atan_op.hpp

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# ifndef CPPAD_LOCAL_ATAN_OP_HPP
# define CPPAD_LOCAL_ATAN_OP_HPP

/* --------------------------------------------------------------------------
CppAD: C++ Algorithmic Differentiation: Copyright (C) 2003-17 Bradley M. Bell

CppAD is distributed under multiple licenses. This distribution is under
the terms of the
                    Eclipse Public License Version 1.0.

A copy of this license is included in the COPYING file of this distribution.
Please visit http://www.coin-or.org/CppAD/ for information on other licenses.
-------------------------------------------------------------------------- */


namespace CppAD { namespace local { // BEGIN_CPPAD_LOCAL_NAMESPACE
/*!
\file atan_op.hpp
Forward and reverse mode calculations for z = atan(x).
*/


/*!
Forward mode Taylor coefficient for result of op = AtanOp.

The C++ source code corresponding to this operation is
\verbatim
     z = atan(x)
\endverbatim
The auxillary result is
\verbatim
     y = 1 + x * x
\endverbatim
The value of y, and its derivatives, are computed along with the value
and derivatives of z.

\copydetails CppAD::local::forward_unary2_op
*/
template <class Base>
inline void forward_atan_op(
     size_t p           ,
     size_t q           ,
     size_t i_z         ,
     size_t i_x         ,
     size_t cap_order   ,
     Base*  taylor      )
{
     // check assumptions
     CPPAD_ASSERT_UNKNOWN( NumArg(AtanOp) == 1 );
     CPPAD_ASSERT_UNKNOWN( NumRes(AtanOp) == 2 );
     CPPAD_ASSERT_UNKNOWN( q < cap_order );
     CPPAD_ASSERT_UNKNOWN( p <= q );

     // Taylor coefficients corresponding to argument and result
     Base* x = taylor + i_x * cap_order;
     Base* z = taylor + i_z * cap_order;
     Base* b = z      -       cap_order;  // called y in documentation

     size_t k;
     if( p == 0 )
     {    z[0] = atan( x[0] );
          b[0] = Base(1.0) + x[0] * x[0];
          p++;
     }
     for(size_t j = p; j <= q; j++)
     {
          b[j] = Base(2.0) * x[0] * x[j];
          z[j] = Base(0.0);
          for(k = 1; k < j; k++)
          {    b[j] += x[k] * x[j-k];
               z[j] -= Base(double(k)) * z[k] * b[j-k];
          }
          z[j] /= Base(double(j));
          z[j] += x[j];
          z[j] /= b[0];
     }
}

/*!
Multiple direction Taylor coefficient for op = AtanOp.

The C++ source code corresponding to this operation is
\verbatim
     z = atan(x)
\endverbatim
The auxillary result is
\verbatim
     y = 1 + x * x
\endverbatim
The value of y, and its derivatives, are computed along with the value
and derivatives of z.

\copydetails CppAD::local::forward_unary2_op_dir
*/
template <class Base>
inline void forward_atan_op_dir(
     size_t q           ,
     size_t r           ,
     size_t i_z         ,
     size_t i_x         ,
     size_t cap_order   ,
     Base*  taylor      )
{
     // check assumptions
     CPPAD_ASSERT_UNKNOWN( NumArg(AtanOp) == 1 );
     CPPAD_ASSERT_UNKNOWN( NumRes(AtanOp) == 2 );
     CPPAD_ASSERT_UNKNOWN( 0 < q );
     CPPAD_ASSERT_UNKNOWN( q < cap_order );

     // Taylor coefficients corresponding to argument and result
     size_t num_taylor_per_var = (cap_order-1) * r + 1;
     Base* x = taylor + i_x * num_taylor_per_var;
     Base* z = taylor + i_z * num_taylor_per_var;
     Base* b = z      -       num_taylor_per_var; // called y in documentation

     size_t m = (q-1) * r + 1;
     for(size_t ell = 0; ell < r; ell++)
     {    b[m+ell] = Base(2.0) * x[m+ell] * x[0];
          z[m+ell] = Base(double(q)) * x[m+ell];
          for(size_t k = 1; k < q; k++)
          {    b[m+ell] += x[(k-1)*r+1+ell] * x[(q-k-1)*r+1+ell];
               z[m+ell] -= Base(double(k)) * z[(k-1)*r+1+ell] * b[(q-k-1)*r+1+ell];
          }
          z[m+ell] /= ( Base(double(q)) * b[0] );
     }
}

/*!
Zero order forward mode Taylor coefficient for result of op = AtanOp.

