\(\newcommand{\W}[1]{ \; #1 \; }\) \(\newcommand{\R}[1]{ {\rm #1} }\) \(\newcommand{\B}[1]{ {\bf #1} }\) \(\newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} }\) \(\newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} }\) \(\newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} }\) \(\newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }\)

atomic_two_eigen_mat_inv.hpp

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atomic_two Eigen Matrix Inversion Class

Purpose

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

Matrix Dimensions

This example puts the matrix dimension \(p\) in the atomic function arguments, instead of the constructor , so it can be different for different calls to the atomic function.

Theory

Forward

The zero order forward mode Taylor coefficient is give by

\[R_0 = A_0^{-1}\]

For \(k = 1 , \ldots\), the k-th order Taylor coefficient of \(A R\) is given by

\[0 = \sum_{\ell=0}^k A_\ell R_{k-\ell}\]

Solving for \(R_k\) in terms of the coefficients for \(A\) and the lower order coefficients for \(R\) we have

\[R_k = - R_0 \left( \sum_{\ell=1}^k A_\ell R_{k-\ell} \right)\]

Furthermore, once we have \(R_k\) we can compute the sum using

\[A_0 R_k = - \left( \sum_{\ell=1}^k A_\ell R_{k-\ell} \right)\]

Product of Three Matrices

Suppose \(\bar{E}\) is the derivative of the scalar value function \(s(E)\) with respect to \(E\); i.e.,

\[\bar{E}_{i,j} = \frac{ \partial s } { \partial E_{i,j} }\]

Also suppose that \(t\) is a scalar valued argument and

\[E(t) = B(t) C(t) D(t)\]

It follows that

\[E'(t) = B'(t) C(t) D(t) + B(t) C'(t) D(t) + B(t) C(t) D'(t)\]

\[(s \circ E)'(t) = \R{tr} [ \bar{E}^\R{T} E'(t) ]\]

\[= \R{tr} [ \bar{E}^\R{T} B'(t) C(t) D(t) ] + \R{tr} [ \bar{E}^\R{T} B(t) C'(t) D(t) ] + \R{tr} [ \bar{E}^\R{T} B(t) C(t) D'(t) ]\]

\[= \R{tr} [ B(t) D(t) \bar{E}^\R{T} B'(t) ] + \R{tr} [ D(t) \bar{E}^\R{T} B(t) C'(t) ] + \R{tr} [ \bar{E}^\R{T} B(t) C(t) D'(t) ]\]

\[\bar{B} = \bar{E} (C D)^\R{T} \W{,} \bar{C} = \B{R}^\R{T} \bar{E} D^\R{T} \W{,} \bar{D} = (B C)^\R{T} \bar{E}\]

Reverse

For \(k > 0\), reverse mode eliminates \(R_k\) and expresses the function values \(s\) in terms of the coefficients of \(A\) and the lower order coefficients of \(R\). The effect on \(\bar{R}_0\) (of eliminating \(R_k\)) is

\[\bar{R}_0 = \bar{R}_0 - \bar{R}_k \left( \sum_{\ell=1}^k A_\ell R_{k-\ell} \right)^\R{T} = \bar{R}_0 + \bar{R}_k ( A_0 R_k )^\R{T}\]

For \(\ell = 1 , \ldots , k\), the effect on \(\bar{R}_{k-\ell}\) and \(A_\ell\) (of eliminating \(R_k\)) is

\[\bar{A}_\ell = \bar{A}_\ell - R_0^\R{T} \bar{R}_k R_{k-\ell}^\R{T}\]

\[\bar{R}_{k-\ell} = \bar{R}_{k-\ell} - ( R_0 A_\ell )^\R{T} \bar{R}_k\]

We note that

\[R_0 '(t) A_0 (t) + R_0 (t) A_0 '(t) = 0\]

\[R_0 '(t) = - R_0 (t) A_0 '(t) R_0 (t)\]

The reverse mode formula that eliminates \(R_0\) is

\[\bar{A}_0 = \bar{A}_0 - R_0^\R{T} \bar{R}_0 R_0^\R{T}\]

Start Class Definition

# include <cppad/cppad.hpp>
# include <Eigen/Core>
# include <Eigen/LU>

Public

Types

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;

Constructor

   // constructor
   atomic_eigen_mat_inv(void) : CppAD::atomic_base<Base>(
      "atom_eigen_mat_inv"                             ,
      CppAD::atomic_base<Base>::set_sparsity_enum
   )
   { }

op

   // 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;
   }

Private

Variables

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_;
   // -------------------------------------------------------------

forward

   // 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( long(nr), long(nr) );
            f_result_[k].resize( long(nr), long(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( long(nr), long(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;
   }

reverse

   // 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( long(nr), long(nr) );
            r_result_[k].resize( long(nr), long(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( long(nr), long(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;
   }

End Class Definition

}; // End of atomic_eigen_mat_inv class

}  // END_EMPTY_NAMESPACE