eigen_mat_inv.hpp

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// $Id$
# ifndef CPPAD_EXAMPLE_EIGEN_MAT_INV_HPP
# define CPPAD_EXAMPLE_EIGEN_MAT_INV_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 atomic_eigen_mat_inv.hpp$$
$spell
     Eigen
     Taylor
$$

$section Atomic Eigen Matrix Inversion Class$$

$head Purpose$$
Construct an atomic operation that computes the matrix inverse
$latex R = A^{-1}$$
for any positive integer $latex p$$
and invertible matrix $latex A \in \B{R}^{p \times p}$$.

$head Matrix Dimensions$$
This example puts the matrix dimension $latex p$$
in the atomic function arguments,
instead of the $cref/constructor/atomic_ctor/$$,
so it can be different for different calls to the atomic function.

$head Theory$$

$subhead Forward$$
The zero order forward mode Taylor coefficient is give by
$latex \[
     R_0 = A_0^{-1}
\]$$
For $latex k = 1 , \ldots$$,
the $th k$$ order Taylor coefficient of $latex A R$$ is given by
$latex \[
     0 = \sum_{\ell=0}^k A_\ell R_{k-\ell}
\] $$
Solving for $latex R_k$$ in terms of the coefficients
for $latex A$$ and the lower order coefficients for $latex R$$ we have
$latex \[
     R_k = - R_0 \left( \sum_{\ell=1}^k A_\ell R_{k-\ell} \right)
\] $$
Furthermore, once we have $latex R_k$$ we can compute the sum using
$latex \[
     A_0 R_k = - \left( \sum_{\ell=1}^k A_\ell R_{k-\ell} \right)
\] $$


$subhead Product of Three Matrices$$
Suppose $latex \bar{E}$$ is the derivative of the
scalar value function $latex s(E)$$ with respect to $latex E$$; i.e.,
$latex \[
     \bar{E}_{i,j} = \frac{ \partial s } { \partial E_{i,j} }
\] $$
Also suppose that $latex t$$ is a scalar valued argument and
$latex \[
     E(t) = B(t) C(t) D(t)
\] $$
It follows that
$latex \[
     E'(t) = B'(t) C(t) D(t) + B(t) C'(t) D(t) +  B(t) C(t) D'(t)
\] $$

$latex \[
     (s \circ E)'(t)
     =
     \R{tr} [ \bar{E}^\R{T} E'(t) ]
\] $$
$latex \[
     =
     \R{tr} [ \bar{E}^\R{T} B'(t) C(t) D(t) ] +
     \R{tr} [ \bar{E}^\R{T} B(t) C'(t) D(t) ] +
     \R{tr} [ \bar{E}^\R{T} B(t) C(t) D'(t) ]
\] $$
$latex \[
     =
     \R{tr} [ B(t) D(t) \bar{E}^\R{T} B'(t) ] +
     \R{tr} [ D(t) \bar{E}^\R{T} B(t) C'(t) ] +
     \R{tr} [ \bar{E}^\R{T} B(t) C(t) D'(t) ]
\] $$
$latex \[
     \bar{B} = \bar{E} (C D)^\R{T} \W{,}
     \bar{C} = B^\R{T} \bar{E} D^\R{T} \W{,}
     \bar{D} = (B C)^\R{T} \bar{E}
\] $$

$subhead Reverse$$
For $latex k > 0$$, reverse mode
eliminates $latex R_k$$ and expresses the function values
$latex s$$ in terms of the coefficients of $latex A$$
and the lower order coefficients of $latex R$$.
The effect on $latex \bar{R}_0$$
(of eliminating $latex R_k$$) is
$latex \[
\bar{R}_0
= \bar{R}_0 - \bar{R}_k \left( \sum_{\ell=1}^k A_\ell R_{k-\ell} \right)^\R{T}
= \bar{R}_0 + \bar{R}_k ( A_0 R_k )^\R{T}
\] $$
For $latex \ell = 1 , \ldots , k$$,
the effect on $latex \bar{R}_{k-\ell}$$ and $latex A_\ell$$
(of eliminating $latex R_k$$) is
$latex \[
\bar{A}_\ell = \bar{A}_\ell - R_0^\R{T} \bar{R}_k R_{k-\ell}^\R{T}
\] $$
$latex \[
\bar{R}_{k-\ell} = \bar{R}_{k-\ell} - ( R_0 A_\ell )^\R{T} \bar{R}_k
\] $$
We note that
$latex \[
     R_0 '(t) A_0 (t) + R_0 (t) A_0 '(t) = 0
\] $$
$latex \[
     R_0 '(t) = - R_0 (t) A_0 '(t) R_0 (t)
\] $$
The reverse mode formula that eliminates $latex R_0$$ is
$latex \[
     \bar{A}_0
     = \bar{A}_0 - R_0^\R{T} \bar{R}_0 R_0^\R{T}
\]$$

$nospell

$head Start Class Definition$$
$srccode%cpp% */
# include <cppad/cppad.hpp>
# include <Eigen/Core>
# include <Eigen/LU>



