CppAD: A C++ Algorithmic Differentiation Package  20130918
atan_op.hpp
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00001 /* $Id$ */
00002 # ifndef CPPAD_ATAN_OP_INCLUDED
00003 # define CPPAD_ATAN_OP_INCLUDED
00004 
00005 /* --------------------------------------------------------------------------
00006 CppAD: C++ Algorithmic Differentiation: Copyright (C) 2003-14 Bradley M. Bell
00007 
00008 CppAD is distributed under multiple licenses. This distribution is under
00009 the terms of the 
00010                     Eclipse Public License Version 1.0.
00011 
00012 A copy of this license is included in the COPYING file of this distribution.
00013 Please visit http://www.coin-or.org/CppAD/ for information on other licenses.
00014 -------------------------------------------------------------------------- */
00015 
00016 
00017 namespace CppAD { // BEGIN_CPPAD_NAMESPACE
00018 /*!
00019 \file atan_op.hpp
00020 Forward and reverse mode calculations for z = atan(x).
00021 */
00022 
00023 
00024 /*!
00025 Forward mode Taylor coefficient for result of op = AtanOp.
00026 
00027 The C++ source code corresponding to this operation is
00028 \verbatim
00029      z = atan(x)
00030 \endverbatim
00031 The auxillary result is
00032 \verbatim
00033      y = 1 + x * x
00034 \endverbatim
00035 The value of y, and its derivatives, are computed along with the value
00036 and derivatives of z.
00037 
00038 \copydetails forward_unary2_op
00039 */
00040 template <class Base>
00041 inline void forward_atan_op(
00042      size_t p           ,
00043      size_t q           ,
00044      size_t i_z         ,
00045      size_t i_x         ,
00046      size_t nc_taylor   , 
00047      Base*  taylor      )
00048 {    
00049      // check assumptions
00050      CPPAD_ASSERT_UNKNOWN( NumArg(AtanOp) == 1 );
00051      CPPAD_ASSERT_UNKNOWN( NumRes(AtanOp) == 2 );
00052      CPPAD_ASSERT_UNKNOWN( i_x + 1 < i_z );
00053      CPPAD_ASSERT_UNKNOWN( q < nc_taylor );
00054      CPPAD_ASSERT_UNKNOWN( p <= q );
00055 
00056      // Taylor coefficients corresponding to argument and result
00057      Base* x = taylor + i_x * nc_taylor;
00058      Base* z = taylor + i_z * nc_taylor;
00059      Base* b = z      -       nc_taylor;  // called y in documentation
00060 
00061      size_t k;
00062      if( p == 0 )
00063      {    z[0] = atan( x[0] );
00064           b[0] = Base(1) + x[0] * x[0];
00065           p++;
00066      }
00067      for(size_t j = p; j <= q; j++)
00068      {
00069           b[j] = Base(2) * x[0] * x[j];
00070           z[j] = Base(0);
00071           for(k = 1; k < j; k++)
00072           {    b[j] += x[k] * x[j-k];
00073                z[j] -= Base(k) * z[k] * b[j-k];
00074           }
00075           z[j] /= Base(j);
00076           z[j] += x[j];
00077           z[j] /= b[0];
00078      }
00079 }
00080 
00081 /*!
00082 Zero order forward mode Taylor coefficient for result of op = AtanOp.
00083 
00084 The C++ source code corresponding to this operation is
00085 \verbatim
00086      z = atan(x)
00087 \endverbatim
00088 The auxillary result is
00089 \verbatim
00090      y = 1 + x * x
00091 \endverbatim
00092 The value of y is computed along with the value of z.
00093 
00094 \copydetails forward_unary2_op_0
00095 */
00096 template <class Base>
00097 inline void forward_atan_op_0(
00098      size_t i_z         ,
00099      size_t i_x         ,
00100      size_t nc_taylor   , 
00101      Base*  taylor      )
00102 {
00103      // check assumptions
00104      CPPAD_ASSERT_UNKNOWN( NumArg(AtanOp) == 1 );
00105      CPPAD_ASSERT_UNKNOWN( NumRes(AtanOp) == 2 );
00106      CPPAD_ASSERT_UNKNOWN( i_x + 1 < i_z );
00107      CPPAD_ASSERT_UNKNOWN( 0 < nc_taylor );
00108 
00109      // Taylor coefficients corresponding to argument and result
00110      Base* x = taylor + i_x * nc_taylor;
00111      Base* z = taylor + i_z * nc_taylor;
00112      Base* b = z      -       nc_taylor; // called y in documentation
00113 
00114      z[0] = atan( x[0] );
00115      b[0] = Base(1) + x[0] * x[0];
00116 }
00117 /*!
00118 Reverse mode partial derivatives for result of op = AtanOp.
00119 
00120 The C++ source code corresponding to this operation is
00121 \verbatim
00122      z = atan(x)
00123 \endverbatim
00124 The auxillary result is
00125 \verbatim
00126      y = 1 + x * x
00127 \endverbatim
00128 The value of y is computed along with the value of z.
00129 
00130 \copydetails reverse_unary2_op
00131 */
00132 
00133 template <class Base>
00134 inline void reverse_atan_op(
00135      size_t      d            ,
00136      size_t      i_z          ,
00137      size_t      i_x          ,
00138      size_t      nc_taylor    , 
00139      const Base* taylor       ,
00140      size_t      nc_partial   ,
00141      Base*       partial      )
00142 {
00143      // check assumptions
00144      CPPAD_ASSERT_UNKNOWN( NumArg(AtanOp) == 1 );
00145      CPPAD_ASSERT_UNKNOWN( NumRes(AtanOp) == 2 );
00146      CPPAD_ASSERT_UNKNOWN( i_x + 1 < i_z );
00147      CPPAD_ASSERT_UNKNOWN( d < nc_taylor );
00148      CPPAD_ASSERT_UNKNOWN( d < nc_partial );
00149 
00150      // Taylor coefficients and partials corresponding to argument
00151      const Base* x  = taylor  + i_x * nc_taylor;
00152      Base* px       = partial + i_x * nc_partial;
00153 
00154      // Taylor coefficients and partials corresponding to first result
00155      const Base* z  = taylor  + i_z * nc_taylor;
00156      Base* pz       = partial + i_z * nc_partial;
00157 
00158      // Taylor coefficients and partials corresponding to auxillary result
00159      const Base* b  = z  - nc_taylor; // called y in documentation
00160      Base* pb       = pz - nc_partial;
00161 
00162      // number of indices to access
00163      size_t j = d;
00164      size_t k;
00165      while(j)
00166      {    // scale partials w.r.t z[j] and b[j]
00167           pz[j] /= b[0];
00168           pb[j] *= Base(2);
00169 
00170           pb[0] -= pz[j] * z[j]; 
00171           px[j] += pz[j] + pb[j] * x[0];
00172           px[0] += pb[j] * x[j];
00173 
00174           // more scaling of partials w.r.t z[j]
00175           pz[j] /= Base(j);
00176 
00177           for(k = 1; k < j; k++)
00178           {    pb[j-k] -= pz[j] * Base(k) * z[k];
00179                pz[k]   -= pz[j] * Base(k) * b[j-k];
00180                px[k]   += pb[j] * x[j-k];
00181           }
00182           --j;
00183      }
00184      px[0] += pz[0] / b[0] + pb[0] * Base(2) * x[0];
00185 }
00186 
00187 } // END_CPPAD_NAMESPACE
00188 # endif
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