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Up-> CppAD AD ADValued atomic atomic_base atomic_tangent.cpp

ADValued-> Arithmetic unary_standard_math binary_math CondExp Discrete numeric_limits atomic

atomic-> checkpoint atomic_base

atomic_base-> atomic_ctor atomic_option atomic_afun atomic_forward atomic_reverse atomic_for_sparse_jac atomic_rev_sparse_jac atomic_for_sparse_hes atomic_rev_sparse_hes atomic_base_clear atomic_get_started.cpp atomic_norm_sq.cpp atomic_reciprocal.cpp atomic_set_sparsity.cpp atomic_tangent.cpp atomic_eigen_mat_mul.cpp atomic_eigen_mat_inv.cpp atomic_eigen_cholesky.cpp atomic_mat_mul.cpp

atomic_tangent.cpp

Headings-> Theory sparsity Start Class Definition Constructor forward reverse for_sparse_jac rev_sparse_jac rev_sparse_hes End Class Definition Use Atomic Function ---..Constructor ---..Recording ---..forward ---..reverse ---..for_sparse_jac ---..rev_sparse_jac ---..rev_sparse_hes ---..Large x Values

# include <cppad/cppad.hpp>
namespace { // Begin empty namespace
using CppAD::vector;
//
// a utility to compute the union of two sets.
using CppAD::set_union;
//
class atomic_tangent : public CppAD::atomic_base<float> {

private:
     const bool hyperbolic_; // is this hyperbolic tangent
public:
     // constructor
     atomic_tangent(const char* name, bool hyperbolic)
     : CppAD::atomic_base<float>(name),
     hyperbolic_(hyperbolic)
     { }
private:

     // forward mode routine called by CppAD
     bool forward(
          size_t                    p ,
          size_t                    q ,
          const vector<bool>&      vx ,
                vector<bool>&     vzy ,
          const vector<float>&     tx ,
                vector<float>&    tzy
     )
     {     size_t q1 = q + 1;
# ifndef NDEBUG
          size_t n  = tx.size()  / q1;
          size_t m  = tzy.size() / q1;
# endif
          assert( n == 1 );
          assert( m == 2 );
          assert( p <= q );
          size_t j, k;

          // check if this is during the call to old_tan(id, ax, ay)
          if( vx.size() > 0 )
          {     // set variable flag for both y an z
               vzy[0] = vx[0];
               vzy[1] = vx[0];
          }

          if( p == 0 )
          {     // z^{(0)} = tan( x^{(0)} ) or tanh( x^{(0)} )
               if( hyperbolic_ )
                    tzy[0] = float( tanh( tx[0] ) );
               else     tzy[0] = float( tan( tx[0] ) );

               // y^{(0)} = z^{(0)} * z^{(0)}
               tzy[q1 + 0] = tzy[0] * tzy[0];

               p++;
          }
          for(j = p; j <= q; j++)
          {     float j_inv = 1.f / float(j);
               if( hyperbolic_ )
                    j_inv = - j_inv;

               // z^{(j)} = x^{(j)} +- sum_{k=1}^j k x^{(k)} y^{(j-k)} / j
               tzy[j] = tx[j];
               for(k = 1; k <= j; k++)
                    tzy[j] += tx[k] * tzy[q1 + j-k] * float(k) * j_inv;

               // y^{(j)} = sum_{k=0}^j z^{(k)} z^{(j-k)}
               tzy[q1 + j] = 0.;
               for(k = 0; k <= j; k++)
                    tzy[q1 + j] += tzy[k] * tzy[j-k];
          }

          // All orders are implemented and there are no possible errors
          return true;
     }

     // reverse mode routine called by CppAD
     virtual bool reverse(
          size_t                    q ,
          const vector<float>&     tx ,
          const vector<float>&    tzy ,
                vector<float>&     px ,
          const vector<float>&    pzy
     )
     {     size_t q1 = q + 1;
# ifndef NDEBUG
          size_t n  = tx.size()  / q1;
          size_t m  = tzy.size() / q1;
# endif
          assert( px.size()  == n * q1 );
          assert( pzy.size() == m * q1 );
          assert( n == 1 );
          assert( m == 2 );

          size_t j, k;

          // copy because partials w.r.t. y and z need to change
          vector<float> qzy = pzy;