/* %$$
$head Public$$

$subhead Types$$
$srccode%cpp% */
namespace { // BEGIN_EMPTY_NAMESPACE

template <class Base>
class atomic_eigen_mat_inv : public CppAD::atomic_base<Base> {
public:
     // -----------------------------------------------------------
     // type of elements during calculation of derivatives
     typedef Base              scalar;
     // type of elements during taping
     typedef CppAD::AD<scalar> ad_scalar;
     // type of matrix during calculation of derivatives
     typedef Eigen::Matrix<
          scalar, Eigen::Dynamic, Eigen::Dynamic, Eigen::RowMajor>     matrix;
     // type of matrix during taping
     typedef Eigen::Matrix<
          ad_scalar, Eigen::Dynamic, Eigen::Dynamic, Eigen::RowMajor > ad_matrix;
/* %$$
$subhead Constructor$$
$srccode%cpp% */
     // constructor
     atomic_eigen_mat_inv(void) : CppAD::atomic_base<Base>(
          "atom_eigen_mat_inv"                             ,
          CppAD::atomic_base<Base>::set_sparsity_enum
     )
     { }
/* %$$
$subhead op$$
$srccode%cpp% */
     // use atomic operation to invert an AD matrix
     ad_matrix op(const ad_matrix& arg)
     {    size_t nr = size_t( arg.rows() );
          size_t ny = nr * nr;
          size_t nx = 1 + ny;
          assert( nr == size_t( arg.cols() ) );
          // -------------------------------------------------------------------
          // packed version of arg
          CPPAD_TESTVECTOR(ad_scalar) packed_arg(nx);
          packed_arg[0] = ad_scalar( nr );
          for(size_t i = 0; i < ny; i++)
               packed_arg[1 + i] = arg.data()[i];
          // -------------------------------------------------------------------
          // packed version of result = arg^{-1}.
          // This is an atomic_base function call that CppAD uses to
          // store the atomic operation on the tape.
          CPPAD_TESTVECTOR(ad_scalar) packed_result(ny);
          (*this)(packed_arg, packed_result);
          // -------------------------------------------------------------------
          // unpack result matrix
          ad_matrix result(nr, nr);
          for(size_t i = 0; i < ny; i++)
               result.data()[i] = packed_result[i];
          return result;
     }
     /* %$$
$head Private$$

