sparse_hes_fun.hpp

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# ifndef CPPAD_SPEED_SPARSE_HES_FUN_HPP
# define CPPAD_SPEED_SPARSE_HES_FUN_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.
-------------------------------------------------------------------------- */
/*
$begin sparse_hes_fun$$
$spell
     hes
     cppad
     hpp
     fp
     CppAD
     namespace
     const
     bool
     exp
     arg
$$

$section Evaluate a Function That Has a Sparse Hessian$$
$mindex sparse_hes_fun$$


$head Syntax$$
$codei%# include <cppad/speed/sparse_hes_fun.hpp>
%$$
$codei%sparse_hes_fun(%n%, %x%, %row%, %col%, %p%, %fp%)%$$

$head Purpose$$
This routine evaluates
$latex f(x)$$, $latex f^{(1)} (x)$$, or $latex f^{(2)} (x)$$
where the Hessian $latex f^{(2)} (x)$$ is sparse.
The function $latex f : \B{R}^n \rightarrow \B{R}$$ only depends on the
size and contents of the index vectors $icode row$$ and $icode col$$.
The non-zero entries in the Hessian of this function have
one of the following forms:
$latex \[
     \DD{f}{x[row[k]]}{x[row[k]]}
     \; , \;
     \DD{f}{x[row[k]]}{x[col[k]]}
     \; , \;
     \DD{f}{x[col[k]]}{x[row[k]]}
     \; , \;
     \DD{f}{x[col[k]]}{x[col[k]]}
\] $$
for some $latex k $$ between zero and $latex K-1 $$.
All the other terms of the Hessian are zero.

$head Inclusion$$
The template function $code sparse_hes_fun$$
is defined in the $code CppAD$$ namespace by including
the file $code cppad/speed/sparse_hes_fun.hpp$$
(relative to the CppAD distribution directory).

$head Float$$
The type $icode Float$$ must be a $cref NumericType$$.
In addition, if $icode y$$ and $icode z$$ are $icode Float$$ objects,
$codei%
     %y% = exp(%z%)
%$$
must set the $icode y$$ equal the exponential of $icode z$$, i.e.,
the derivative of $icode y$$ with respect to $icode z$$ is equal to $icode y$$.

$head FloatVector$$
The type $icode FloatVector$$ is any
$cref SimpleVector$$, or it can be a raw pointer,
with elements of type $icode Float$$.

$head n$$
The argument $icode n$$ has prototype
$codei%
     size_t %n%
%$$
It specifies the dimension for the domain space for $latex f(x)$$.

$head x$$
The argument $icode x$$ has prototype
$codei%
     const %FloatVector%& %x%
%$$
It contains the argument value for which the function,
or its derivative, is being evaluated.
We use $latex n$$ to denote the size of the vector $icode x$$.

$head row$$
The argument $icode row$$ has prototype
$codei%
      const CppAD::vector<size_t>& %row%
%$$
It specifies one of the first
index of $latex x$$ for each non-zero Hessian term
(see $cref/purpose/sparse_hes_fun/Purpose/$$ above).
All the elements of $icode row$$ must be between zero and $icode%n%-1%$$.
The value $latex K$$ is defined by $icode%K% = %row%.size()%$$.

$head col$$
The argument $icode col$$ has prototype
$codei%
      const CppAD::vector<size_t>& %col%
%$$
and its size must be $latex K$$; i.e., the same as for $icode col$$.
It specifies the second
index of $latex x$$ for the non-zero Hessian terms.
All the elements of $icode col$$ must be between zero and $icode%n%-1%$$.
There are no duplicated entries requested, to be specific,
if $icode%k1% != %k2%$$ then
$codei%
     ( %row%[%k1%] , %col%[%k1%] ) != ( %row%[%k2%] , %col%[%k2%] )
%$$

$head p$$
The argument $icode p$$ has prototype
$codei%
     size_t %p%
%$$
It is either zero or two and
specifies the order of the derivative of $latex f$$
that is being evaluated, i.e., $latex f^{(p)} (x)$$ is evaluated.

