operation.hpp

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            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;