text: //
  //  Copyright (c) 2000-2002
  //  Joerg Walter, Mathias Koch
  //
  //  Distributed under the Boost Software License, Version 1.0. (See
  //  accompanying file LICENSE_1_0.txt or copy at
  //  http://www.boost.org/LICENSE_1_0.txt)
  //
  //  The authors gratefully acknowledge the support of
  //  GeNeSys mbH & Co. KG in producing this work.
  //
  
  #ifndef _BOOST_UBLAS_VECTOR_EXPRESSION_
  #define _BOOST_UBLAS_VECTOR_EXPRESSION_
  
  #include <boost/numeric/ublas/expression_types.hpp>
  
  
  // Expression templates based on ideas of Todd Veldhuizen and Geoffrey Furnish
  // Iterators based on ideas of Jeremy Siek
  //
  // Classes that model the Vector Expression concept
  
  namespace geofeatures_boost {} namespace boost = geofeatures_boost; namespace geofeatures_boost { namespace numeric { namespace ublas {
  
      template<class E>
      class vector_reference:
          public vector_expression<vector_reference<E> > {
  
          typedef vector_reference<E> self_type;
      public:
  #ifdef BOOST_UBLAS_ENABLE_PROXY_SHORTCUTS
          using vector_expression<vector_reference<E> >::operator ();
  #endif
          typedef typename E::size_type size_type;
          typedef typename E::difference_type difference_type;
          typedef typename E::value_type value_type;
          typedef typename E::const_reference const_reference;
          typedef typename geofeatures_boost::mpl::if_<geofeatures_boost::is_const<E>,
                                            typename E::const_reference,
                                            typename E::reference>::type reference;
          typedef E referred_type;
          typedef const self_type const_closure_type;
          typedef self_type closure_type;
          typedef typename E::storage_category storage_category;
  
          // Construction and destruction
          BOOST_UBLAS_INLINE
          explicit vector_reference (referred_type &e):
              e_ (e) {}
  
          // Accessors
          BOOST_UBLAS_INLINE
          size_type size () const {
              return expression ().size ();
          }
  
      public:
          // Expression accessors - const correct
          BOOST_UBLAS_INLINE
          const referred_type &expression () const {
              return e_;
          }
          BOOST_UBLAS_INLINE
          referred_type &expression () {
              return e_;
          }
  
      public:
          // Element access
  #ifndef BOOST_UBLAS_REFERENCE_CONST_MEMBER
          BOOST_UBLAS_INLINE
          const_reference operator () (size_type i) const {
              return expression () (i);
          }
          BOOST_UBLAS_INLINE
          reference operator () (size_type i) {
              return expression () (i);
          }
  
          BOOST_UBLAS_INLINE
          const_reference operator [] (size_type i) const {
              return expression () [i];
          }
          BOOST_UBLAS_INLINE
          reference operator [] (size_type i) {
              return expression () [i];
          }
  #else
          BOOST_UBLAS_INLINE
          reference operator () (size_type i) const {
              return expression () (i);
          }
  
          BOOST_UBLAS_INLINE
          reference operator [] (size_type i) const {
              return expression () [i];
          }
  #endif
  
          // Assignment
          BOOST_UBLAS_INLINE
          vector_reference &operator = (const vector_reference &v) {
              expression ().operator = (v);
              return *this;
          }
          template<class AE>
          BOOST_UBLAS_INLINE
          vector_reference &operator = (const vector_expression<AE> &ae) {
              expression ().operator = (ae);
              return *this;
          }
          template<class AE>
          BOOST_UBLAS_INLINE
          vector_reference &assign (const vector_expression<AE> &ae) {
              expression ().assign (ae);
              return *this;
          }
          template<class AE>
          BOOST_UBLAS_INLINE
          vector_reference &operator += (const vector_expression<AE> &ae) {
              expression ().operator += (ae);
              return *this;
          }
          template<class AE>
          BOOST_UBLAS_INLINE
          vector_reference &plus_assign (const vector_expression<AE> &ae) {
              expression ().plus_assign (ae);
              return *this;
          }
          template<class AE>
          BOOST_UBLAS_INLINE
          vector_reference &operator -= (const vector_expression<AE> &ae) {
              expression ().operator -= (ae);
              return *this;
          }
          template<class AE>
          BOOST_UBLAS_INLINE
          vector_reference &minus_assign (const vector_expression<AE> &ae) {
              expression ().minus_assign (ae);
              return *this;
          }
          template<class AT>
          BOOST_UBLAS_INLINE
          vector_reference &operator *= (const AT &at) {
              expression ().operator *= (at);
              return *this;
          }
          template<class AT>
          BOOST_UBLAS_INLINE
          vector_reference &operator /= (const AT &at) {
              expression ().operator /= (at);
              return *this;
          }
  
          // Swapping
          BOOST_UBLAS_INLINE
          void swap (vector_reference &v) {
              expression ().swap (v.expression ());
          }
  
          // Closure comparison
          BOOST_UBLAS_INLINE
          bool same_closure (const vector_reference &vr) const {
              return &(*this).e_ == &vr.e_;
          }
  
          // Iterator types
          typedef typename E::const_iterator const_iterator;
          typedef typename geofeatures_boost::mpl::if_<geofeatures_boost::is_const<E>,
                                            typename E::const_iterator,
                                            typename E::iterator>::type iterator;
  
          // Element lookup
          BOOST_UBLAS_INLINE
          const_iterator find (size_type i) const {
              return expression ().find (i);
          }
          BOOST_UBLAS_INLINE
          iterator find (size_type i) {
              return expression ().find (i);
          }
  
          // Iterator is the iterator of the referenced expression.
  