          // initialize accumultion of reverse mode partials
          for(k = 0; k < q1; k++)
               px[k] = 0.;

          // eliminate positive orders
          for(j = q; j > 0; j--)
          {     float j_inv = 1.f / float(j);
               if( hyperbolic_ )
                    j_inv = - j_inv;

               // H_{x^{(k)}} += delta(j-k) +- H_{z^{(j)} y^{(j-k)} * k / j
               px[j] += qzy[j];
               for(k = 1; k <= j; k++)
                    px[k] += qzy[j] * tzy[q1 + j-k] * float(k) * j_inv;

               // H_{y^{j-k)} += +- H_{z^{(j)} x^{(k)} * k / j
               for(k = 1; k <= j; k++)
                    qzy[q1 + j-k] += qzy[j] * tx[k] * float(k) * j_inv;

               // H_{z^{(k)}} += H_{y^{(j-1)}} * z^{(j-k-1)} * 2.
               for(k = 0; k < j; k++)
                    qzy[k] += qzy[q1 + j-1] * tzy[j-k-1] * 2.f;
          }

          // eliminate order zero
          if( hyperbolic_ )
               px[0] += qzy[0] * (1.f - tzy[q1 + 0]);
          else
               px[0] += qzy[0] * (1.f + tzy[q1 + 0]);

          return true;
     }

     // forward Jacobian sparsity routine called by CppAD
     virtual bool for_sparse_jac(
          size_t                                p ,
          const vector<bool>&                   r ,
                vector<bool>&                   s ,
          const vector<float>&                  x )
     {
# ifndef NDEBUG
          size_t n = r.size() / p;
          size_t m = s.size() / p;
# endif
          assert( n == x.size() );
          assert( n == 1 );
          assert( m == 2 );

          // sparsity for S(x) = f'(x) * R
          for(size_t j = 0; j < p; j++)
          {     s[0 * p + j] = r[j];
               s[1 * p + j] = r[j];
          }

          return true;
     }
     // forward Jacobian sparsity routine called by CppAD
     virtual bool for_sparse_jac(
          size_t                                p ,
          const vector< std::set<size_t> >&     r ,
                vector< std::set<size_t> >&     s ,
          const vector<float>&                  x )
     {
# ifndef NDEBUG
          size_t n = r.size();
          size_t m = s.size();
# endif
          assert( n == x.size() );
          assert( n == 1 );
          assert( m == 2 );

          // sparsity for S(x) = f'(x) * R
          s[0] = r[0];
          s[1] = r[0];

          return true;
     }

     // reverse Jacobian sparsity routine called by CppAD
     virtual bool rev_sparse_jac(
          size_t                                p ,
          const vector<bool>&                  rt ,
                vector<bool>&                  st ,
          const vector<float>&                  x )
     {
# ifndef NDEBUG
          size_t n = st.size() / p;
          size_t m = rt.size() / p;
# endif
          assert( n == 1 );
          assert( m == 2 );
          assert( n == x.size() );

          // sparsity for S(x)^T = f'(x)^T * R^T
          for(size_t j = 0; j < p; j++)
               st[j] = rt[0 * p + j] | rt[1 * p + j];

          return true;
     }
     // reverse Jacobian sparsity routine called by CppAD
     virtual bool rev_sparse_jac(
          size_t                                p ,
          const vector< std::set<size_t> >&    rt ,
                vector< std::set<size_t> >&    st ,
          const vector<float>&                  x )
     {
# ifndef NDEBUG
          size_t n = st.size();
          size_t m = rt.size();
# endif
          assert( n == 1 );
          assert( m == 2 );
          assert( n == x.size() );

          // sparsity for S(x)^T = f'(x)^T * R^T
          st[0] = set_union(rt[0], rt[1]);
          return true;
     }

     // reverse Hessian sparsity routine called by CppAD
     virtual bool rev_sparse_hes(
          const vector<bool>&                   vx,
          const vector<bool>&                   s ,
                vector<bool>&                   t ,
          size_t                                p ,
          const vector<bool>&                   r ,
          const vector<bool>&                   u ,
                vector<bool>&                   v ,
          const vector<float>&                  x )
     {
# ifndef NDEBUG
          size_t m = s.size();
          size_t n = t.size();
# endif
          assert( x.size() == n );
          assert( r.size() == n * p );
          assert( u.size() == m * p );
          assert( v.size() == n * p );
          assert( n == 1 );
          assert( m == 2 );

          // There are no cross term second derivatives for this case,
          // so it is not necessary to vx.

