//

//  Copyright (c) 2000-2002

//  Joerg Walter, Mathias Koch

//

//  Permission to use, copy, modify, distribute and sell this software

//  and its documentation for any purpose is hereby granted without fee,

//  provided that the above copyright notice appear in all copies and

//  that both that copyright notice and this permission notice appear

//  in supporting documentation.  The authors make no representations

//  about the suitability of this software for any purpose.

//  It is provided "as is" without express or implied warranty.

//

//  The authors gratefully acknowledge the support of

//  GeNeSys mbH & Co. KG in producing this work.

//

#ifndef _BOOST_UBLAS_OPERATION_

#define _BOOST_UBLAS_OPERATION_

#include <boost/numeric/ublas/matrix_proxy.hpp>

/** \file operation.hpp

 *  \brief This file contains some specialized products.

 */

// axpy-based products

// Alexei Novakov had a lot of ideas to improve these. Thanks.

// Hendrik Kueck proposed some new kernel. Thanks again.

namespace boost { namespace numeric { namespace ublas {

    template<class V, class T1, class IA1, class TA1, class E2>

    BOOST_UBLAS_INLINE

    V &

    axpy_prod (const compressed_matrix<T1, row_major, 0, IA1, TA1> &e1,

               const vector_expression<E2> &e2,

               V &v, row_major_tag) {

        typedef typename V::size_type size_type;

        typedef typename V::value_type value_type;

        for (size_type i = 0; i < e1.filled1 () -1; ++ i) {

            size_type begin = e1.index1_data () [i];

            size_type end = e1.index1_data () [i + 1];

            value_type t (v (i));

            for (size_type j = begin; j < end; ++ j)

                t += e1.value_data () [j] * e2 () (e1.index2_data () [j]);

            v (i) = t;

        }

        return v;

    }

    template<class V, class T1, class IA1, class TA1, class E2>

    BOOST_UBLAS_INLINE

    V &

    axpy_prod (const compressed_matrix<T1, column_major, 0, IA1, TA1> &e1,

               const vector_expression<E2> &e2,

               V &v, column_major_tag) {

        typedef typename V::size_type size_type;

        for (size_type j = 0; j < e1.filled1 () -1; ++ j) {

            size_type begin = e1.index1_data () [j];

            size_type end = e1.index1_data () [j + 1];

            for (size_type i = begin; i < end; ++ i)

                v (e1.index2_data () [i]) += e1.value_data () [i] * e2 () (j);

        }

        return v;

    }

    // Dispatcher

    template<class V, class T1, class L1, class IA1, class TA1, class E2>

    BOOST_UBLAS_INLINE

    V &

    axpy_prod (const compressed_matrix<T1, L1, 0, IA1, TA1> &e1,

               const vector_expression<E2> &e2,

               V &v, bool init = true) {

        typedef typename V::value_type value_type;

        typedef typename L1::orientation_category orientation_category;

        if (init)

            v.assign (zero_vector<value_type> (e1.size1 ()));

#if BOOST_UBLAS_TYPE_CHECK

        vector<value_type> cv (v);

        typedef typename type_traits<value_type>::real_type real_type;

        real_type verrorbound (norm_1 (v) + norm_1 (e1) * norm_1 (e2));

        indexing_vector_assign<scalar_plus_assign> (cv, prod (e1, e2));

#endif

        axpy_prod (e1, e2, v, orientation_category ());

#if BOOST_UBLAS_TYPE_CHECK

        BOOST_UBLAS_CHECK (norm_1 (v - cv) <= 2 * std::numeric_limits<real_type>::epsilon () * verrorbound, internal_logic ());

#endif

        return v;

    }

    template<class V, class T1, class L1, class IA1, class TA1, class E2>

    BOOST_UBLAS_INLINE

    V

    axpy_prod (const compressed_matrix<T1, L1, 0, IA1, TA1> &e1,

               const vector_expression<E2> &e2) {

        typedef V vector_type;

        vector_type v (e1.size1 ());

        return axpy_prod (e1, e2, v, true);

    }

    template<class V, class T1, class L1, class IA1, class TA1, class E2>

    BOOST_UBLAS_INLINE

    V &

    axpy_prod (const coordinate_matrix<T1, L1, 0, IA1, TA1> &e1,

               const vector_expression<E2> &e2,

               V &v, bool init = true) {

        typedef typename V::size_type size_type;

        typedef typename V::value_type value_type;

        typedef L1 layout_type;

        size_type size1 = e1.size1();

        size_type size2 = e1.size2();

        if (init) {

            noalias(v) = zero_vector<value_type>(size1);

