation sequence

    // declare independent variables and start recording operation sequence
    CppAD::Independent(U);

    // evaluate our exponential approximation
    AD<double> x       = U[0];
    AD<double> epsilon = U[1];
    AD<double> apx = exp_eps(x, epsilon);

    // range space vector
    size_t m = 1;  // dimension of the range space
    vector< AD<double> > Y(m);
    Y[0] = apx;    // variable that represents only range space component

    // Create f: U -> Y corresponding to this operation sequence
    // and stop recording. This also executes a zero order forward
    // mode sweep using values in U for x and e.
    CppAD::ADFun<double> f(U, Y);

    // first order forward mode sweep that computes partial w.r.t x
    vector<double> du(n);      // differential in domain space
    vector<double> dy(m);      // differential in range space
    du[0] = 1.;                // x direction in domain space
    du[1] = 0.;
    dy    = f.Forward(1, du);  // partial w.r.t. x
    double check = 1.5;
    ok   &= NearEqual(dy[0], check, 1e-10, 1e-10);

    // first order reverse mode sweep that computes the derivative
    vector<double>  w(m);     // weights for components of the range
    vector<double> dw(n);     // derivative of the weighted function
    w[0] = 1.;                // there is only one weight
    dw   = f.Reverse(1, w);   // derivative of w[0] * exp_eps(x, epsilon)
    check = 1.5;              // partial w.r.t. x
    ok   &= NearEqual(dw[0], check, 1e-10, 1e-10);
    check = 0.;               // partial w.r.t. epsilon
    ok   &= NearEqual(dw[1], check, 1e-10, 1e-10);

    // second order forward sweep that computes
    // second partial of exp_eps(x, epsilon) w.r.t. x
    vector<double> x2(n);     // second order Taylor coefficients
    vector<double> y2(m);
    x2[0] = 0.;               // evaluate partial w.r.t x
    x2[1] = 0.;
    y2    = f.Forward(2, x2);
    check = 0.5 * 1.;         // Taylor coef is 1/2 second derivative
    ok   &= NearEqual(y2[0], check, 1e-10, 1e-10);

    // second order reverse sweep that computes
    // derivative of partial of exp_eps(x, epsilon) w.r.t. x
    dw.resize(2 * n);         // space for first and second derivative
    dw    = f.Reverse(2, w);
    check = 1.;               // result should be second derivative
    ok   &= NearEqual(dw[0*2+1], check, 1e-10, 1e-10);

    return ok;
}