          BOOST_UBLAS_INLINE
          const_iterator begin () const {
              return expression ().begin ();
          }
          BOOST_UBLAS_INLINE
          const_iterator cbegin () const {
              return begin ();
          }
          BOOST_UBLAS_INLINE
          const_iterator end () const {
              return expression ().end ();
          }
          BOOST_UBLAS_INLINE
          const_iterator cend () const {
              return end ();
          }
  
          BOOST_UBLAS_INLINE
          iterator begin () {
              return expression ().begin ();
          }
          BOOST_UBLAS_INLINE
          iterator end () {
              return expression ().end ();
          }
  
          // Reverse iterator
          typedef reverse_iterator_base<const_iterator> const_reverse_iterator;
          typedef reverse_iterator_base<iterator> reverse_iterator;
  
          BOOST_UBLAS_INLINE
          const_reverse_iterator rbegin () const {
              return const_reverse_iterator (end ());
          }
          BOOST_UBLAS_INLINE
          const_reverse_iterator crbegin () const {
              return rbegin ();
          }
          BOOST_UBLAS_INLINE
          const_reverse_iterator rend () const {
              return const_reverse_iterator (begin ());
          }
          BOOST_UBLAS_INLINE
          const_reverse_iterator crend () const {
              return rend ();
          }
  
          BOOST_UBLAS_INLINE
          reverse_iterator rbegin () {
              return reverse_iterator (end ());
          }
          BOOST_UBLAS_INLINE
          reverse_iterator rend () {
              return reverse_iterator (begin ());
          }
  
      private:
          referred_type &e_;
      };
  
  
      template<class E, class F>
      class vector_unary:
          public vector_expression<vector_unary<E, F> > {
  
          typedef F functor_type;
          typedef typename geofeatures_boost::mpl::if_<geofeatures_boost::is_same<F, scalar_identity<typename E::value_type> >,
                                            E,
                                            const E>::type expression_type;
          typedef typename geofeatures_boost::mpl::if_<geofeatures_boost::is_const<expression_type>,
                                            typename E::const_closure_type,
                                            typename E::closure_type>::type expression_closure_type;
          typedef vector_unary<E, F> self_type;
      public:
  #ifdef BOOST_UBLAS_ENABLE_PROXY_SHORTCUTS
          using vector_expression<vector_unary<E, F> >::operator ();
  #endif
          typedef typename E::size_type size_type;
          typedef typename E::difference_type difference_type;
          typedef typename F::result_type value_type;
          typedef value_type const_reference;
          typedef typename geofeatures_boost::mpl::if_<geofeatures_boost::is_same<F, scalar_identity<value_type> >,
                                            typename E::reference,
                                            value_type>::type reference;
          typedef const self_type const_closure_type;
          typedef self_type closure_type;
          typedef unknown_storage_tag storage_category;
  
          // Construction and destruction
          BOOST_UBLAS_INLINE
          // May be used as mutable expression.
          explicit vector_unary (expression_type &e):
              e_ (e) {}
  
          // Accessors
          BOOST_UBLAS_INLINE
          size_type size () const {
              return e_.size ();
          }
  
      public:
          // Expression accessors
          BOOST_UBLAS_INLINE
          const expression_closure_type &expression () const {
              return e_;
          }
  
      public:
          // Element access
          BOOST_UBLAS_INLINE
          const_reference operator () (size_type i) const {
              return functor_type::apply (e_ (i));
          }
          BOOST_UBLAS_INLINE
          reference operator () (size_type i) {
              BOOST_STATIC_ASSERT ((geofeatures_boost::is_same<functor_type, scalar_identity<value_type > >::value));
              return e_ (i);
          }
  
          BOOST_UBLAS_INLINE
          const_reference operator [] (size_type i) const {
              return functor_type::apply (e_ [i]);
          }
          BOOST_UBLAS_INLINE
          reference operator [] (size_type i) {
              BOOST_STATIC_ASSERT ((geofeatures_boost::is_same<functor_type, scalar_identity<value_type > >::value));
              return e_ [i];
          }
  
          // Closure comparison
          BOOST_UBLAS_INLINE
          bool same_closure (const vector_unary &vu) const {
              return (*this).expression ().same_closure (vu.expression ());
          }
  
          // Iterator types
      private:
          typedef typename E::const_iterator const_subiterator_type;
          typedef const value_type *const_pointer;
  
      public:
  #ifdef BOOST_UBLAS_USE_INDEXED_ITERATOR
          typedef indexed_const_iterator<const_closure_type, typename const_subiterator_type::iterator_category> const_iterator;
          typedef const_iterator iterator;
  #else
          class const_iterator;
          typedef const_iterator iterator;
  #endif
  
          // Element lookup
          BOOST_UBLAS_INLINE
          const_iterator find (size_type i) const {
  #ifdef BOOST_UBLAS_USE_INDEXED_ITERATOR
              const_subiterator_type it (e_.find (i));
              return const_iterator (*this, it.index ());
  #else
              return const_iterator (*this, e_.find (i));
  #endif
          }
  
          // Iterator enhances the iterator of the referenced expression
          // with the unary functor.
  
  #ifndef BOOST_UBLAS_USE_INDEXED_ITERATOR
          class const_iterator:
              public container_const_reference<vector_unary>,
              public iterator_base_traits<typename E::const_iterator::iterator_category>::template
                          iterator_base<const_iterator, value_type>::type {
          public:
              typedef typename E::const_iterator::iterator_category iterator_category;
              typedef typename vector_unary::difference_type difference_type;
              typedef typename vector_unary::value_type value_type;
              typedef typename vector_unary::const_reference reference;
              typedef typename vector_unary::const_pointer pointer;
  
              // Construction and destruction
              BOOST_UBLAS_INLINE
              const_iterator ():
                  container_const_reference<self_type> (), it_ () {}
              BOOST_UBLAS_INLINE
              const_iterator (const self_type &vu, const const_subiterator_type &it):
                  container_const_reference<self_type> (vu), it_ (it) {}
  
              // Arithmetic
              BOOST_UBLAS_INLINE
              const_iterator &operator ++ () {
                  ++ it_;
                  return *this;
              }
              BOOST_UBLAS_INLINE
              const_iterator &operator -- () {
                  -- it_;
                  return *this;
              }
              BOOST_UBLAS_INLINE
              const_iterator &operator += (difference_type n) {
                  it_ += n;
                  return *this;
              }
              BOOST_UBLAS_INLINE
              const_iterator &operator -= (difference_type n) {
                  it_ -= n;
                  return *this;
              }
              BOOST_UBLAS_INLINE
              difference_type operator - (const const_iterator &it) const {
                  BOOST_UBLAS_CHECK ((*this) ().same_closure (it ()), external_logic ());
                  return it_ - it.it_;
              }
  
              // Dereference
              BOOST_UBLAS_INLINE
              const_reference operator * () const {
                  return functor_type::apply (*it_);
              }
              BOOST_UBLAS_INLINE
              const_reference operator [] (difference_type n) const {
                  return *(*this + n);
              }
  