        }

        for (size_type i = 0; i < e1.nnz(); ++i) {

            size_type row_index = layout_type::element1( e1.index1_data () [i], size1, e1.index2_data () [i], size2 );

            size_type col_index = layout_type::element2( e1.index1_data () [i], size1, e1.index2_data () [i], size2 );

            v( row_index ) += e1.value_data () [i] * e2 () (col_index);

        }

        return v;

    }

    template<class V, class E1, class E2>

    BOOST_UBLAS_INLINE

    V &

    axpy_prod (const matrix_expression<E1> &e1,

               const vector_expression<E2> &e2,

               V &v, packed_random_access_iterator_tag, row_major_tag) {

        typedef const E1 expression1_type;

        typedef const E2 expression2_type;

        typedef typename V::size_type size_type;

        typename expression1_type::const_iterator1 it1 (e1 ().begin1 ());

        typename expression1_type::const_iterator1 it1_end (e1 ().end1 ());

        while (it1 != it1_end) {

            size_type index1 (it1.index1 ());

#ifndef BOOST_UBLAS_NO_NESTED_CLASS_RELATION

            typename expression1_type::const_iterator2 it2 (it1.begin ());

            typename expression1_type::const_iterator2 it2_end (it1.end ());

#else

            typename expression1_type::const_iterator2 it2 (boost::numeric::ublas::begin (it1, iterator1_tag ()));

            typename expression1_type::const_iterator2 it2_end (boost::numeric::ublas::end (it1, iterator1_tag ()));

#endif

            while (it2 != it2_end) {

                v (index1) += *it2 * e2 () (it2.index2 ());

                ++ it2;

            }

            ++ it1;

        }

        return v;

    }

    template<class V, class E1, class E2>

    BOOST_UBLAS_INLINE

    V &

    axpy_prod (const matrix_expression<E1> &e1,

               const vector_expression<E2> &e2,

               V &v, packed_random_access_iterator_tag, column_major_tag) {

        typedef const E1 expression1_type;

        typedef const E2 expression2_type;

        typedef typename V::size_type size_type;

        typename expression1_type::const_iterator2 it2 (e1 ().begin2 ());

        typename expression1_type::const_iterator2 it2_end (e1 ().end2 ());

        while (it2 != it2_end) {

            size_type index2 (it2.index2 ());

#ifndef BOOST_UBLAS_NO_NESTED_CLASS_RELATION

            typename expression1_type::const_iterator1 it1 (it2.begin ());

            typename expression1_type::const_iterator1 it1_end (it2.end ());

#else

            typename expression1_type::const_iterator1 it1 (boost::numeric::ublas::begin (it2, iterator2_tag ()));

            typename expression1_type::const_iterator1 it1_end (boost::numeric::ublas::end (it2, iterator2_tag ()));

#endif

            while (it1 != it1_end) {

                v (it1.index1 ()) += *it1 * e2 () (index2);

                ++ it1;

            }

            ++ it2;

        }

        return v;

    }

    template<class V, class E1, class E2>

    BOOST_UBLAS_INLINE

    V &

    axpy_prod (const matrix_expression<E1> &e1,

               const vector_expression<E2> &e2,

               V &v, sparse_bidirectional_iterator_tag) {

        typedef const E1 expression1_type;

        typedef const E2 expression2_type;

        typedef typename V::size_type size_type;

        typename expression2_type::const_iterator it (e2 ().begin ());

        typename expression2_type::const_iterator it_end (e2 ().end ());

        while (it != it_end) {

            v.plus_assign (column (e1 (), it.index ()) * *it);

            ++ it;

        }

        return v;

    }

    // Dispatcher

    template<class V, class E1, class E2>

    BOOST_UBLAS_INLINE

    V &

    axpy_prod (const matrix_expression<E1> &e1,

               const vector_expression<E2> &e2,

               V &v, packed_random_access_iterator_tag) {

        typedef typename E1::orientation_category orientation_category;

        return axpy_prod (e1, e2, v, packed_random_access_iterator_tag (), orientation_category ());

    }

  /** \brief computes <tt>v += A x</tt> or <tt>v = A x</tt> in an

          optimized fashion.

