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# ifndef CPPAD_REV_TWO_INCLUDED
# define CPPAD_REV_TWO_INCLUDED

/* --------------------------------------------------------------------------
CppAD: C++ Algorithmic Differentiation: Copyright (C) 2003-06 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 RevTwo$$
$spell
        ddw
        typename
        Taylor
        const
$$


$index partial, second order driver$$
$index second, order partial driver$$
$index driver, second order partial$$

$index easy, partial$$
$index driver, easy partial$$
$index partial, easy$$


$section Reverse Mode Second Partial Derivative Driver$$

$head Syntax$$
$syntax%%ddw% = %f%.RevTwo(%x%, %i%, %j%)%$$


$head Purpose$$
We use $latex F : B^n \rightarrow B^m$$ to denote the
$xref/glossary/AD Function/AD function/$$ corresponding to $italic f$$.
The syntax above sets 
$latex \[
        ddw [ k * p + \ell ]
        =
        \DD{ F_{i[ \ell ]} }{ x_{j[ \ell ]} }{ x_k } (x) 
\] $$
for $latex k = 0 , \ldots , n-1$$
and $latex \ell = 0 , \ldots , p$$,
where $latex p$$ is the size of the vectors $italic i$$ and $italic j$$.

$head f$$
The object $italic f$$ has prototype
$syntax%
        ADFun<%Base%> %f%
%$$
Note that the $xref/ADFun/$$ object $italic f$$ is not $code const$$
(see $xref/RevTwo/RevTwo Uses Forward/RevTwo Uses Forward/$$ below).

$head x$$
The argument $italic x$$ has prototype
$syntax%
        const %VectorBase% &%x%
%$$
(see $xref/RevTwo/VectorBase/VectorBase/$$ below)
and its size 
must be equal to $italic n$$, the dimension of the
$xref/SeqProperty/Domain/domain/$$ space for $italic f$$.
It specifies
that point at which to evaluate the partial derivatives listed above.

$head i$$
The argument $italic i$$ has prototype
$syntax%
        const %VectorSize_t% &%i%
%$$
(see $xref/RevTwo/VectorSize_t/VectorSize_t/$$ below)
We use $italic p$$ to denote the size of the vector $italic i$$.
All of the indices in $italic i$$ 
must be less than $italic m$$, the dimension of the
$xref/SeqProperty/Range/range/$$ space for $italic f$$; i.e.,
for $latex \ell = 0 , \ldots , p-1$$, $latex i[ \ell ]  < m$$.

$head j$$
The argument $italic j$$ has prototype
$syntax%
        const %VectorSize_t% &%j%
%$$
(see $xref/RevTwo/VectorSize_t/VectorSize_t/$$ below)
and its size must be equal to $italic p$$,
the size of the vector $italic i$$.
All of the indices in $italic j$$ 
must be less than $italic n$$; i.e.,
for $latex \ell = 0 , \ldots , p-1$$, $latex j[ \ell ]  < n$$.

$head ddw$$
The result $italic ddw$$ has prototype
$syntax%
        %VectorBase% %ddw%
%$$
(see $xref/RevTwo/VectorBase/VectorBase/$$ below)
and its size is $latex n * p$$.
It contains the requested partial derivatives; to be specific,
for $latex k = 0 , \ldots , n - 1 $$ 
and $latex \ell = 0 , \ldots , p - 1$$
$latex \[
        ddw [ k * p + \ell ]
        =
        \DD{ F_{i[ \ell ]} }{ x_{j[ \ell ]} }{ x_k } (x) 
\] $$

$head VectorBase$$
The type $italic VectorBase$$ must be a $xref/SimpleVector/$$ class with
$xref/SimpleVector/Elements of Specified Type/elements of type Base/$$.
The routine $xref/CheckSimpleVector/$$ will generate an error message
if this is not the case.

$head VectorSize_t$$
The type $italic VectorSize_t$$ must be a $xref/SimpleVector/$$ class with
$xref/SimpleVector/Elements of Specified Type/elements of type size_t/$$.
The routine $xref/CheckSimpleVector/$$ will generate an error message
if this is not the case.

$head RevTwo Uses Forward$$
After each call to $xref/Forward/$$,
the object $italic f$$ contains the corresponding 
$xref/glossary/Taylor Coefficient/Taylor coefficients/$$.
After $code RevTwo$$,
the previous calls to $xref/Forward/$$ are undefined.

$head Examples$$
$children%
        example/rev_two.cpp
%$$
The routine 
$xref/RevTwo.cpp//RevTwo/$$ is both an example and test.
It returns $code true$$, if it succeeds and $code false$$ otherwise.

$end
-----------------------------------------------------------------------------
*/

//  BEGIN CppAD namespace
namespace CppAD {

template <typename Base>
template <typename VectorBase, typename VectorSize_t>
VectorBase ADFun<Base>::RevTwo(
        const VectorBase   &x,
        const VectorSize_t &i,
        const VectorSize_t &j)
{       size_t i1;
        size_t j1;
        size_t k;
        size_t l;

        size_t n = Domain();
        size_t m = Range();
        size_t p = i.size();

        // check VectorBase is Simple Vector class with Base elements
        CheckSimpleVector<Base, VectorBase>();

        // check VectorSize_t is Simple Vector class with size_t elements
        CheckSimpleVector<size_t, VectorSize_t>();

        CPPAD_ASSERT_KNOWN(
                x.size() == n,
                "RevTwo: Length of x not equal domain dimension for f."
        ); 
        CPPAD_ASSERT_KNOWN(
                i.size() == j.size(),
                "RevTwo: Lenght of the i and j vectors are not equal."
        );
        // point at which we are evaluating the second partials
        Forward(0, x);

        // dimension the return value
        VectorBase ddw(n * p);

        // direction vector in argument space
        VectorBase dx(n);
        for(j1 = 0; j1 < n; j1++)
                dx[j1] = Base(0);

        // direction vector in range space
        VectorBase w(m);
        for(i1 = 0; i1 < m; i1++)
                w[i1] = Base(0);

        // place to hold the results of a reverse calculation
        VectorBase r(n * 2);

        // check the indices in i and j
        for(l = 0; l < p; l++)
        {       i1 = i[l];
                j1 = j[l];
                CPPAD_ASSERT_KNOWN(
                i1 < m,
                "RevTwo: an eleemnt of i not less than range dimension for f."
                );
                CPPAD_ASSERT_KNOWN(
                j1 < n,
                "RevTwo: an element of j not less than domain dimension for f."
                );
        }

        // loop over all forward directions 
        for(j1 = 0; j1 < n; j1++) 
        {       // first order forward mode calculation done 
                bool first_done = false;
                for(l = 0; l < p; l++) if( j[l] == j1 )
                {       if( ! first_done )
                        {       first_done = true;

                                // first order forward mode in j1 direction
                                dx[j1] = Base(1);
                                Forward(1, dx);
                                dx[j1] = Base(0);
                        }
                        // execute a reverse in this component direction
                        i1    = i[l];
                        w[i1] = Base(1);
                        r     = Reverse(2, w);
                        w[i1] = Base(0);

                        // place the reverse result in return value
                        for(k = 0; k < n; k++)
                                ddw[k * p + l] = r[k * 2 + 1];
                }
        }
        return ddw;
}

} // END CppAD namespace

# endif

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