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colpack_hes.cpp

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# include <cppad/cppad.hpp>
bool colpack_hes(void)
{   bool ok = true;
    using CppAD::AD;
    using CppAD::NearEqual;
    typedef CPPAD_TESTVECTOR(AD<double>)            a_vector;
    typedef CPPAD_TESTVECTOR(double)                d_vector;
    typedef CppAD::vector<size_t>                   i_vector;
    typedef CppAD::sparse_rc<i_vector>              sparsity;
    typedef CppAD::sparse_rcv<i_vector, d_vector>   sparse_matrix;
    double eps = 10. * CppAD::numeric_limits<double>::epsilon();
    //
    // domain space vector
    size_t n = 5;
    a_vector  a_x(n);
    for(size_t j = 0; j < n; j++)
        a_x[j] = AD<double> (0);
    //
    // declare independent variables and starting recording
    CppAD::Independent(a_x);

    // colpack example case where hessian is a spear head
    // i.e, H(i, j) non zero implies i = 0, j = 0, or i = j
    AD<double> sum = 0.0;
    // partial_0 partial_j = x[j]
    // partial_j partial_j = x[0]
    for(size_t j = 1; j < n; j++)
        sum += a_x[0] * a_x[j] * a_x[j] / 2.0;
    //
    // partial_i partial_i = 2 * x[i]
    for(size_t i = 0; i < n; i++)
        sum += a_x[i] * a_x[i] * a_x[i] / 3.0;

    // declare dependent variables
    size_t m = 1;
    a_vector  a_y(m);
    a_y[0] = sum;

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

    // new value for the independent variable vector
    d_vector x(n);
    for(size_t j = 0; j < n; j++)
        x[j] = double(j + 1);

    /*
          [ 2  2  3  4  5 ]
    hes = [ 2  5  0  0  0 ]
          [ 3  0  7  0  0 ]
          [ 4  0  0  9  0 ]
          [ 5  0  0  0 11 ]
    */
    // Normally one would use CppAD to compute sparsity pattern, but for this
    // example we set it directly
    size_t nr  = n;
    size_t nc  = n;
    size_t nnz = n + 2 * (n - 1);
    sparsity pattern(nr, nc, nnz);
    for(size_t k = 0; k < n; k++)
    {   size_t r = k;
        size_t c = k;
        pattern.set(k, r, c);
    }
    for(size_t i = 1; i < n; i++)
    {   size_t k = n + 2 * (i - 1);
        size_t r = i;
        size_t c = 0;
        pattern.set(k,   r, c);
        pattern.set(k+1, c, r);
    }

    // subset of elements to compute
    // (only compute lower traingle)
    nnz = n + (n - 1);
    sparsity lower_triangle(nr, nc, nnz);
    d_vector check(nnz);
    for(size_t k = 0; k < n; k++)
    {   size_t r = k;
        size_t c = k;
        lower_triangle.set(k, r, c);
        check[k] = 2.0 * x[k];
        if( k > 0 )
            check[k] += x[0];
    }
    for(size_t j = 1; j < n; j++)
    {   size_t k = n + (j - 1);
        size_t r = 0;
        size_t c = j;
        lower_triangle.set(k, r, c);
        check[k] = x[c];
    }
    sparse_matrix subset( lower_triangle );

    // check results for both CppAD and Colpack
    for(size_t i_method = 0; i_method < 4; i_method++)
    {   // coloring method
        std::string coloring;
        switch(i_method)
        {   case 0:
            coloring = "cppad.symmetric";
            break;

            case 1:
            coloring = "cppad.general";
            break;

            case 2:
            coloring = "colpack.symmetric";
            break;

            case 3:
            coloring = "colpack.general";
            break;
        }
        //
        // compute Hessian
        CppAD::sparse_hes_work work;
        d_vector w(m);
        w[0] = 1.0;
        size_t n_sweep = f.sparse_hes(
            x, w, subset, pattern, coloring, work
        );
        //
        // check result
        const d_vector& hes( subset.val() );
        for(size_t k = 0; k < nnz; k++)
            ok &= NearEqual(check[k], hes[k], eps, eps);
        if(
            coloring == "cppad.symmetric"
        ||  coloring == "colpack.symmetric"
        )
            ok &= n_sweep == 2;
        else
            ok &= n_sweep == 5;
    }

    return ok;
}