          \param e1 the matrix expression \c A

          \param e2 the vector expression \c x

          \param v  the result vector \c v

          \param init a boolean parameter

          <tt>axpy_prod(A, x, v, init)</tt> implements the well known

          axpy-product.  Setting \a init to \c true is equivalent to call

          <tt>v.clear()</tt> before <tt>axpy_prod</tt>. Currently \a init

          defaults to \c true, but this may change in the future.

          Up to now there are some specialisation for compressed

          matrices that give a large speed up compared to prod.

          \ingroup blas2

          \internal

          template parameters:

          \param V type of the result vector \c v

          \param E1 type of a matrix expression \c A

          \param E2 type of a vector expression \c x

  */

    template<class V, class E1, class E2>

    BOOST_UBLAS_INLINE

    V &

    axpy_prod (const matrix_expression<E1> &e1,

               const vector_expression<E2> &e2,

               V &v, bool init = true) {

        typedef typename V::value_type value_type;

        typedef typename E2::const_iterator::iterator_category iterator_category;

        if (init)

            v.assign (zero_vector<value_type> (e1 ().size1 ()));

#if BOOST_UBLAS_TYPE_CHECK

        vector<value_type> cv (v);

        typedef typename type_traits<value_type>::real_type real_type;

        real_type verrorbound (norm_1 (v) + norm_1 (e1) * norm_1 (e2));

        indexing_vector_assign<scalar_plus_assign> (cv, prod (e1, e2));

#endif

        axpy_prod (e1, e2, v, iterator_category ());

#if BOOST_UBLAS_TYPE_CHECK

        BOOST_UBLAS_CHECK (norm_1 (v - cv) <= 2 * std::numeric_limits<real_type>::epsilon () * verrorbound, internal_logic ());

#endif

        return v;

    }

    template<class V, class E1, class E2>

    BOOST_UBLAS_INLINE

    V

    axpy_prod (const matrix_expression<E1> &e1,

               const vector_expression<E2> &e2) {

        typedef V vector_type;

        vector_type v (e1 ().size1 ());

        return axpy_prod (e1, e2, v, true);

    }

    template<class V, class E1, class T2, class IA2, class TA2>

    BOOST_UBLAS_INLINE

    V &

    axpy_prod (const vector_expression<E1> &e1,

               const compressed_matrix<T2, column_major, 0, IA2, TA2> &e2,

               V &v, column_major_tag) {

        typedef typename V::size_type size_type;

        typedef typename V::value_type value_type;

        for (size_type j = 0; j < e2.filled1 () -1; ++ j) {

            size_type begin = e2.index1_data () [j];

            size_type end = e2.index1_data () [j + 1];

            value_type t (v (j));

            for (size_type i = begin; i < end; ++ i)

                t += e2.value_data () [i] * e1 () (e2.index2_data () [i]);

            v (j) = t;

        }

        return v;

    }

    template<class V, class E1, class T2, class IA2, class TA2>

    BOOST_UBLAS_INLINE

    V &

    axpy_prod (const vector_expression<E1> &e1,

               const compressed_matrix<T2, row_major, 0, IA2, TA2> &e2,

               V &v, row_major_tag) {

        typedef typename V::size_type size_type;

        for (size_type i = 0; i < e2.filled1 () -1; ++ i) {

            size_type begin = e2.index1_data () [i];

            size_type end = e2.index1_data () [i + 1];

            for (size_type j = begin; j < end; ++ j)

                v (e2.index2_data () [j]) += e2.value_data () [j] * e1 () (i);

        }

        return v;

    }

    // Dispatcher

    template<class V, class E1, class T2, class L2, class IA2, class TA2>

    BOOST_UBLAS_INLINE

    V &

    axpy_prod (const vector_expression<E1> &e1,

               const compressed_matrix<T2, L2, 0, IA2, TA2> &e2,

               V &v, bool init = true) {

        typedef typename V::value_type value_type;

        typedef typename L2::orientation_category orientation_category;

        if (init)

            v.assign (zero_vector<value_type> (e2.size2 ()));

#if BOOST_UBLAS_TYPE_CHECK

        vector<value_type> cv (v);

        typedef typename type_traits<value_type>::real_type real_type;

        real_type verrorbound (norm_1 (v) + norm_1 (e1) * norm_1 (e2));

        indexing_vector_assign<scalar_plus_assign> (cv, prod (e1, e2));