$subhead Variables$$
$srccode%cpp% */
private:
     // -------------------------------------------------------------
     // one forward mode vector of matrices for argument and result
     CppAD::vector<matrix> f_arg_, f_result_;
     // one reverse mode vector of matrices for argument and result
     CppAD::vector<matrix> r_arg_, r_result_;
     // -------------------------------------------------------------
/* %$$
$subhead forward$$
$srccode%cpp% */
     // forward mode routine called by CppAD
     virtual bool forward(
          // lowest order Taylor coefficient we are evaluating
          size_t                          p ,
          // highest order Taylor coefficient we are evaluating
          size_t                          q ,
          // which components of x are variables
          const CppAD::vector<bool>&      vx ,
          // which components of y are variables
          CppAD::vector<bool>&            vy ,
          // tx [ j * (q+1) + k ] is x_j^k
          const CppAD::vector<scalar>&    tx ,
          // ty [ i * (q+1) + k ] is y_i^k
          CppAD::vector<scalar>&          ty
     )
     {    size_t n_order = q + 1;
          size_t nr      = size_t( CppAD::Integer( tx[ 0 * n_order + 0 ] ) );
          size_t ny      = nr * nr;
# ifndef NDEBUG
          size_t nx      = 1 + ny;
# endif
          assert( vx.size() == 0 || nx == vx.size() );
          assert( vx.size() == 0 || ny == vy.size() );
          assert( nx * n_order == tx.size() );
          assert( ny * n_order == ty.size() );
          //
          // -------------------------------------------------------------------
          // make sure f_arg_ and f_result_ are large enough
          assert( f_arg_.size() == f_result_.size() );
          if( f_arg_.size() < n_order )
          {    f_arg_.resize(n_order);
               f_result_.resize(n_order);
               //
               for(size_t k = 0; k < n_order; k++)
               {    f_arg_[k].resize(nr, nr);
                    f_result_[k].resize(nr, nr);
               }
          }
          // -------------------------------------------------------------------
          // unpack tx into f_arg_
          for(size_t k = 0; k < n_order; k++)
          {    // unpack arg values for this order
               for(size_t i = 0; i < ny; i++)
                    f_arg_[k].data()[i] = tx[ (1 + i) * n_order + k ];
          }
          // -------------------------------------------------------------------
          // result for each order
          // (we could avoid recalculting f_result_[k] for k=0,...,p-1)
          //
          f_result_[0] = f_arg_[0].inverse();
          for(size_t k = 1; k < n_order; k++)
          {    // initialize sum
               matrix f_sum = matrix::Zero(nr, nr);
               // compute sum
               for(size_t ell = 1; ell <= k; ell++)
                    f_sum -= f_arg_[ell] * f_result_[k-ell];
               // result_[k] = arg_[0]^{-1} * sum_
               f_result_[k] = f_result_[0] * f_sum;
          }
          // -------------------------------------------------------------------
          // pack result_ into ty
          for(size_t k = 0; k < n_order; k++)
          {    for(size_t i = 0; i < ny; i++)
                    ty[ i * n_order + k ] = f_result_[k].data()[i];
          }
          // -------------------------------------------------------------------
          // check if we are computing vy
          if( vx.size() == 0 )
               return true;
          // ------------------------------------------------------------------
          // This is a very dumb algorithm that over estimates which
          // elements of the inverse are variables (which is not efficient).
          bool var = false;
          for(size_t i = 0; i < ny; i++)
               var |= vx[1 + i];
          for(size_t i = 0; i < ny; i++)
               vy[i] = var;
          return true;
     }
/* %$$
$subhead reverse$$
$srccode%cpp% */
     // reverse mode routine called by CppAD
     virtual bool reverse(
          // highest order Taylor coefficient that we are computing derivative of
          size_t                     q ,
          // forward mode Taylor coefficients for x variables
          const CppAD::vector<double>&     tx ,
          // forward mode Taylor coefficients for y variables
          const CppAD::vector<double>&     ty ,
          // upon return, derivative of G[ F[ {x_j^k} ] ] w.r.t {x_j^k}
          CppAD::vector<double>&           px ,
          // derivative of G[ {y_i^k} ] w.r.t. {y_i^k}
          const CppAD::vector<double>&     py
     )
     {    size_t n_order = q + 1;
          size_t nr      = size_t( CppAD::Integer( tx[ 0 * n_order + 0 ] ) );
          size_t ny      = nr * nr;
# ifndef NDEBUG
          size_t nx      = 1 + ny;
# endif
          //
          assert( nx * n_order == tx.size() );
          assert( ny * n_order == ty.size() );
          assert( px.size()    == tx.size() );
          assert( py.size()    == ty.size() );
          // -------------------------------------------------------------------
          // make sure f_arg_ is large enough
          assert( f_arg_.size() == f_result_.size() );
          // must have previous run forward with order >= n_order
          assert( f_arg_.size() >= n_order );
          // -------------------------------------------------------------------
          // make sure r_arg_, r_result_ are large enough
          assert( r_arg_.size() == r_result_.size() );
          if( r_arg_.size() < n_order )
          {    r_arg_.resize(n_order);
               r_result_.resize(n_order);
               //
               for(size_t k = 0; k < n_order; k++)
               {    r_arg_[k].resize(nr, nr);
                    r_result_[k].resize(nr, nr);
               }
          }
          // -------------------------------------------------------------------
          // unpack tx into f_arg_
          for(size_t k = 0; k < n_order; k++)
          {    // unpack arg values for this order
               for(size_t i = 0; i < ny; i++)
                    f_arg_[k].data()[i] = tx[ (1 + i) * n_order + k ];
          }
          // -------------------------------------------------------------------
          // unpack py into r_result_
          for(size_t k = 0; k < n_order; k++)
          {    for(size_t i = 0; i < ny; i++)
                    r_result_[k].data()[i] = py[ i * n_order + k ];
          }
          // -------------------------------------------------------------------
          // initialize r_arg_ as zero
          for(size_t k = 0; k < n_order; k++)
               r_arg_[k]   = matrix::Zero(nr, nr);
          // -------------------------------------------------------------------
          // matrix reverse mode calculation
          //
          for(size_t k1 = n_order; k1 > 1; k1--)
          {    size_t k = k1 - 1;
               // bar{R}_0 = bar{R}_0 + bar{R}_k (A_0 R_k)^T
               r_result_[0] +=
               r_result_[k] * f_result_[k].transpose() * f_arg_[0].transpose();
               //
               for(size_t ell = 1; ell <= k; ell++)
               {    // bar{A}_l = bar{A}_l - R_0^T bar{R}_k R_{k-l}^T
                    r_arg_[ell] -= f_result_[0].transpose()
                         * r_result_[k] * f_result_[k-ell].transpose();
                    // bar{R}_{k-l} = bar{R}_{k-1} - (R_0 A_l)^T bar{R}_k
                    r_result_[k-ell] -= f_arg_[ell].transpose()
                         * f_result_[0].transpose() * r_result_[k];
               }
          }
          r_arg_[0] -=
          f_result_[0].transpose() * r_result_[0] * f_result_[0].transpose();
          // -------------------------------------------------------------------
          // pack r_arg into px
          for(size_t k = 0; k < n_order; k++)
          {    for(size_t i = 0; i < ny; i++)
                    px[ (1 + i) * n_order + k ] = r_arg_[k].data()[i];
          }
          //
          return true;
     }
/* %$$
$head End Class Definition$$
$srccode%cpp% */
}; // End of atomic_eigen_mat_inv class

}  // END_EMPTY_NAMESPACE
/* %$$
$$ $comment end nospell$$
$end
*/


# endif