              // Index
              BOOST_UBLAS_INLINE
              size_type index () const {
                  return it_.index ();
              }
  
              // Assignment
              BOOST_UBLAS_INLINE
              const_iterator &operator = (const const_iterator &it) {
                  container_const_reference<self_type>::assign (&it ());
                  it_ = it.it_;
                  return *this;
              }
  
              // Comparison
              BOOST_UBLAS_INLINE
              bool operator == (const const_iterator &it) const {
                  BOOST_UBLAS_CHECK ((*this) ().same_closure (it ()), external_logic ());
                  return it_ == it.it_;
              }
              BOOST_UBLAS_INLINE
              bool operator < (const const_iterator &it) const {
                  BOOST_UBLAS_CHECK ((*this) ().same_closure (it ()), external_logic ());
                  return it_ < it.it_;
              }
  
          private:
              const_subiterator_type it_;
          };
  #endif
  
          BOOST_UBLAS_INLINE
          const_iterator begin () const {
              return find (0); 
          }
          BOOST_UBLAS_INLINE
          const_iterator cbegin () const {
              return begin ();
          }
          BOOST_UBLAS_INLINE
          const_iterator end () const {
              return find (size ());
          }
          BOOST_UBLAS_INLINE
          const_iterator cend () const {
              return end ();
          }
  
          // Reverse iterator
          typedef reverse_iterator_base<const_iterator> const_reverse_iterator;
  
          BOOST_UBLAS_INLINE
          const_reverse_iterator rbegin () const {
              return const_reverse_iterator (end ());
          }
          BOOST_UBLAS_INLINE
          const_reverse_iterator crbegin () const {
              return rbegin ();
          }
          BOOST_UBLAS_INLINE
          const_reverse_iterator rend () const {
              return const_reverse_iterator (begin ());
          }
          BOOST_UBLAS_INLINE
          const_reverse_iterator crend () const {
              return rend ();
          }
  
      private:
          expression_closure_type e_;
      };
  
      template<class E, class F>
      struct vector_unary_traits {
          typedef vector_unary<E, F> expression_type;
  //FIXME
  // #ifndef BOOST_UBLAS_SIMPLE_ET_DEBUG
          typedef expression_type result_type;
  // #else
  //         typedef typename E::vector_temporary_type result_type;
  // #endif
      };
  
      // (- v) [i] = - v [i]
      template<class E> 
      BOOST_UBLAS_INLINE
      typename vector_unary_traits<E, scalar_negate<typename E::value_type> >::result_type
      operator - (const vector_expression<E> &e) {
          typedef typename vector_unary_traits<E, scalar_negate<typename E::value_type> >::expression_type expression_type;
          return expression_type (e ());
      }
  
      // (conj v) [i] = conj (v [i])
      template<class E> 
      BOOST_UBLAS_INLINE
      typename vector_unary_traits<E, scalar_conj<typename E::value_type> >::result_type
      conj (const vector_expression<E> &e) {
          typedef typename vector_unary_traits<E, scalar_conj<typename E::value_type> >::expression_type expression_type;
          return expression_type (e ());
      }
  
      // (real v) [i] = real (v [i])
      template<class E>
      BOOST_UBLAS_INLINE
      typename vector_unary_traits<E, scalar_real<typename E::value_type> >::result_type
      real (const vector_expression<E> &e) {
          typedef typename vector_unary_traits<E, scalar_real<typename E::value_type> >::expression_type expression_type;
          return expression_type (e ());
      }
  
      // (imag v) [i] = imag (v [i])
      template<class E>
      BOOST_UBLAS_INLINE
      typename vector_unary_traits<E, scalar_imag<typename E::value_type> >::result_type
      imag (const vector_expression<E> &e) {
          typedef typename vector_unary_traits<E, scalar_imag<typename E::value_type> >::expression_type expression_type;
          return expression_type (e ());
      }
  
      // (trans v) [i] = v [i]
      template<class E>
      BOOST_UBLAS_INLINE
      typename vector_unary_traits<const E, scalar_identity<typename E::value_type> >::result_type
      trans (const vector_expression<E> &e) {
          typedef typename vector_unary_traits<const E, scalar_identity<typename E::value_type> >::expression_type expression_type;
          return expression_type (e ());
      }
      template<class E>
      BOOST_UBLAS_INLINE
      typename vector_unary_traits<E, scalar_identity<typename E::value_type> >::result_type
      trans (vector_expression<E> &e) {
          typedef typename vector_unary_traits<E, scalar_identity<typename E::value_type> >::expression_type expression_type;
          return expression_type (e ());
      }
  
      // (herm v) [i] = conj (v [i])
      template<class E>
      BOOST_UBLAS_INLINE
      typename vector_unary_traits<E, scalar_conj<typename E::value_type> >::result_type
      herm (const vector_expression<E> &e) {
          typedef typename vector_unary_traits<E, scalar_conj<typename E::value_type> >::expression_type expression_type;
          return expression_type (e ());
      }
  
      template<class E1, class E2, class F>
      class vector_binary:
          public vector_expression<vector_binary<E1, E2, F> > {
  
          typedef E1 expression1_type;
          typedef E2 expression2_type;
          typedef F functor_type;
          typedef typename E1::const_closure_type expression1_closure_type;
          typedef typename E2::const_closure_type expression2_closure_type;
          typedef vector_binary<E1, E2, F> self_type;
      public:
  #ifdef BOOST_UBLAS_ENABLE_PROXY_SHORTCUTS
          using vector_expression<vector_binary<E1, E2, F> >::operator ();
  #endif
          typedef typename promote_traits<typename E1::size_type, typename E2::size_type>::promote_type size_type;
          typedef typename promote_traits<typename E1::difference_type, typename E2::difference_type>::promote_type difference_type;
          typedef typename F::result_type value_type;
          typedef value_type const_reference;
          typedef const_reference reference;
          typedef const self_type const_closure_type;
          typedef const_closure_type closure_type;
          typedef unknown_storage_tag storage_category;
  
          // Construction and destruction
          BOOST_UBLAS_INLINE
          vector_binary (const expression1_type &e1, const expression2_type &e2):
              e1_ (e1), e2_ (e2) {}
  