#endif

        axpy_prod (e1, e2, v, orientation_category ());

#if BOOST_UBLAS_TYPE_CHECK

        BOOST_UBLAS_CHECK (norm_1 (v - cv) <= 2 * std::numeric_limits<real_type>::epsilon () * verrorbound, internal_logic ());

#endif

        return v;

    }

    template<class V, class E1, class T2, class L2, class IA2, class TA2>

    BOOST_UBLAS_INLINE

    V

    axpy_prod (const vector_expression<E1> &e1,

               const compressed_matrix<T2, L2, 0, IA2, TA2> &e2) {

        typedef V vector_type;

        vector_type v (e2.size2 ());

        return axpy_prod (e1, e2, v, true);

    }

    template<class V, class E1, class E2>

    BOOST_UBLAS_INLINE

    V &

    axpy_prod (const vector_expression<E1> &e1,

               const matrix_expression<E2> &e2,

               V &v, packed_random_access_iterator_tag, column_major_tag) {

        typedef const E1 expression1_type;

        typedef const E2 expression2_type;

        typedef typename V::size_type size_type;

        typename expression2_type::const_iterator2 it2 (e2 ().begin2 ());

        typename expression2_type::const_iterator2 it2_end (e2 ().end2 ());

        while (it2 != it2_end) {

            size_type index2 (it2.index2 ());

#ifndef BOOST_UBLAS_NO_NESTED_CLASS_RELATION

            typename expression2_type::const_iterator1 it1 (it2.begin ());

            typename expression2_type::const_iterator1 it1_end (it2.end ());

#else

            typename expression2_type::const_iterator1 it1 (boost::numeric::ublas::begin (it2, iterator2_tag ()));

            typename expression2_type::const_iterator1 it1_end (boost::numeric::ublas::end (it2, iterator2_tag ()));

#endif

            while (it1 != it1_end) {

                v (index2) += *it1 * e1 () (it1.index1 ());

                ++ it1;

            }

            ++ it2;

        }

        return v;

    }

    template<class V, class E1, class E2>

    BOOST_UBLAS_INLINE

    V &

    axpy_prod (const vector_expression<E1> &e1,

               const matrix_expression<E2> &e2,

               V &v, packed_random_access_iterator_tag, row_major_tag) {

        typedef const E1 expression1_type;

        typedef const E2 expression2_type;

        typedef typename V::size_type size_type;

        typename expression2_type::const_iterator1 it1 (e2 ().begin1 ());

        typename expression2_type::const_iterator1 it1_end (e2 ().end1 ());

        while (it1 != it1_end) {

            size_type index1 (it1.index1 ());

#ifndef BOOST_UBLAS_NO_NESTED_CLASS_RELATION

            typename expression2_type::const_iterator2 it2 (it1.begin ());

            typename expression2_type::const_iterator2 it2_end (it1.end ());

#else

            typename expression2_type::const_iterator2 it2 (boost::numeric::ublas::begin (it1, iterator1_tag ()));

            typename expression2_type::const_iterator2 it2_end (boost::numeric::ublas::end (it1, iterator1_tag ()));

#endif

            while (it2 != it2_end) {

                v (it2.index2 ()) += *it2 * e1 () (index1);

                ++ it2;

            }

            ++ it1;

        }

        return v;

    }

    template<class V, class E1, class E2>

    BOOST_UBLAS_INLINE

    V &

    axpy_prod (const vector_expression<E1> &e1,

               const matrix_expression<E2> &e2,

               V &v, sparse_bidirectional_iterator_tag) {

        typedef const E1 expression1_type;

        typedef const E2 expression2_type;

        typedef typename V::size_type size_type;

        typename expression1_type::const_iterator it (e1 ().begin ());

        typename expression1_type::const_iterator it_end (e1 ().end ());

        while (it != it_end) {

            v.plus_assign (*it * row (e2 (), it.index ()));

            ++ it;

        }

        return v;

    }

    // Dispatcher

    template<class V, class E1, class E2>

    BOOST_UBLAS_INLINE

    V &

    axpy_prod (const vector_expression<E1> &e1,

               const matrix_expression<E2> &e2,

               V &v, packed_random_access_iterator_tag) {

        typedef typename E2::orientation_category orientation_category;

        return axpy_prod (e1, e2, v, packed_random_access_iterator_tag (), orientation_category ());

    }

  /** \brief computes <tt>v += A<sup>T</sup> x</tt> or <tt>v = A<sup>T</sup> x</tt> in an

          optimized fashion.

