Atomic Function Jacobian Sparsity: Example and Test

atomic_three_jac_sparsity.cpp

@(@\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 . Atomic Function Jacobian Sparsity: Example and Test
Purpose
This example demonstrates calculation of a Jacobian sparsity pattern using an atomic operation.

Function
For this example, the atomic function @(@ g : \B{R}^3 \rightarrow \B{R}^2 @)@ is defined by @[@ g(x) = \left( \begin{array}{c} x_2 * x_2 \\ x_0 * x_1 \end{array} \right) @]@

Jacobian
The corresponding Jacobian is @[@ g^{(1)} (x) = \left( \begin{array}{ccc} 0 & 0 & 2 x_2 \\ x_1 & x_0 & 0 \end{array} \right) @]@

Start Class Definition

# include <cppad/cppad.hpp>
namespace {          // begin empty namespace
using CppAD::vector; // abbreviate CppAD::vector as vector
//
class atomic_jac_sparsity : public CppAD::atomic_three<double> {

Constructor

public:
    atomic_jac_sparsity(const std::string& name) :
    CppAD::atomic_three<double>(name)
    { }
private:

for_type

    // calculate type_y
    bool for_type(
        const vector<double>&               parameter_x ,
        const vector<CppAD::ad_type_enum>&  type_x      ,
        vector<CppAD::ad_type_enum>&        type_y      ) override
    {   assert( parameter_x.size() == type_x.size() );
        bool ok = type_x.size() == 3; // n
        ok     &= type_y.size() == 2; // m
        if( ! ok )
            return false;

        type_y[0]  = type_x[2];
        type_y[1] = std::max(type_x[0], type_x[1]);
        return true;
    }

forward

    // forward mode routine called by CppAD
    bool forward(
        const vector<double>&              parameter_x  ,
        const vector<CppAD::ad_type_enum>& type_x       ,
        size_t                             need_y       ,
        size_t                             order_low    ,
        size_t                             order_up     ,
        const vector<double>&              taylor_x     ,
        vector<double>&                    taylor_y     ) override
    {
# ifndef NDEBUG
        size_t n = taylor_x.size() / (order_up + 1);
        size_t m = taylor_y.size() / (order_up + 1);
# endif
        assert( n == 3 );
        assert( m == 2 );
        assert( order_low <= order_up );

        // return flag
        bool ok = order_up == 0;
        if( ! ok )
            return ok;

        // Order zero forward mode must always be implemented
        taylor_y[0] = taylor_x[2] * taylor_x[2];
        taylor_y[1] = taylor_x[0] * taylor_x[1];

        return ok;
    }

jac_sparsity

    // Jacobian sparsity routine called by CppAD
    bool jac_sparsity(
        const vector<double>&               parameter_x ,
        const vector<CppAD::ad_type_enum>&  type_x      ,
        bool                                dependency  ,
        const vector<bool>&                 select_x    ,
        const vector<bool>&                 select_y    ,
        CppAD::sparse_rc< vector<size_t> >& pattern_out ) override
    {
        size_t n = select_x.size();
        size_t m = select_y.size();
        assert( parameter_x.size() == n );
        assert( n == 3 );
        assert( m == 2 );

        // count number of non-zeros in sparsity pattern
        size_t nnz = 0;
        // row 0
        if( select_y[0] & select_x[2] )
            ++nnz;
        // row 1
        if( select_y[1] )
        {   // column 0
            if( select_x[0] )
                ++nnz;
            // column 1
            if( select_x[1] )
                ++nnz;
        }

        // size of pattern_out
        size_t nr = m;
        size_t nc = n;
        pattern_out.resize(nr, nc, nnz);

        // set the values in pattern_out using index k
        size_t k = 0;

        // y_0 depends and has possibly non-zeron partial w.r.t x_2
        if( select_y[0] & select_x[2] )
            pattern_out.set(k++, 0, 2);
        if( select_y[1] )
        {   // y_1 depends and has possibly non-zero partial w.r.t x_0
            if( select_x[0] )
                pattern_out.set(k++, 1, 0);
            // y_1 depends and has possibly non-zero partial w.r.t x_1
            if( select_x[1] )
                pattern_out.set(k++, 1, 1);
        }
        assert( k == nnz );
        //
        return true;
    }
}; // End of atomic_three_jac_sparsity class