          // Accessors
          BOOST_UBLAS_INLINE
          size_type size () const { 
              return BOOST_UBLAS_SAME (e1_.size (), e2_.size ()); 
          }
  
      private:
          // Accessors
          BOOST_UBLAS_INLINE
          const expression1_closure_type &expression1 () const {
              return e1_;
          }
          BOOST_UBLAS_INLINE
          const expression2_closure_type &expression2 () const {
              return e2_;
          }
  
      public:
          // Element access
          BOOST_UBLAS_INLINE
          const_reference operator () (size_type i) const {
              return functor_type::apply (e1_ (i), e2_ (i));
          }
  
          BOOST_UBLAS_INLINE
          const_reference operator [] (size_type i) const {
              return functor_type::apply (e1_ [i], e2_ [i]);
          }
  
          // Closure comparison
          BOOST_UBLAS_INLINE
          bool same_closure (const vector_binary &vb) const {
              return (*this).expression1 ().same_closure (vb.expression1 ()) &&
                     (*this).expression2 ().same_closure (vb.expression2 ());
          }
  
          // Iterator types
      private:
          typedef typename E1::const_iterator const_subiterator1_type;
          typedef typename E2::const_iterator const_subiterator2_type;
          typedef const value_type *const_pointer;
  
      public:
  #ifdef BOOST_UBLAS_USE_INDEXED_ITERATOR
          typedef typename iterator_restrict_traits<typename const_subiterator1_type::iterator_category,
                                                    typename const_subiterator2_type::iterator_category>::iterator_category iterator_category;
          typedef indexed_const_iterator<const_closure_type, iterator_category> const_iterator;
          typedef const_iterator iterator;
  #else
          class const_iterator;
          typedef const_iterator iterator;
  #endif
  
          // Element lookup
          BOOST_UBLAS_INLINE
          const_iterator find (size_type i) const {
              const_subiterator1_type it1 (e1_.find (i));
              const_subiterator1_type it1_end (e1_.find (size ()));
              const_subiterator2_type it2 (e2_.find (i));
              const_subiterator2_type it2_end (e2_.find (size ()));
              i = (std::min) (it1 != it1_end ? it1.index () : size (),
                            it2 != it2_end ? it2.index () : size ());
  #ifdef BOOST_UBLAS_USE_INDEXED_ITERATOR
              return const_iterator (*this, i);
  #else
              return const_iterator (*this, i, it1, it1_end, it2, it2_end);
  #endif
          }
  
          // Iterator merges the iterators of the referenced expressions and
          // enhances them with the binary functor.
  
  #ifndef BOOST_UBLAS_USE_INDEXED_ITERATOR
          class const_iterator:
              public container_const_reference<vector_binary>,
              public iterator_base_traits<typename iterator_restrict_traits<typename E1::const_iterator::iterator_category,
                                                                            typename E2::const_iterator::iterator_category>::iterator_category>::template
                          iterator_base<const_iterator, value_type>::type {
          public:
              typedef typename iterator_restrict_traits<typename E1::const_iterator::iterator_category,
                                                        typename E2::const_iterator::iterator_category>::iterator_category iterator_category;
              typedef typename vector_binary::difference_type difference_type;
              typedef typename vector_binary::value_type value_type;
              typedef typename vector_binary::const_reference reference;
              typedef typename vector_binary::const_pointer pointer;
  
              // Construction and destruction
              BOOST_UBLAS_INLINE
              const_iterator ():
                  container_const_reference<self_type> (), i_ (), it1_ (), it1_end_ (), it2_ (), it2_end_ () {}
              BOOST_UBLAS_INLINE
              const_iterator (const self_type &vb, size_type i,
                              const const_subiterator1_type &it1, const const_subiterator1_type &it1_end,
                              const const_subiterator2_type &it2, const const_subiterator2_type &it2_end):
                  container_const_reference<self_type> (vb), i_ (i), it1_ (it1), it1_end_ (it1_end), it2_ (it2), it2_end_ (it2_end) {}
  
          private: 
              // Dense specializations
              BOOST_UBLAS_INLINE
              void increment (dense_random_access_iterator_tag) {
                  ++ i_; ++ it1_; ++ it2_;
              }
              BOOST_UBLAS_INLINE
              void decrement (dense_random_access_iterator_tag) {
                  -- i_; -- it1_; -- it2_;
              }
              BOOST_UBLAS_INLINE
              void increment (dense_random_access_iterator_tag, difference_type n) {
                  i_ += n; it1_ += n; it2_ += n;
              }
              BOOST_UBLAS_INLINE
              void decrement (dense_random_access_iterator_tag, difference_type n) {
                  i_ -= n; it1_ -= n; it2_ -= n;
              }
              BOOST_UBLAS_INLINE
              value_type dereference (dense_random_access_iterator_tag) const {
                  return functor_type::apply (*it1_, *it2_);
              }
  
              // Packed specializations
              BOOST_UBLAS_INLINE
              void increment (packed_random_access_iterator_tag) {
                  if (it1_ != it1_end_)
                      if (it1_.index () <= i_)
                          ++ it1_;
                  if (it2_ != it2_end_)
                      if (it2_.index () <= i_)
                          ++ it2_;
                  ++ i_;
              }
              BOOST_UBLAS_INLINE
              void decrement (packed_random_access_iterator_tag) {
                  if (it1_ != it1_end_)
                      if (i_ <= it1_.index ())
                          -- it1_;
                  if (it2_ != it2_end_)
                      if (i_ <= it2_.index ())
                          -- it2_;
                  -- i_;
              }
              BOOST_UBLAS_INLINE
              void increment (packed_random_access_iterator_tag, difference_type n) {
                  while (n > 0) {
                      increment (packed_random_access_iterator_tag ());
                      --n;
                  }
                  while (n < 0) {
                      decrement (packed_random_access_iterator_tag ());
                      ++n;
                  }
              }
              BOOST_UBLAS_INLINE
              void decrement (packed_random_access_iterator_tag, difference_type n) {
                  while (n > 0) {
                      decrement (packed_random_access_iterator_tag ());
                      --n;
                  }
                  while (n < 0) {
                      increment (packed_random_access_iterator_tag ());
                      ++n;
                  }
              }
              BOOST_UBLAS_INLINE
              value_type dereference (packed_random_access_iterator_tag) const {
                  value_type t1 = value_type/*zero*/();
                  if (it1_ != it1_end_)
                      if (it1_.index () == i_)
                          t1 = *it1_;
                  value_type t2 = value_type/*zero*/();
                  if (it2_ != it2_end_)
                      if (it2_.index () == i_)
                          t2 = *it2_;
                  return functor_type::apply (t1, t2);
              }
  