          // sparsity for T(x) = S(x) * f'(x)
          t[0] =  s[0] | s[1];

          // V(x) = f'(x)^T * g''(y) * f'(x) * R  +  g'(y) * f''(x) * R
          // U(x) = g''(y) * f'(x) * R
          // S(x) = g'(y)

          // back propagate the sparsity for U, note both components
          // of f'(x) may be non-zero;
          size_t j;
          for(j = 0; j < p; j++)
               v[j] = u[ 0 * p + j ] | u[ 1 * p + j ];

          // include forward Jacobian sparsity in Hessian sparsity
          // (note sparsty for f''(x) * R same as for R)
          if( s[0] | s[1] )
          {     for(j = 0; j < p; j++)
               {     // Visual Studio 2013 generates warning without bool below
                    v[j] |= bool( r[j] );
               }
          }

          return true;
     }
     // reverse Hessian sparsity routine called by CppAD
     virtual bool rev_sparse_hes(
          const vector<bool>&                   vx,
          const vector<bool>&                   s ,
                vector<bool>&                   t ,
          size_t                                p ,
          const vector< std::set<size_t> >&     r ,
          const vector< std::set<size_t> >&     u ,
                vector< std::set<size_t> >&     v ,
          const vector<float>&                  x )
     {
# ifndef NDEBUG
          size_t m = s.size();
          size_t n = t.size();
# endif
          assert( x.size() == n );
          assert( r.size() == n );
          assert( u.size() == m );
          assert( v.size() == n );
          assert( n == 1 );
          assert( m == 2 );

          // There are no cross term second derivatives for this case,
          // so it is not necessary to vx.

          // sparsity for T(x) = S(x) * f'(x)
          t[0] =  s[0] | s[1];

          // V(x) = f'(x)^T * g''(y) * f'(x) * R  +  g'(y) * f''(x) * R
          // U(x) = g''(y) * f'(x) * R
          // S(x) = g'(y)

          // back propagate the sparsity for U, note both components
          // of f'(x) may be non-zero;
          v[0] = set_union(u[0], u[1]);

          // include forward Jacobian sparsity in Hessian sparsity
          // (note sparsty for f''(x) * R same as for R)
          if( s[0] | s[1] )
               v[0] = set_union(v[0], r[0]);

          return true;
     }

}; // End of atomic_tangent class
}  // End empty namespace

bool tangent(void)
{     bool ok = true;
     using CppAD::AD;
     using CppAD::NearEqual;
     float eps = 10.f * CppAD::numeric_limits<float>::epsilon();

     // --------------------------------------------------------------------
     // Creater a tan and tanh object
     atomic_tangent my_tan("my_tan", false), my_tanh("my_tanh", true);

     // domain space vector
     size_t n  = 1;
     float  x0 = 0.5;
     CppAD::vector< AD<float> > ax(n);
     ax[0]     = x0;

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

     // range space vector
     size_t m = 3;
     CppAD::vector< AD<float> > af(m);

     // temporary vector for computations
     // (my_tan and my_tanh computes tan or tanh and its square)
     CppAD::vector< AD<float> > az(2);

     // call atomic tan function and store tan(x) in f[0] (ignore tan(x)^2)
     my_tan(ax, az);
     af[0] = az[0];

     // call atomic tanh function and store tanh(x) in f[1] (ignore tanh(x)^2)
     my_tanh(ax, az);
     af[1] = az[0];

     // put a constant in f[2] = tanh(1.) (for sparsity pattern testing)
     CppAD::vector< AD<float> > one(1);
     one[0] = 1.;
     my_tanh(one, az);
     af[2] = az[0];

     // create f: x -> f and stop tape recording
     CppAD::ADFun<float> F;
     F.Dependent(ax, af);

     // check function value
     float tan = std::tan(x0);
     ok &= NearEqual(af[0] , tan,  eps, eps);
     float tanh = std::tanh(x0);
     ok &= NearEqual(af[1] , tanh,  eps, eps);