The C++ source code corresponding to this operation is
\verbatim
     z = atan(x)
\endverbatim
The auxillary result is
\verbatim
     y = 1 + x * x
\endverbatim
The value of y is computed along with the value of z.

\copydetails CppAD::local::forward_unary2_op_0
*/
template <class Base>
inline void forward_atan_op_0(
     size_t i_z         ,
     size_t i_x         ,
     size_t cap_order   ,
     Base*  taylor      )
{
     // check assumptions
     CPPAD_ASSERT_UNKNOWN( NumArg(AtanOp) == 1 );
     CPPAD_ASSERT_UNKNOWN( NumRes(AtanOp) == 2 );
     CPPAD_ASSERT_UNKNOWN( 0 < cap_order );

     // Taylor coefficients corresponding to argument and result
     Base* x = taylor + i_x * cap_order;
     Base* z = taylor + i_z * cap_order;
     Base* b = z      -       cap_order; // called y in documentation

     z[0] = atan( x[0] );
     b[0] = Base(1.0) + x[0] * x[0];
}
/*!
Reverse mode partial derivatives for result of op = AtanOp.

The C++ source code corresponding to this operation is
\verbatim
     z = atan(x)
\endverbatim
The auxillary result is
\verbatim
     y = 1 + x * x
\endverbatim
The value of y is computed along with the value of z.

\copydetails CppAD::local::reverse_unary2_op
*/

template <class Base>
inline void reverse_atan_op(
     size_t      d            ,
     size_t      i_z          ,
     size_t      i_x          ,
     size_t      cap_order    ,
     const Base* taylor       ,
     size_t      nc_partial   ,
     Base*       partial      )
{
     // check assumptions
     CPPAD_ASSERT_UNKNOWN( NumArg(AtanOp) == 1 );
     CPPAD_ASSERT_UNKNOWN( NumRes(AtanOp) == 2 );
     CPPAD_ASSERT_UNKNOWN( d < cap_order );
     CPPAD_ASSERT_UNKNOWN( d < nc_partial );

     // Taylor coefficients and partials corresponding to argument
     const Base* x  = taylor  + i_x * cap_order;
     Base* px       = partial + i_x * nc_partial;

     // Taylor coefficients and partials corresponding to first result
     const Base* z  = taylor  + i_z * cap_order;
     Base* pz       = partial + i_z * nc_partial;

     // Taylor coefficients and partials corresponding to auxillary result
     const Base* b  = z  - cap_order; // called y in documentation
     Base* pb       = pz - nc_partial;

     Base inv_b0 = Base(1.0) / b[0];

     // number of indices to access
     size_t j = d;
     size_t k;
     while(j)
     {    // scale partials w.r.t z[j] and b[j]
          pz[j]  = azmul(pz[j], inv_b0);
          pb[j] *= Base(2.0);

          pb[0] -= azmul(pz[j], z[j]);
          px[j] += pz[j] + azmul(pb[j], x[0]);
          px[0] += azmul(pb[j], x[j]);

          // more scaling of partials w.r.t z[j]
          pz[j] /= Base(double(j));

          for(k = 1; k < j; k++)
          {    pb[j-k] -= Base(double(k)) * azmul(pz[j], z[k]);
               pz[k]   -= Base(double(k)) * azmul(pz[j], b[j-k]);
               px[k]   += azmul(pb[j], x[j-k]);
          }
          --j;
     }
     px[0] += azmul(pz[0], inv_b0) + Base(2.0) * azmul(pb[0], x[0]);
}

} } // END_CPPAD_LOCAL_NAMESPACE
# endif