              // Sparse specializations
              BOOST_UBLAS_INLINE
              void increment (sparse_bidirectional_iterator_tag) {
                  size_type index1 = (*this) ().size ();
                  if (it1_ != it1_end_) {
                      if  (it1_.index () <= i_)
                          ++ it1_;
                      if (it1_ != it1_end_)
                          index1 = it1_.index ();
                  }
                  size_type index2 = (*this) ().size ();
                  if (it2_ != it2_end_) {
                      if (it2_.index () <= i_)
                          ++ it2_;
                      if (it2_ != it2_end_)
                          index2 = it2_.index ();
                  }
                  i_ = (std::min) (index1, index2);
              }
              BOOST_UBLAS_INLINE
              void decrement (sparse_bidirectional_iterator_tag) {
                  size_type index1 = (*this) ().size ();
                  if (it1_ != it1_end_) {
                      if (i_ <= it1_.index ())
                          -- it1_;
                      if (it1_ != it1_end_)
                          index1 = it1_.index ();
                  }
                  size_type index2 = (*this) ().size ();
                  if (it2_ != it2_end_) {
                      if (i_ <= it2_.index ())
                          -- it2_;
                      if (it2_ != it2_end_)
                          index2 = it2_.index ();
                  }
                  i_ = (std::max) (index1, index2);
              }
              BOOST_UBLAS_INLINE
              void increment (sparse_bidirectional_iterator_tag, difference_type n) {
                  while (n > 0) {
                      increment (sparse_bidirectional_iterator_tag ());
                      --n;
                  }
                  while (n < 0) {
                      decrement (sparse_bidirectional_iterator_tag ());
                      ++n;
                  }
              }
              BOOST_UBLAS_INLINE
              void decrement (sparse_bidirectional_iterator_tag, difference_type n) {
                  while (n > 0) {
                      decrement (sparse_bidirectional_iterator_tag ());
                      --n;
                  }
                  while (n < 0) {
                      increment (sparse_bidirectional_iterator_tag ());
                      ++n;
                  }
              }
              BOOST_UBLAS_INLINE
              value_type dereference (sparse_bidirectional_iterator_tag) const {
                  value_type t1 = value_type/*zero*/();
                  if (it1_ != it1_end_)
                      if (it1_.index () == i_)
                          t1 = *it1_;
                  value_type t2 = value_type/*zero*/();
                  if (it2_ != it2_end_)
                      if (it2_.index () == i_)
                          t2 = *it2_;
                  return functor_type::apply (t1, t2);
              }
  
          public: 
              // Arithmetic
              BOOST_UBLAS_INLINE
              const_iterator &operator ++ () {
                  increment (iterator_category ());
                  return *this;
              }
              BOOST_UBLAS_INLINE
              const_iterator &operator -- () {
                  decrement (iterator_category ());
                  return *this;
              }
              BOOST_UBLAS_INLINE
              const_iterator &operator += (difference_type n) {
                  increment (iterator_category (), n);
                  return *this;
              }
              BOOST_UBLAS_INLINE
              const_iterator &operator -= (difference_type n) {
                  decrement (iterator_category (), n);
                  return *this;
              }
              BOOST_UBLAS_INLINE
              difference_type operator - (const const_iterator &it) const {
                  BOOST_UBLAS_CHECK ((*this) ().same_closure (it ()), external_logic ());
                  return index () - it.index ();
              }
  
              // Dereference
              BOOST_UBLAS_INLINE
              const_reference operator * () const {
                  return dereference (iterator_category ());
              }
              BOOST_UBLAS_INLINE
              const_reference operator [] (difference_type n) const {
                  return *(*this + n);
              }
  
              // Index
              BOOST_UBLAS_INLINE
              size_type index () const {
                  return i_;
              }
  
              // Assignment
              BOOST_UBLAS_INLINE
              const_iterator &operator = (const const_iterator &it) {
                  container_const_reference<self_type>::assign (&it ());
                  i_ = it.i_;
                  it1_ = it.it1_;
                  it1_end_ = it.it1_end_;
                  it2_ = it.it2_;
                  it2_end_ = it.it2_end_;
                  return *this;
              }
  
              // Comparison
              BOOST_UBLAS_INLINE
              bool operator == (const const_iterator &it) const {
                  BOOST_UBLAS_CHECK ((*this) ().same_closure (it ()), external_logic ());
                  return index () == it.index ();
              }
              BOOST_UBLAS_INLINE
              bool operator < (const const_iterator &it) const {
                  BOOST_UBLAS_CHECK ((*this) ().same_closure (it ()), external_logic ());
                  return index () < it.index ();
              }
  
          private:
              size_type i_;
              const_subiterator1_type it1_;
              const_subiterator1_type it1_end_;
              const_subiterator2_type it2_;
              const_subiterator2_type it2_end_;
          };
  #endif
  
          BOOST_UBLAS_INLINE
          const_iterator begin () const {
              return find (0);
          }
          BOOST_UBLAS_INLINE
          const_iterator cbegin () const {
              return begin ();
          }
          BOOST_UBLAS_INLINE
          const_iterator end () const {
              return find (size ());
          }
          BOOST_UBLAS_INLINE
          const_iterator cend () const {
              return end ();
          }
  
          // Reverse iterator
          typedef reverse_iterator_base<const_iterator> const_reverse_iterator;
  
          BOOST_UBLAS_INLINE
          const_reverse_iterator rbegin () const {
              return const_reverse_iterator (end ());
          }
          BOOST_UBLAS_INLINE
          const_reverse_iterator crbegin () const {
              return rbegin ();
          }
          BOOST_UBLAS_INLINE
          const_reverse_iterator rend () const {
              return const_reverse_iterator (begin ());
          }
          BOOST_UBLAS_INLINE
          const_reverse_iterator crend () const {
              return rend ();
          }
  
      private:
          expression1_closure_type e1_;
          expression2_closure_type e2_;
      };
  
      template<class E1, class E2, class F>
      struct vector_binary_traits {
          typedef vector_binary<E1, E2, F> expression_type;
  #ifndef BOOST_UBLAS_SIMPLE_ET_DEBUG
          typedef expression_type result_type; 
  #else
          typedef typename E1::vector_temporary_type result_type;
  #endif
      };
  