          \param e1 the vector expression \c x

          \param e2 the matrix expression \c A

          \param v  the result vector \c v

          \param init a boolean parameter

          <tt>axpy_prod(x, A, v, init)</tt> implements the well known

          axpy-product.  Setting \a init to \c true is equivalent to call

          <tt>v.clear()</tt> before <tt>axpy_prod</tt>. Currently \a init

          defaults to \c true, but this may change in the future.

          Up to now there are some specialisation for compressed

          matrices that give a large speed up compared to prod.

          \ingroup blas2

          \internal

          template parameters:

          \param V type of the result vector \c v

          \param E1 type of a vector expression \c x

          \param E2 type of a matrix expression \c A

  */

    template<class V, class E1, class E2>

    BOOST_UBLAS_INLINE

    V &

    axpy_prod (const vector_expression<E1> &e1,

               const matrix_expression<E2> &e2,

               V &v, bool init = true) {

        typedef typename V::value_type value_type;

        typedef typename E1::const_iterator::iterator_category iterator_category;

        if (init)

            v.assign (zero_vector<value_type> (e2 ().size2 ()));

#if BOOST_UBLAS_TYPE_CHECK

        vector<value_type> cv (v);

        typedef typename type_traits<value_type>::real_type real_type;

        real_type verrorbound (norm_1 (v) + norm_1 (e1) * norm_1 (e2));

        indexing_vector_assign<scalar_plus_assign> (cv, prod (e1, e2));

#endif

        axpy_prod (e1, e2, v, iterator_category ());

#if BOOST_UBLAS_TYPE_CHECK

        BOOST_UBLAS_CHECK (norm_1 (v - cv) <= 2 * std::numeric_limits<real_type>::epsilon () * verrorbound, internal_logic ());

#endif

        return v;

    }

    template<class V, class E1, class E2>

    BOOST_UBLAS_INLINE

    V

    axpy_prod (const vector_expression<E1> &e1,

               const matrix_expression<E2> &e2) {

        typedef V vector_type;

        vector_type v (e2 ().size2 ());

        return axpy_prod (e1, e2, v, true);

    }

    template<class M, class E1, class E2, class TRI>

    BOOST_UBLAS_INLINE

    M &

    axpy_prod (const matrix_expression<E1> &e1,

               const matrix_expression<E2> &e2,

               M &m, TRI,

               dense_proxy_tag, row_major_tag) {

        typedef M matrix_type;

        typedef const E1 expression1_type;

        typedef const E2 expression2_type;

        typedef typename M::size_type size_type;

        typedef typename M::value_type value_type;

#if BOOST_UBLAS_TYPE_CHECK

        matrix<value_type, row_major> cm (m);

        typedef typename type_traits<value_type>::real_type real_type;

        real_type merrorbound (norm_1 (m) + norm_1 (e1) * norm_1 (e2));

        indexing_matrix_assign<scalar_plus_assign> (cm, prod (e1, e2), row_major_tag ());

#endif

        size_type size1 (e1 ().size1 ());

        size_type size2 (e1 ().size2 ());

        for (size_type i = 0; i < size1; ++ i)

            for (size_type j = 0; j < size2; ++ j)

                row (m, i).plus_assign (e1 () (i, j) * row (e2 (), j));

#if BOOST_UBLAS_TYPE_CHECK

        BOOST_UBLAS_CHECK (norm_1 (m - cm) <= 2 * std::numeric_limits<real_type>::epsilon () * merrorbound, internal_logic ());

#endif

        return m;

    }

    template<class M, class E1, class E2, class TRI>

    BOOST_UBLAS_INLINE

    M &

    axpy_prod (const matrix_expression<E1> &e1,

               const matrix_expression<E2> &e2,

               M &m, TRI,

               sparse_proxy_tag, row_major_tag) {

        typedef M matrix_type;

        typedef TRI triangular_restriction;

        typedef const E1 expression1_type;

        typedef const E2 expression2_type;

        typedef typename M::size_type size_type;

        typedef typename M::value_type value_type;

#if BOOST_UBLAS_TYPE_CHECK

        matrix<value_type, row_major> cm (m);

        typedef typename type_traits<value_type>::real_type real_type;

        real_type merrorbound (norm_1 (m) + norm_1 (e1) * norm_1 (e2));

        indexing_matrix_assign<scalar_plus_assign> (cm, prod (e1, e2), row_major_tag ());

