Reverse Mode Hessian Sparsity: Example and Test

rev_sparse_hes.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 . Reverse Mode Hessian Sparsity: Example and Test


# include <cppad/cppad.hpp>
namespace { // -------------------------------------------------------------

// expected sparsity pattern
bool check_f0[] = {
    false, false, false,  // partials w.r.t x0 and (x0, x1, x2)
    false, false, false,  // partials w.r.t x1 and (x0, x1, x2)
    false, false, true    // partials w.r.t x2 and (x0, x1, x2)
};
bool check_f1[] = {
    false,  true, false,  // partials w.r.t x0 and (x0, x1, x2)
    true,  false, false,  // partials w.r.t x1 and (x0, x1, x2)
    false, false, false   // partials w.r.t x2 and (x0, x1, x2)
};

// define the template function BoolCases<Vector> in empty namespace
template <class Vector> // vector class, elements of type bool
bool BoolCases(void)
{   bool ok = true;
    using CppAD::AD;

    // domain space vector
    size_t n = 3;
    CPPAD_TESTVECTOR(AD<double>) ax(n);
    ax[0] = 0.;
    ax[1] = 1.;
    ax[2] = 2.;

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

    // range space vector
    size_t m = 2;
    CPPAD_TESTVECTOR(AD<double>) ay(m);
    ay[0] = sin( ax[2] );
    ay[1] = ax[0] * ax[1];

    // create f: x -> y and stop tape recording
    CppAD::ADFun<double> f(ax, ay);

    // sparsity pattern for the identity matrix
    Vector r(n * n);
    size_t i, j;
    for(i = 0; i < n; i++)
    {   for(j = 0; j < n; j++)
            r[ i * n + j ] = (i == j);
    }

    // compute sparsity pattern for J(x) = F^{(1)} (x)
    f.ForSparseJac(n, r);

    // compute sparsity pattern for H(x) = F_0^{(2)} (x)
    Vector s(m);
    for(i = 0; i < m; i++)
        s[i] = false;
    s[0] = true;
    Vector h(n * n);
    h    = f.RevSparseHes(n, s);

    // check values
    for(i = 0; i < n; i++)
        for(j = 0; j < n; j++)
            ok &= (h[ i * n + j ] == check_f0[ i * n + j ] );

    // compute sparsity pattern for H(x) = F_1^{(2)} (x)
    for(i = 0; i < m; i++)
        s[i] = false;
    s[1] = true;
    h    = f.RevSparseHes(n, s);

    // check values
    for(i = 0; i < n; i++)
        for(j = 0; j < n; j++)
            ok &= (h[ i * n + j ] == check_f1[ i * n + j ] );

    // call that transposed the result
    bool transpose = true;
    h    = f.RevSparseHes(n, s, transpose);

    // This h is symmetric, because R is symmetric, not really testing here
    for(i = 0; i < n; i++)
        for(j = 0; j < n; j++)
            ok &= (h[ j * n + i ] == check_f1[ i * n + j ] );

    return ok;
}
// define the template function SetCases<Vector> in empty namespace
template <class Vector> // vector class, elements of type std::set<size_t>
bool SetCases(void)
{   bool ok = true;
    using CppAD::AD;

    // domain space vector
    size_t n = 3;
    CPPAD_TESTVECTOR(AD<double>) ax(n);
    ax[0] = 0.;
    ax[1] = 1.;
    ax[2] = 2.;

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

    // range space vector
    size_t m = 2;
    CPPAD_TESTVECTOR(AD<double>) ay(m);
    ay[0] = sin( ax[2] );
    ay[1] = ax[0] * ax[1];

    // create f: x -> y and stop tape recording
    CppAD::ADFun<double> f(ax, ay);

    // sparsity pattern for the identity matrix
    Vector r(n);
    size_t i;
    for(i = 0; i < n; i++)
    {   assert( r[i].empty() );
        r[i].insert(i);
    }

    // compute sparsity pattern for J(x) = F^{(1)} (x)
    f.ForSparseJac(n, r);

    // compute sparsity pattern for H(x) = F_0^{(2)} (x)
    Vector s(1);
    assert( s[0].empty() );
    s[0].insert(0);
    Vector h(n);
    h    = f.RevSparseHes(n, s);

    // check values
    std::set<size_t>::iterator itr;
    size_t j;
    for(i = 0; i < n; i++)
    {   for(j = 0; j < n; j++)
        {   bool found = h[i].find(j) != h[i].end();
            ok        &= (found == check_f0[i * n + j]);
        }
    }

    // compute sparsity pattern for H(x) = F_1^{(2)} (x)
    s[0].clear();
    assert( s[0].empty() );
    s[0].insert(1);
    h    = f.RevSparseHes(n, s);

    // check values
    for(i = 0; i < n; i++)
    {   for(j = 0; j < n; j++)
        {   bool found = h[i].find(j) != h[i].end();
            ok        &= (found == check_f1[i * n + j]);
        }
    }

    // call that transposed the result
    bool transpose = true;
    h    = f.RevSparseHes(n, s, transpose);

    // This h is symmetric, because R is symmetric, not really testing here
    for(i = 0; i < n; i++)
    {   for(j = 0; j < n; j++)
        {   bool found = h[j].find(i) != h[j].end();
            ok        &= (found == check_f1[i * n + j]);
        }
    }

    return ok;
}
} // End empty namespace

# include <vector>
# include <valarray>
bool rev_sparse_hes(void)
{   bool ok = true;
    // Run with Vector equal to four different cases
    // all of which are Simple Vectors with elements of type bool.
    ok &= BoolCases< CppAD::vector  <bool> >();
    ok &= BoolCases< CppAD::vectorBool     >();
    ok &= BoolCases< std::vector    <bool> >();
    ok &= BoolCases< std::valarray  <bool> >();

    // Run with Vector equal to two different cases both of which are
    // Simple Vectors with elements of type std::set<size_t>
    typedef std::set<size_t> set;
    ok &= SetCases< CppAD::vector  <set> >();
    ok &= SetCases< std::vector    <set> >();

    // Do not use valarray because its element access in the const case
    // returns a copy instead of a reference
    // ok &= SetCases< std::valarray  <set> >();

    return ok;
}

Input File: example/sparse/rev_sparse_hes.cpp