      // (v1 + v2) [i] = v1 [i] + v2 [i]
      template<class E1, class E2>
      BOOST_UBLAS_INLINE
      typename vector_binary_traits<E1, E2, scalar_plus<typename E1::value_type, 
                                                        typename E2::value_type> >::result_type
      operator + (const vector_expression<E1> &e1,
                  const vector_expression<E2> &e2) {
          typedef typename vector_binary_traits<E1, E2, scalar_plus<typename E1::value_type,
                                                                                typename E2::value_type> >::expression_type expression_type;
          return expression_type (e1 (), e2 ());
      }
  
      // (v1 - v2) [i] = v1 [i] - v2 [i]
      template<class E1, class E2>
      BOOST_UBLAS_INLINE
      typename vector_binary_traits<E1, E2, scalar_minus<typename E1::value_type,
                                                         typename E2::value_type> >::result_type
      operator - (const vector_expression<E1> &e1,
                  const vector_expression<E2> &e2) {
          typedef typename vector_binary_traits<E1, E2, scalar_minus<typename E1::value_type,
                                                                                 typename E2::value_type> >::expression_type expression_type;
          return expression_type (e1 (), e2 ());
      }
  
      // (v1 * v2) [i] = v1 [i] * v2 [i]
      template<class E1, class E2>
      BOOST_UBLAS_INLINE
      typename vector_binary_traits<E1, E2, scalar_multiplies<typename E1::value_type,
                                                              typename E2::value_type> >::result_type
      element_prod (const vector_expression<E1> &e1,
                    const vector_expression<E2> &e2) {
          typedef typename vector_binary_traits<E1, E2, scalar_multiplies<typename E1::value_type,
                                                                                      typename E2::value_type> >::expression_type expression_type;
          return expression_type (e1 (), e2 ());
      }
  
      // (v1 / v2) [i] = v1 [i] / v2 [i]
      template<class E1, class E2>
      BOOST_UBLAS_INLINE
      typename vector_binary_traits<E1, E2, scalar_divides<typename E1::value_type,
                                                           typename E2::value_type> >::result_type
      element_div (const vector_expression<E1> &e1,
                   const vector_expression<E2> &e2) {
          typedef typename vector_binary_traits<E1, E2, scalar_divides<typename E1::value_type,
                                                                                   typename E2::value_type> >::expression_type expression_type;
          return expression_type (e1 (), e2 ());
      }
  
  
      template<class E1, class E2, class F>
      class vector_binary_scalar1:
          public vector_expression<vector_binary_scalar1<E1, E2, F> > {
  
          typedef F functor_type;
          typedef E1 expression1_type;
          typedef E2 expression2_type;
      public:
          typedef const E1& expression1_closure_type;
          typedef typename E2::const_closure_type expression2_closure_type;
      private:
          typedef vector_binary_scalar1<E1, E2, F> self_type;
      public:
  #ifdef BOOST_UBLAS_ENABLE_PROXY_SHORTCUTS
          using vector_expression<vector_binary_scalar1<E1, E2, F> >::operator ();
  #endif
          typedef typename E2::size_type size_type;
          typedef typename E2::difference_type difference_type;
          typedef typename F::result_type value_type;
          typedef value_type const_reference;
          typedef const_reference reference;
          typedef const self_type const_closure_type;
          typedef const_closure_type closure_type;
          typedef unknown_storage_tag storage_category;
  
          // Construction and destruction
          BOOST_UBLAS_INLINE
          vector_binary_scalar1 (const expression1_type &e1, const expression2_type &e2):
              e1_ (e1), e2_ (e2) {}
  
          // Accessors
          BOOST_UBLAS_INLINE
          size_type size () const {
              return e2_.size ();
          }
  
      public:
          // Element access
          BOOST_UBLAS_INLINE
          const_reference operator () (size_type i) const {
              return functor_type::apply (e1_, e2_ (i));
          }
  
          BOOST_UBLAS_INLINE
          const_reference operator [] (size_type i) const {
              return functor_type::apply (e1_, e2_ [i]);
          }
  
          // Closure comparison
          BOOST_UBLAS_INLINE
          bool same_closure (const vector_binary_scalar1 &vbs1) const {
              return &e1_ == &(vbs1.e1_) &&
                     (*this).e2_.same_closure (vbs1.e2_);
          }
  
          // Iterator types
      private:
          typedef expression1_type const_subiterator1_type;
          typedef typename expression2_type::const_iterator const_subiterator2_type;
          typedef const value_type *const_pointer;
  
      public:
  #ifdef BOOST_UBLAS_USE_INDEXED_ITERATOR
          typedef indexed_const_iterator<const_closure_type, typename const_subiterator2_type::iterator_category> const_iterator;
          typedef const_iterator iterator;
  #else
          class const_iterator;
          typedef const_iterator iterator;
  #endif
  
          // Element lookup
          BOOST_UBLAS_INLINE
          const_iterator find (size_type i) const {
  #ifdef BOOST_UBLAS_USE_INDEXED_ITERATOR
              const_subiterator2_type it (e2_.find (i));
              return const_iterator (*this, it.index ());
  #else
              return const_iterator (*this, const_subiterator1_type (e1_), e2_.find (i));
  #endif
          }
  
          // Iterator enhances the iterator of the referenced vector expression
          // with the binary functor.
  
  #ifndef BOOST_UBLAS_USE_INDEXED_ITERATOR
          class const_iterator:
              public container_const_reference<vector_binary_scalar1>,
              public iterator_base_traits<typename E2::const_iterator::iterator_category>::template
                          iterator_base<const_iterator, value_type>::type {
          public:
              typedef typename E2::const_iterator::iterator_category iterator_category;
              typedef typename vector_binary_scalar1::difference_type difference_type;
              typedef typename vector_binary_scalar1::value_type value_type;
              typedef typename vector_binary_scalar1::const_reference reference;
              typedef typename vector_binary_scalar1::const_pointer pointer;
  
              // Construction and destruction
              BOOST_UBLAS_INLINE
              const_iterator ():
                  container_const_reference<self_type> (), it1_ (), it2_ () {}
              BOOST_UBLAS_INLINE
              const_iterator (const self_type &vbs, const const_subiterator1_type &it1, const const_subiterator2_type &it2):
                  container_const_reference<self_type> (vbs), it1_ (it1), it2_ (it2) {}
  