#endif

        typename expression1_type::const_iterator1 it1 (e1 ().begin1 ());

        typename expression1_type::const_iterator1 it1_end (e1 ().end1 ());

        while (it1 != it1_end) {

#ifndef BOOST_UBLAS_NO_NESTED_CLASS_RELATION

            typename expression1_type::const_iterator2 it2 (it1.begin ());

            typename expression1_type::const_iterator2 it2_end (it1.end ());

#else

            typename expression1_type::const_iterator2 it2 (boost::numeric::ublas::begin (it1, iterator1_tag ()));

            typename expression1_type::const_iterator2 it2_end (boost::numeric::ublas::end (it1, iterator1_tag ()));

#endif

            while (it2 != it2_end) {

                // row (m, it1.index1 ()).plus_assign (*it2 * row (e2 (), it2.index2 ()));

                matrix_row<expression2_type> mr (e2 (), it2.index2 ());

                typename matrix_row<expression2_type>::const_iterator itr (mr.begin ());

                typename matrix_row<expression2_type>::const_iterator itr_end (mr.end ());

                while (itr != itr_end) {

                    if (triangular_restriction::other (it1.index1 (), itr.index ()))

                        m (it1.index1 (), itr.index ()) += *it2 * *itr;

                    ++ itr;

                }

                ++ it2;

            }

            ++ it1;

        }

#if BOOST_UBLAS_TYPE_CHECK

        BOOST_UBLAS_CHECK (norm_1 (m - cm) <= 2 * std::numeric_limits<real_type>::epsilon () * merrorbound, internal_logic ());

#endif

        return m;

    }

    template<class M, class E1, class E2, class TRI>

    BOOST_UBLAS_INLINE

    M &

    axpy_prod (const matrix_expression<E1> &e1,

               const matrix_expression<E2> &e2,

               M &m, TRI,

               dense_proxy_tag, column_major_tag) {

        typedef M matrix_type;

        typedef const E1 expression1_type;

        typedef const E2 expression2_type;

        typedef typename M::size_type size_type;

        typedef typename M::value_type value_type;

#if BOOST_UBLAS_TYPE_CHECK

        matrix<value_type, column_major> cm (m);

        typedef typename type_traits<value_type>::real_type real_type;

        real_type merrorbound (norm_1 (m) + norm_1 (e1) * norm_1 (e2));

        indexing_matrix_assign<scalar_plus_assign> (cm, prod (e1, e2), column_major_tag ());

#endif

        size_type size1 (e2 ().size1 ());

        size_type size2 (e2 ().size2 ());

        for (size_type j = 0; j < size2; ++ j)

            for (size_type i = 0; i < size1; ++ i)

                column (m, j).plus_assign (e2 () (i, j) * column (e1 (), i));

#if BOOST_UBLAS_TYPE_CHECK

        BOOST_UBLAS_CHECK (norm_1 (m - cm) <= 2 * std::numeric_limits<real_type>::epsilon () * merrorbound, internal_logic ());

#endif

        return m;

    }

    template<class M, class E1, class E2, class TRI>

    BOOST_UBLAS_INLINE

    M &

    axpy_prod (const matrix_expression<E1> &e1,

               const matrix_expression<E2> &e2,

               M &m, TRI,

               sparse_proxy_tag, column_major_tag) {

        typedef M matrix_type;

        typedef TRI triangular_restriction;

        typedef const E1 expression1_type;

        typedef const E2 expression2_type;

        typedef typename M::size_type size_type;

        typedef typename M::value_type value_type;

#if BOOST_UBLAS_TYPE_CHECK

        matrix<value_type, column_major> cm (m);

        typedef typename type_traits<value_type>::real_type real_type;

        real_type merrorbound (norm_1 (m) + norm_1 (e1) * norm_1 (e2));

        indexing_matrix_assign<scalar_plus_assign> (cm, prod (e1, e2), column_major_tag ());

#endif

        typename expression2_type::const_iterator2 it2 (e2 ().begin2 ());

        typename expression2_type::const_iterator2 it2_end (e2 ().end2 ());

        while (it2 != it2_end) {

#ifndef BOOST_UBLAS_NO_NESTED_CLASS_RELATION

            typename expression2_type::const_iterator1 it1 (it2.begin ());

            typename expression2_type::const_iterator1 it1_end (it2.end ());