Use Atomic Function

bool use_jac_sparsity(bool x_1_variable, bool forward)
{   bool ok = true;
    using CppAD::AD;
    using CppAD::NearEqual;
    double eps = 10. * CppAD::numeric_limits<double>::epsilon();
    //
    // Create the atomic_jac_sparsity object correspnding to g(x)
    atomic_jac_sparsity afun("atomic_jac_sparsity");
    //
    // Create the function f(u) = g(u) for this example.
    //
    // domain space vector
    size_t n  = 3;
    double u_0 = 1.00;
    double u_1 = 2.00;
    double u_2 = 3.00;
    vector< AD<double> > au(n);
    au[0] = u_0;
    au[1] = u_1;
    au[2] = u_2;

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

    // range space vector
    size_t m = 2;
    vector< AD<double> > ay(m);

    // call atomic function
    vector< AD<double> > ax(n);
    ax[0] = au[0];
    ax[2] = au[2];
    if( x_1_variable )
    {   ok   &= Variable( au[1] );
        ax[1] = au[1];
    }
    else
    {   AD<double> ap = u_1;
        ok   &= Parameter(ap);
        ok   &= ap == au[1];
        ax[1] = u_1;
    }
    // x_1_variable true:  y = [ u_2 * u_2 ,  u_0 * u_1 ]^T
    // x_1_variable false: y = [ u_2 * u_2 ,  u_0 * p   ]^T
    afun(ax, ay);

    // create f: u -> y and stop tape recording
    CppAD::ADFun<double> f;
    f.Dependent (au, ay);  // f(u) = y
    //
    // check function value
    double check = u_2 * u_2;
    ok &= NearEqual( Value(ay[0]) , check,  eps, eps);
    check = u_0 * u_1;
    ok &= NearEqual( Value(ay[1]) , check,  eps, eps);

    // check zero order forward mode
    size_t q;
    vector<double> xq(n), yq(m);
    q     = 0;
    xq[0] = u_0;
    xq[1] = u_1;
    xq[2] = u_2;
    yq    = f.Forward(q, xq);
    check = u_2 * u_2;
    ok &= NearEqual(yq[0] , check,  eps, eps);
    check = u_0 * u_1;
    ok &= NearEqual(yq[1] , check,  eps, eps);

    // sparsity pattern for identity matrix
    size_t nnz;
    if( forward )
        nnz = n;
    else
        nnz = m;
    CppAD::sparse_rc< CPPAD_TESTVECTOR(size_t) > pattern_in(nnz, nnz, nnz);
    for(size_t k = 0; k < nnz; ++k)
        pattern_in.set(k, k, k);

    // Jacobian sparsity for f(u)
    bool transpose     = false;
    bool dependency    = false;
    bool internal_bool = false;
    CppAD::sparse_rc< CPPAD_TESTVECTOR(size_t) > pattern_out;
    if( forward )
    {   f.for_jac_sparsity(
            pattern_in, transpose, dependency, internal_bool, pattern_out
        );
    }
    else
    {   f.rev_jac_sparsity(
            pattern_in, transpose, dependency, internal_bool, pattern_out
        );
    }
    const CPPAD_TESTVECTOR(size_t)& row = pattern_out.row();
    const CPPAD_TESTVECTOR(size_t)& col = pattern_out.col();
    CPPAD_TESTVECTOR(size_t) row_major  = pattern_out.row_major();
    //
    // first element in row major order has index (0, 2)
    size_t k = 0;
    size_t r = row[ row_major[k] ];
    size_t c = col[ row_major[k] ];
    ok      &= r == 0 && c == 2;
    //
    // second element in row major order has index (1, 0)
    ++k;
    r   = row[ row_major[k] ];
    c   = col[ row_major[k] ];
    ok &= r == 1 && c == 0;
    //
    if( x_1_variable )
    {   // third element in row major order has index (1, 1)
        ++k;
        r   = row[ row_major[k] ];
        c   = col[ row_major[k] ];
        ok &= r == 1 && c == 1;
    }
    // k + 1 should be the number of values in sparsity pattern
    ok &= k + 1 == pattern_out.nnz();
    //
    return ok;
}
}  // End empty namespace

Test with u_1 Both a Variable and a Parameter

bool jac_sparsity(void)
{   bool ok           = true;
    //
    bool u_1_variable = true;
    bool forward      = true;
    ok               &= use_jac_sparsity(u_1_variable, forward);
    //
    u_1_variable      = true;
    forward           = false;
    ok               &= use_jac_sparsity(u_1_variable, forward);
    //
    u_1_variable      = false;
    forward           = true;
    ok               &= use_jac_sparsity(u_1_variable, forward);
    //
    u_1_variable      = false;
    forward           = false;
    ok               &= use_jac_sparsity(u_1_variable, forward);
    //
    return ok;
}

Input File: example/atomic_three/jac_sparsity.cpp