# ifndef CPPAD_ATAN_OP_INCLUDED # define CPPAD_ATAN_OP_INCLUDED /* -------------------------------------------------------------------------- CppAD: C++ Algorithmic Differentiation: Copyright (C) 2003-07 Bradley M. Bell CppAD is distributed under multiple licenses. This distribution is under the terms of the Common 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. -------------------------------------------------------------------------- */ /* $begin ForAtanOp$$comment CppAD Developer Documentation$$$spell atan Taylor const inline Op $$index forward, atan$$ $index atan, forward$$index ForAtanOp$$$section Forward Mode Atan Function$$head Syntax$$ $syntax%inline void ForAtanOp(size_t %d%, %Base% *%z%, %Base% *%b%, const %Base% *%x%)%$$head Description$$ Computes the$italic d$$order Taylor coefficient for latex Z$$ and $latex B$$where syntax% %Z% = atan(%X%) %B% = 1 + %X% * %X% %$$$head x$$The vector italic x$$ has length $latex d+1$$and contains the th d$$ order Taylor coefficient row vector for$italic X$$. head z$$ The vector $italic z$$has length latex d+1$$. On input it contains the$th d-1$$order Taylor coefficient row vector for italic Z$$. On output it contains the $th d$$order Taylor coefficient row vector for italic Z$$; i.e.,$syntax%%z%[%d%]%$$is set equal to the th d$$ Taylor coefficient for the function $italic S$$. head b$$ The vector$italic c$$has length latex d+1$$. On input it contains the $th d-1$$order Taylor coefficient row vector for italic B$$. On output it contains the$th d$$order Taylor coefficient row vector for italic B$$; i.e., $syntax%%b%[%d%]%$$is set equal to the th d$$ Taylor coefficient for the function$italic B$$. end ------------------------------------------------------------------------------ begin RevAtanOp$$ $comment CppAD Developer Documentation$$spell ps Atan pb Sin Taylor const inline Op px py pz$$$index reverse, atan$$index atan, reverse$$ $index RevAtanOp$$section Reverse Mode Atan Function$$$head Syntax$$syntax%inline void RevAtanOp(size_t %d%, const %Base% *%z%, const %Base% *%b%, const %Base% *%x%, %Base% *%pz%, %Base% *%pb%, %Base% *%px%)%$$ $head Description$$We are given the partial derivatives for a function latex G(z, b, x)$$ and we wish to compute the partial derivatives for the function$latex $H(x) = G [ Z(x) , B(x) , x ]$ $$where latex Z(x)$$ and $latex B(x)$$are defined as the th d$$ order Taylor coefficient row vector for$latex \arctan(x)$$and latex 1 + x * x$$ as a function of the corresponding row vector for $italic X$$; i.e., latex $\begin{array}{rcl} Z & = & \arctan(X) \\ B & = & 1 + X * X \end{array}$$$ Note that$italic Z$$and latex B$$ have been used both the original functions and for the corresponding mapping of Taylor coefficients. $head x$$The vector italic x$$ has length$latex d+1$$and contains the th d$$ order Taylor coefficient row vector for $italic X$$. head z$$ The vector$italic z$$has length latex d+1$$ and contains $th d$$order Taylor coefficient row vector for italic z$$.$head b$$The vector italic b$$ has length $latex d+1$$and contains th d$$ order Taylor coefficient row vector for$italic B$$. head On Input$$ $subhead px$$The vector italic px$$ has length$latex d+1$$and syntax%%px%[%j%]%$$ contains the partial for $italic G$$with respect to the th j$$ order Taylor coefficient for$italic X$$. subhead pz$$ The vector $italic pz$$has length latex d+1$$ and$syntax%%pz%[%j%]%$$contains the partial for italic G$$ with respect to the $th j$$order Taylor coefficient for italic Z$$.$subhead pb$$The vector italic pb$$ has length $latex d+1$$and syntax%%pb%[%j%]%$$ contains the partial for$italic G$$with respect to the th j$$ order Taylor coefficient for $italic B$$. head On Output$$$subhead px$$The vector italic px$$ has length $latex d+1$$and syntax%%px%[%j%]%$$ contains the partial for$italic H$$with respect to the th j$$ order Taylor coefficient for $italic X$$. subhead pz$$ The vector$italic ps$$has length latex d+1$$ and its contents are no longer specified; i.e., it has been used for work space. $subhead pb$$The vector italic pb$$ has length$latex d+1 and its contents are no longer specified; i.e., it has been used for work space. \$end ------------------------------------------------------------------------------ */ // BEGIN CppAD namespace namespace CppAD { template inline void ForAtanOp(size_t j, Base *z, Base *b, const Base *x) { size_t k; if( j == 0 ) { z[j] = atan( x[0] ); b[j] = Base(1) + x[0] * x[0]; } else { b[j] = Base(2) * x[0] * x[j]; z[j] = Base(0); for(k = 1; k < j; k++) { b[j] += x[k] * x[j-k]; z[j] -= Base(k) * z[k] * b[j-k]; } z[j] /= Base(j); z[j] += x[j]; z[j] /= b[0]; } } template inline void RevAtanOp(size_t d, const Base *z, const Base *b, const Base *x, Base *pz, Base *pb, Base *px) { size_t k; // number of indices to access size_t j = d; while(j) { // scale partials w.r.t z[j] and b[j] pz[j] /= b[0]; pb[j] *= Base(2); pb[0] -= pz[j] * z[j]; px[j] += pz[j] + pb[j] * x[0]; px[0] += pb[j] * x[j]; // more scaling of partials w.r.t z[j] pz[j] /= Base(j); for(k = 1; k < j; k++) { pb[j-k] -= pz[j] * Base(k) * z[k]; pz[k] -= pz[j] * Base(k) * b[j-k]; px[k] += pb[j] * x[j-k]; } --j; } px[0] += pz[0] / b[0] + pb[0] * Base(2) * x[0]; } } // END CppAD namespace # endif