              // Arithmetic
              BOOST_UBLAS_INLINE
              const_iterator &operator ++ () {
                  ++ it2_;
                  return *this;
              }
              BOOST_UBLAS_INLINE
              const_iterator &operator -- () {
                  -- it2_;
                  return *this;
              }
              BOOST_UBLAS_INLINE
              const_iterator &operator += (difference_type n) {
                  it2_ += n;
                  return *this;
              }
              BOOST_UBLAS_INLINE
              const_iterator &operator -= (difference_type n) {
                  it2_ -= n;
                  return *this;
              }
              BOOST_UBLAS_INLINE
              difference_type operator - (const const_iterator &it) const {
                  BOOST_UBLAS_CHECK ((*this) ().same_closure (it ()), external_logic ());
                  // FIXME we shouldn't compare floats
                  // BOOST_UBLAS_CHECK (it1_ == it.it1_, external_logic ());
                  return it2_ - it.it2_;
              }
  
              // Dereference
              BOOST_UBLAS_INLINE
              const_reference operator * () const {
                  return functor_type::apply (it1_, *it2_);
              }
              BOOST_UBLAS_INLINE
              const_reference operator [] (difference_type n) const {
                  return *(*this + n);
              }
  
              // Index
              BOOST_UBLAS_INLINE
              size_type index () const {
                  return it2_.index ();
              }
  
              // Assignment 
              BOOST_UBLAS_INLINE
              const_iterator &operator = (const const_iterator &it) {
                  container_const_reference<self_type>::assign (&it ());
                  it1_ = it.it1_;
                  it2_ = it.it2_;
                  return *this;
              }
  
              // Comparison
              BOOST_UBLAS_INLINE
              bool operator == (const const_iterator &it) const {
                  BOOST_UBLAS_CHECK ((*this) ().same_closure (it ()), external_logic ());
                  // FIXME we shouldn't compare floats
                  // BOOST_UBLAS_CHECK (it1_ == it.it1_, external_logic ());
                  return it2_ == it.it2_;
              }
              BOOST_UBLAS_INLINE
              bool operator < (const const_iterator &it) const {
                  BOOST_UBLAS_CHECK ((*this) ().same_closure (it ()), external_logic ());
                  // FIXME we shouldn't compare floats
                  // BOOST_UBLAS_CHECK (it1_ == it.it1_, external_logic ());
                  return it2_ < it.it2_;
              }
  
          private:
              const_subiterator1_type it1_;
              const_subiterator2_type it2_;
          };
  #endif
  
          BOOST_UBLAS_INLINE
          const_iterator begin () const {
              return find (0); 
          }
          BOOST_UBLAS_INLINE
          const_iterator cbegin () const {
              return begin ();
          }
          BOOST_UBLAS_INLINE
          const_iterator end () const {
              return find (size ()); 
          }
          BOOST_UBLAS_INLINE
          const_iterator cend () const {
              return end ();
          }
  
          // Reverse iterator
          typedef reverse_iterator_base<const_iterator> const_reverse_iterator;
  
          BOOST_UBLAS_INLINE
          const_reverse_iterator rbegin () const {
              return const_reverse_iterator (end ());
          }
          BOOST_UBLAS_INLINE
          const_reverse_iterator crbegin () const {
              return rbegin ();
          }
          BOOST_UBLAS_INLINE
          const_reverse_iterator rend () const {
              return const_reverse_iterator (begin ());
          }
          BOOST_UBLAS_INLINE
          const_reverse_iterator crend () const {
              return end ();
          }
  
      private:
          expression1_closure_type e1_;
          expression2_closure_type e2_;
      };
  
      template<class E1, class E2, class F>
      struct vector_binary_scalar1_traits {
          typedef vector_binary_scalar1<E1, E2, F> expression_type;   // allow E1 to be builtin type
  #ifndef BOOST_UBLAS_SIMPLE_ET_DEBUG
          typedef expression_type result_type;
  #else
          typedef typename E2::vector_temporary_type result_type;
  #endif
      };
  
      // (t * v) [i] = t * v [i]
      template<class T1, class E2>
      BOOST_UBLAS_INLINE
      typename enable_if< is_convertible<T1, typename E2::value_type >,    
      typename vector_binary_scalar1_traits<const T1, E2, scalar_multiplies<T1, typename E2::value_type> >::result_type
      >::type
      operator * (const T1 &e1,
                  const vector_expression<E2> &e2) {
          typedef typename vector_binary_scalar1_traits<const T1, E2, scalar_multiplies<T1, typename E2::value_type> >::expression_type expression_type;
          return expression_type (e1, e2 ());
      }
  
  
      template<class E1, class E2, class F>
      class vector_binary_scalar2:
          public vector_expression<vector_binary_scalar2<E1, E2, F> > {
  
          typedef F functor_type;
          typedef E1 expression1_type;
          typedef E2 expression2_type;
          typedef typename E1::const_closure_type expression1_closure_type;
          typedef const E2& expression2_closure_type;
          typedef vector_binary_scalar2<E1, E2, F> self_type;
      public:
  #ifdef BOOST_UBLAS_ENABLE_PROXY_SHORTCUTS
          using vector_expression<vector_binary_scalar2<E1, E2, F> >::operator ();
  #endif
          typedef typename E1::size_type size_type;
          typedef typename E1::difference_type difference_type;
          typedef typename F::result_type value_type;
          typedef value_type const_reference;
          typedef const_reference reference;
          typedef const self_type const_closure_type;
          typedef const_closure_type closure_type;
          typedef unknown_storage_tag storage_category;
  
          // Construction and destruction
          BOOST_UBLAS_INLINE
          vector_binary_scalar2 (const expression1_type &e1, const expression2_type &e2):
              e1_ (e1), e2_ (e2) {}
  
          // Accessors
          BOOST_UBLAS_INLINE
          size_type size () const {
              return e1_.size (); 
          }
  
      public:
          // Element access
          BOOST_UBLAS_INLINE
          const_reference operator () (size_type i) const {
              return functor_type::apply (e1_ (i), e2_);
          }
  
          BOOST_UBLAS_INLINE
          const_reference operator [] (size_type i) const {
              return functor_type::apply (e1_ [i], e2_);
          }
  
          // Closure comparison
          BOOST_UBLAS_INLINE
          bool same_closure (const vector_binary_scalar2 &vbs2) const {
              return (*this).e1_.same_closure (vbs2.e1_) &&
                     &e2_ == &(vbs2.e2_);
          }
  