     // check zero order forward
     CppAD::vector<float> x(n), f(m);
     x[0] = x0;
     f    = F.Forward(0, x);
     ok &= NearEqual(f[0] , tan,  eps, eps);
     ok &= NearEqual(f[1] , tanh,  eps, eps);

     // compute first partial of f w.r.t. x[0] using forward mode
     CppAD::vector<float> dx(n), df(m);
     dx[0] = 1.;
     df    = F.Forward(1, dx);

     // compute derivative of tan - tanh using reverse mode
     CppAD::vector<float> w(m), dw(n);
     w[0]  = 1.;
     w[1]  = 1.;
     w[2]  = 0.;
     dw    = F.Reverse(1, w);

     // tan'(x)   = 1 + tan(x)  * tan(x)
     // tanh'(x)  = 1 - tanh(x) * tanh(x)
     float tanp  = 1.f + tan * tan;
     float tanhp = 1.f - tanh * tanh;
     ok   &= NearEqual(df[0], tanp, eps, eps);
     ok   &= NearEqual(df[1], tanhp, eps, eps);
     ok   &= NearEqual(dw[0], w[0]*tanp + w[1]*tanhp, eps, eps);

     // compute second partial of f w.r.t. x[0] using forward mode
     CppAD::vector<float> ddx(n), ddf(m);
     ddx[0] = 0.;
     ddf    = F.Forward(2, ddx);

     // compute second derivative of tan - tanh using reverse mode
     CppAD::vector<float> ddw(2);
     ddw   = F.Reverse(2, w);

     // tan''(x)   = 2 *  tan(x) * tan'(x)
     // tanh''(x)  = - 2 * tanh(x) * tanh'(x)
     // Note that second order Taylor coefficient for u half the
     // corresponding second derivative.
     float two    = 2;
     float tanpp  =   two * tan * tanp;
     float tanhpp = - two * tanh * tanhp;
     ok   &= NearEqual(two * ddf[0], tanpp, eps, eps);
     ok   &= NearEqual(two * ddf[1], tanhpp, eps, eps);
     ok   &= NearEqual(ddw[0], w[0]*tanp  + w[1]*tanhp , eps, eps);
     ok   &= NearEqual(ddw[1], w[0]*tanpp + w[1]*tanhpp, eps, eps);

     // Forward mode computation of sparsity pattern for F.
     size_t p = n;
     // user vectorBool because m and n are small
     CppAD::vectorBool r1(p), s1(m * p);
     r1[0] = true;            // propagate sparsity for x[0]
     s1    = F.ForSparseJac(p, r1);
     ok  &= (s1[0] == true);  // f[0] depends on x[0]
     ok  &= (s1[1] == true);  // f[1] depends on x[0]
     ok  &= (s1[2] == false); // f[2] does not depend on x[0]

     // Reverse mode computation of sparsity pattern for F.
     size_t q = m;
     CppAD::vectorBool s2(q * m), r2(q * n);
     // Sparsity pattern for identity matrix
     size_t i, j;
     for(i = 0; i < q; i++)
     {     for(j = 0; j < m; j++)
               s2[i * q + j] = (i == j);
     }
     r2   = F.RevSparseJac(q, s2);
     ok  &= (r2[0] == true);  // f[0] depends on x[0]
     ok  &= (r2[1] == true);  // f[1] depends on x[0]
     ok  &= (r2[2] == false); // f[2] does not depend on x[0]

     // Hessian sparsity for f[0]
     CppAD::vectorBool s3(m), h(p * n);
     s3[0] = true;
     s3[1] = false;
     s3[2] = false;
     h    = F.RevSparseHes(p, s3);
     ok  &= (h[0] == true);  // Hessian is non-zero

     // Hessian sparsity for f[2]
     s3[0] = false;
     s3[2] = true;
     h    = F.RevSparseHes(p, s3);
     ok  &= (h[0] == false);  // Hessian is zero

     // check tanh results for a large value of x
     x[0]  = std::numeric_limits<float>::max() / two;
     f     = F.Forward(0, x);
     tanh  = 1.;
     ok   &= NearEqual(f[1], tanh, eps, eps);
     df    = F.Forward(1, dx);
     tanhp = 0.;
     ok   &= NearEqual(df[1], tanhp, eps, eps);

     return ok;
}