#else

            typename expression2_type::const_iterator1 it1 (boost::numeric::ublas::begin (it2, iterator2_tag ()));

            typename expression2_type::const_iterator1 it1_end (boost::numeric::ublas::end (it2, iterator2_tag ()));

#endif

            while (it1 != it1_end) {

                // column (m, it2.index2 ()).plus_assign (*it1 * column (e1 (), it1.index1 ()));

                matrix_column<expression1_type> mc (e1 (), it1.index1 ());

                typename matrix_column<expression1_type>::const_iterator itc (mc.begin ());

                typename matrix_column<expression1_type>::const_iterator itc_end (mc.end ());

                while (itc != itc_end) {

                    if (triangular_restriction::functor_type ().other (itc.index (), it2.index2 ()))

                        m (itc.index (), it2.index2 ()) += *it1 * *itc;

                    ++ itc;

                }

                ++ it1;

            }

            ++ it2;

        }

#if BOOST_UBLAS_TYPE_CHECK

        BOOST_UBLAS_CHECK (norm_1 (m - cm) <= 2 * std::numeric_limits<real_type>::epsilon () * merrorbound, internal_logic ());

#endif

        return m;

    }

    // Dispatcher

    template<class M, class E1, class E2, class TRI>

    BOOST_UBLAS_INLINE

    M &

    axpy_prod (const matrix_expression<E1> &e1,

               const matrix_expression<E2> &e2,

               M &m, TRI, bool init = true) {

        typedef typename M::value_type value_type;

        typedef typename M::storage_category storage_category;

        typedef typename M::orientation_category orientation_category;

        typedef TRI triangular_restriction;

        if (init)

            m.assign (zero_matrix<value_type> (e1 ().size1 (), e2 ().size2 ()));

        return axpy_prod (e1, e2, m, triangular_restriction (), storage_category (), orientation_category ());

    }

    template<class M, class E1, class E2, class TRI>

    BOOST_UBLAS_INLINE

    M

    axpy_prod (const matrix_expression<E1> &e1,

               const matrix_expression<E2> &e2,

               TRI) {

        typedef M matrix_type;

        typedef TRI triangular_restriction;

        matrix_type m (e1 ().size1 (), e2 ().size2 ());

        return axpy_prod (e1, e2, m, triangular_restriction (), true);

    }

  /** \brief computes <tt>M += A X</tt> or <tt>M = A X</tt> in an

          optimized fashion.

          \param e1 the matrix expression \c A

          \param e2 the matrix expression \c X

          \param m  the result matrix \c M

          \param init a boolean parameter

          <tt>axpy_prod(A, X, M, init)</tt> implements the well known

          axpy-product.  Setting \a init to \c true is equivalent to call

          <tt>M.clear()</tt> before <tt>axpy_prod</tt>. Currently \a init

          defaults to \c true, but this may change in the future.

          Up to now there are no specialisations.

          \ingroup blas3

          \internal

          template parameters:

          \param M type of the result matrix \c M

          \param E1 type of a matrix expression \c A

          \param E2 type of a matrix expression \c X

  */

    template<class M, class E1, class E2>

    BOOST_UBLAS_INLINE

    M &

    axpy_prod (const matrix_expression<E1> &e1,

               const matrix_expression<E2> &e2,

               M &m, bool init = true) {

        typedef typename M::value_type value_type;

        typedef typename M::storage_category storage_category;

        typedef typename M::orientation_category orientation_category;

        if (init)

            m.assign (zero_matrix<value_type> (e1 ().size1 (), e2 ().size2 ()));

        return axpy_prod (e1, e2, m, full (), storage_category (), orientation_category ());

    }

    template<class M, class E1, class E2>

    BOOST_UBLAS_INLINE

    M

    axpy_prod (const matrix_expression<E1> &e1,

               const matrix_expression<E2> &e2) {

        typedef M matrix_type;

        matrix_type m (e1 ().size1 (), e2 ().size2 ());

        return axpy_prod (e1, e2, m, full (), true);

    }

    template<class M, class E1, class E2>

    BOOST_UBLAS_INLINE

    M &

    opb_prod (const matrix_expression<E1> &e1,

              const matrix_expression<E2> &e2,

              M &m,

              dense_proxy_tag, row_major_tag) {

        typedef M matrix_type;

        typedef const E1 expression1_type;

        typedef const E2 expression2_type;

        typedef typename M::size_type size_type;

        typedef typename M::value_type value_type;