$head fp$$
The argument $icode fp$$ has prototype
$codei%
     %FloatVector%& %fp%
%$$
The input value of the elements of $icode fp$$ does not matter.

$subhead Function$$
If $icode p$$ is zero, $icode fp$$ has size one and
$icode%fp%[0]%$$ is the value of $latex f(x)$$.

$subhead Hessian$$
If $icode p$$ is two, $icode fp$$ has size $icode K$$ and
for $latex k = 0 , \ldots , K-1$$,
$latex \[
     \DD{f}{ x[ \R{row}[k] ] }{ x[ \R{col}[k] ]} = fp [k]
\] $$

$children%
     speed/example/sparse_hes_fun.cpp%
     omh/sparse_hes_fun.omh
%$$

$head Example$$
The file
$cref sparse_hes_fun.cpp$$
contains an example and test  of $code sparse_hes_fun.hpp$$.
It returns true if it succeeds and false otherwise.

$head Source Code$$
The file
$cref sparse_hes_fun.hpp$$
contains the source code for this template function.

$end
------------------------------------------------------------------------------
*/
// BEGIN C++
# include <cppad/core/cppad_assert.hpp>
# include <cppad/utility/check_numeric_type.hpp>
# include <cppad/utility/vector.hpp>

// following needed by gcc under fedora 17 so that exp(double) is defined
# include <cppad/base_require.hpp>

namespace CppAD {
     template <class Float, class FloatVector>
     void sparse_hes_fun(
          size_t                       n    ,
          const FloatVector&           x    ,
          const CppAD::vector<size_t>& row  ,
          const CppAD::vector<size_t>& col  ,
          size_t                       p    ,
          FloatVector&                fp    )
     {
          // check numeric type specifications
          CheckNumericType<Float>();

          // check value of p
          CPPAD_ASSERT_KNOWN(
               p == 0 || p == 2,
               "sparse_hes_fun: p != 0 and p != 2"
          );

          size_t K = row.size();
          size_t i, j, k;
          if( p == 0 )
               fp[0] = Float(0);
          else
          {    for(k = 0; k < K; k++)
                    fp[k] = Float(0);
          }

          // determine which diagonal entries are present in row[k], col[k]
          CppAD::vector<size_t> diagonal(n);
          for(i = 0; i < n; i++)
               diagonal[i] = K;   // no diagonal entry for this row
          for(k = 0; k < K; k++)
          {    if( row[k] == col[k] )
               {    CPPAD_ASSERT_UNKNOWN( diagonal[row[k]] == K );
                    // index of the diagonal entry
                    diagonal[ row[k] ] = k;
               }
          }

          // determine which entries must be multiplied by a factor of two
          CppAD::vector<Float> factor(K);
          for(k = 0; k < K; k++)
          {    factor[k] = Float(1);
               for(size_t k1 = 0; k1 < K; k1++)
               {    bool reflected = true;
                    reflected &= k != k1;
                    reflected &= row[k] != col[k];
                    reflected &= row[k] == col[k1];
                    reflected &= col[k] == row[k1];
                    if( reflected )
                         factor[k] = Float(2);
               }
          }

          Float t;
          for(k = 0; k < K; k++)
          {    i    = row[k];
               j    = col[k];
               t    = exp( x[i] * x[j] );
               switch(p)
               {
                    case 0:
                    fp[0] += t;
                    break;

                    case 2:
                    if( i == j )
                    {    // dt_dxi = 2.0 * xi * t
                         fp[k] += ( Float(2) + Float(4) * x[i] * x[i] ) * t;
                    }
                    else
                    {    // dt_dxi = xj * t
                         fp[k] += factor[k] * ( Float(1) + x[i] * x[j] ) * t;
                         if( diagonal[i] != K )
                         {    size_t ki = diagonal[i];
                              fp[ki] += x[j] * x[j] * t;
                         }
                         if( diagonal[j] != K )
                         {    size_t kj = diagonal[j];
                              fp[kj] += x[i] * x[i] * t;
                         }
                    }
                    break;
               }
          }

     }
}
// END C++
# endif