Hessian: Example and Test

hessian.cpp

Headings

@(@\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}} }@)@ This is cppad-20221105 documentation. Here is a link to its current documentation . Hessian: Example and Test


# include <cppad/cppad.hpp>
namespace { // ---------------------------------------------------------
// define the template function HessianCases<Vector> in empty namespace
template <class Vector>
bool HessianCases()
{   bool ok = true;
    using CppAD::AD;
    using CppAD::NearEqual;
    double eps99 = 99.0 * std::numeric_limits<double>::epsilon();
    using CppAD::exp;
    using CppAD::sin;
    using CppAD::cos;

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

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

    // a calculation between the domain and range values
    AD<double> Square = X[0] * X[0];

    // range space vector
    size_t m = 3;
    CPPAD_TESTVECTOR(AD<double>)  Y(m);
    Y[0] = Square * exp( X[1] );
    Y[1] = Square * sin( X[1] );
    Y[2] = Square * cos( X[1] );

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

    // new value for the independent variable vector
    Vector x(n);
    x[0] = 2.;
    x[1] = 1.;

    // second derivative of y[1]
    Vector hes( n * n );
    hes = f.Hessian(x, 1);
    /*
    F_1       = x[0] * x[0] * sin(x[1])

    F_1^{(1)} = [ 2 * x[0] * sin(x[1]) , x[0] * x[0] * cos(x[1]) ]

    F_1^{(2)} = [        2 * sin(x[1]) ,      2 * x[0] * cos(x[1]) ]
                [ 2 * x[0] * cos(x[1]) , - x[0] * x[0] * sin(x[1]) ]
    */
    ok &=  NearEqual(          2.*sin(x[1]), hes[0*n+0], eps99, eps99);
    ok &=  NearEqual(     2.*x[0]*cos(x[1]), hes[0*n+1], eps99, eps99);
    ok &=  NearEqual(     2.*x[0]*cos(x[1]), hes[1*n+0], eps99, eps99);
    ok &=  NearEqual( - x[0]*x[0]*sin(x[1]), hes[1*n+1], eps99, eps99);

    return ok;
}
} // End empty namespace
# include <vector>
# include <valarray>
bool Hessian(void)
{   bool ok = true;
    // Run with Vector equal to three different cases
    // all of which are Simple Vectors with elements of type double.
    ok &= HessianCases< CppAD::vector  <double> >();
    ok &= HessianCases< std::vector    <double> >();
    ok &= HessianCases< std::valarray  <double> >();
    return ok;
}

Input File: example/general/hessian.cpp