#if BOOST_UBLAS_TYPE_CHECK

        matrix<value_type, row_major> cm (m);

        typedef typename type_traits<value_type>::real_type real_type;

        real_type merrorbound (norm_1 (m) + norm_1 (e1) * norm_1 (e2));

        indexing_matrix_assign<scalar_plus_assign> (cm, prod (e1, e2), row_major_tag ());

#endif

        size_type size (BOOST_UBLAS_SAME (e1 ().size2 (), e2 ().size1 ()));

        for (size_type k = 0; k < size; ++ k) {

            vector<value_type> ce1 (column (e1 (), k));

            vector<value_type> re2 (row (e2 (), k));

            m.plus_assign (outer_prod (ce1, re2));

        }

#if BOOST_UBLAS_TYPE_CHECK

        BOOST_UBLAS_CHECK (norm_1 (m - cm) <= 2 * std::numeric_limits<real_type>::epsilon () * merrorbound, internal_logic ());

#endif

        return m;

    }

    template<class M, class E1, class E2>

    BOOST_UBLAS_INLINE

    M &

    opb_prod (const matrix_expression<E1> &e1,

              const matrix_expression<E2> &e2,

              M &m,

              dense_proxy_tag, column_major_tag) {

        typedef M matrix_type;

        typedef const E1 expression1_type;

        typedef const E2 expression2_type;

        typedef typename M::size_type size_type;

        typedef typename M::value_type value_type;

#if BOOST_UBLAS_TYPE_CHECK

        matrix<value_type, column_major> cm (m);

        typedef typename type_traits<value_type>::real_type real_type;

        real_type merrorbound (norm_1 (m) + norm_1 (e1) * norm_1 (e2));

        indexing_matrix_assign<scalar_plus_assign> (cm, prod (e1, e2), column_major_tag ());

#endif

        size_type size (BOOST_UBLAS_SAME (e1 ().size2 (), e2 ().size1 ()));

        for (size_type k = 0; k < size; ++ k) {

            vector<value_type> ce1 (column (e1 (), k));

            vector<value_type> re2 (row (e2 (), k));

            m.plus_assign (outer_prod (ce1, re2));

        }

#if BOOST_UBLAS_TYPE_CHECK

        BOOST_UBLAS_CHECK (norm_1 (m - cm) <= 2 * std::numeric_limits<real_type>::epsilon () * merrorbound, internal_logic ());

#endif

        return m;

    }

    // Dispatcher

  /** \brief computes <tt>M += A X</tt> or <tt>M = A X</tt> in an

          optimized fashion.

          \param e1 the matrix expression \c A

          \param e2 the matrix expression \c X

          \param m  the result matrix \c M

          \param init a boolean parameter

          <tt>opb_prod(A, X, M, init)</tt> implements the well known

          axpy-product. Setting \a init to \c true is equivalent to call

          <tt>M.clear()</tt> before <tt>opb_prod</tt>. Currently \a init

          defaults to \c true, but this may change in the future.

          This function may give a speedup if \c A has less columns than

          rows, because the product is computed as a sum of outer

          products.

          \ingroup blas3

          \internal

          template parameters:

          \param M type of the result matrix \c M

          \param E1 type of a matrix expression \c A

          \param E2 type of a matrix expression \c X

  */

    template<class M, class E1, class E2>

    BOOST_UBLAS_INLINE

    M &

    opb_prod (const matrix_expression<E1> &e1,

              const matrix_expression<E2> &e2,

              M &m, bool init = true) {

        typedef typename M::value_type value_type;

        typedef typename M::storage_category storage_category;

        typedef typename M::orientation_category orientation_category;

        if (init)

            m.assign (zero_matrix<value_type> (e1 ().size1 (), e2 ().size2 ()));

        return opb_prod (e1, e2, m, storage_category (), orientation_category ());

    }

    template<class M, class E1, class E2>

    BOOST_UBLAS_INLINE

    M

    opb_prod (const matrix_expression<E1> &e1,

              const matrix_expression<E2> &e2) {

        typedef M matrix_type;

        matrix_type m (e1 ().size1 (), e2 ().size2 ());

        return opb_prod (e1, e2, m, true);

    }

}}}

#endif