          // Iterator types
      private:
          typedef typename expression1_type::const_iterator const_subiterator1_type;
          typedef expression2_type const_subiterator2_type;
          typedef const value_type *const_pointer;
  
      public:
  #ifdef BOOST_UBLAS_USE_INDEXED_ITERATOR
          typedef indexed_const_iterator<const_closure_type, typename const_subiterator2_type::iterator_category> const_iterator;
          typedef const_iterator iterator;
  #else
          class const_iterator;
          typedef const_iterator iterator;
  #endif
  
          // Element lookup
          BOOST_UBLAS_INLINE
          const_iterator find (size_type i) const {
  #ifdef BOOST_UBLAS_USE_INDEXED_ITERATOR
              const_subiterator1_type it (e1_.find (i));
              return const_iterator (*this, it.index ());
  #else
              return const_iterator (*this, e1_.find (i), const_subiterator2_type (e2_));
  #endif
          }
  
          // Iterator enhances the iterator of the referenced vector expression
          // with the binary functor.
  
  #ifndef BOOST_UBLAS_USE_INDEXED_ITERATOR
          class const_iterator:
              public container_const_reference<vector_binary_scalar2>,
              public iterator_base_traits<typename E1::const_iterator::iterator_category>::template
                          iterator_base<const_iterator, value_type>::type {
          public:
              typedef typename E1::const_iterator::iterator_category iterator_category;
              typedef typename vector_binary_scalar2::difference_type difference_type;
              typedef typename vector_binary_scalar2::value_type value_type;
              typedef typename vector_binary_scalar2::const_reference reference;
              typedef typename vector_binary_scalar2::const_pointer pointer;
  
              // Construction and destruction
              BOOST_UBLAS_INLINE
              const_iterator ():
                  container_const_reference<self_type> (), it1_ (), it2_ () {}
              BOOST_UBLAS_INLINE
              const_iterator (const self_type &vbs, const const_subiterator1_type &it1, const const_subiterator2_type &it2):
                  container_const_reference<self_type> (vbs), it1_ (it1), it2_ (it2) {}
  
              // Arithmetic
              BOOST_UBLAS_INLINE
              const_iterator &operator ++ () {
                  ++ it1_;
                  return *this;
              }
              BOOST_UBLAS_INLINE
              const_iterator &operator -- () {
                  -- it1_;
                  return *this;
              }
              BOOST_UBLAS_INLINE
              const_iterator &operator += (difference_type n) {
                  it1_ += n;
                  return *this;
              }
              BOOST_UBLAS_INLINE
              const_iterator &operator -= (difference_type n) {
                  it1_ -= n;
                  return *this;
              }
              BOOST_UBLAS_INLINE
              difference_type operator - (const const_iterator &it) const {
                  BOOST_UBLAS_CHECK ((*this) ().same_closure (it ()), external_logic ());
                  // FIXME we shouldn't compare floats
                  // BOOST_UBLAS_CHECK (it2_ == it.it2_, external_logic ());
                  return it1_ - it.it1_;
              }
  
              // Dereference
              BOOST_UBLAS_INLINE
              const_reference operator * () const {
                  return functor_type::apply (*it1_, it2_);
              }
              BOOST_UBLAS_INLINE
              const_reference operator [] (difference_type n) const {
                  return *(*this + n);
              }
  
              // Index
              BOOST_UBLAS_INLINE
              size_type index () const {
                  return it1_.index ();
              }
  
              // Assignment
              BOOST_UBLAS_INLINE
              const_iterator &operator = (const const_iterator &it) {
                  container_const_reference<self_type>::assign (&it ());
                  it1_ = it.it1_;
                  it2_ = it.it2_;
                  return *this;
              }
  
              // Comparison
              BOOST_UBLAS_INLINE
              bool operator == (const const_iterator &it) const {
                  BOOST_UBLAS_CHECK ((*this) ().same_closure (it ()), external_logic ());
                  // FIXME we shouldn't compare floats
                  // BOOST_UBLAS_CHECK (it2_ == it.it2_, external_logic ());
                  return it1_ == it.it1_;
              }
              BOOST_UBLAS_INLINE
              bool operator < (const const_iterator &it) const {
                  BOOST_UBLAS_CHECK ((*this) ().same_closure (it ()), external_logic ());
                  // FIXME we shouldn't compare floats
                  // BOOST_UBLAS_CHECK (it2_ == it.it2_, external_logic ());
                  return it1_ < it.it1_;
              }
  
          private:
              const_subiterator1_type it1_;
              const_subiterator2_type it2_;
          };
  #endif
  
          BOOST_UBLAS_INLINE
          const_iterator begin () const {
              return find (0);
          }
          BOOST_UBLAS_INLINE
          const_iterator cbegin () const {
              return begin ();
          }
          BOOST_UBLAS_INLINE
          const_iterator end () const {
              return find (size ());
          }
          BOOST_UBLAS_INLINE
          const_iterator cend () const {
              return end ();
          }
  
          // Reverse iterator
          typedef reverse_iterator_base<const_iterator> const_reverse_iterator;
  
          BOOST_UBLAS_INLINE
          const_reverse_iterator rbegin () const {
              return const_reverse_iterator (end ());
          }
          BOOST_UBLAS_INLINE
          const_reverse_iterator crbegin () const {
              return rbegin ();
          }
          BOOST_UBLAS_INLINE
          const_reverse_iterator rend () const {
              return const_reverse_iterator (begin ());
          }
          BOOST_UBLAS_INLINE
          const_reverse_iterator crend () const {
              return rend ();
          }
  
      private:
          expression1_closure_type e1_;
          expression2_closure_type e2_;
      };
  
      template<class E1, class E2, class F>
      struct vector_binary_scalar2_traits {
          typedef vector_binary_scalar2<E1, E2, F> expression_type;   // allow E2 to be builtin type
  #ifndef BOOST_UBLAS_SIMPLE_ET_DEBUG
          typedef expression_type result_type;
  #else
          typedef typename E1::vector_temporary_type result_type;
  #endif
      };
  
      // (v * t) [i] = v [i] * t
      template<class E1, class T2>
      BOOST_UBLAS_INLINE
      typename enable_if< is_convertible<T2, typename E1::value_type >,    
      typename vector_binary_scalar2_traits<E1, const T2, scalar_multiplies<typename E1::value_type, T2> >::result_type
      >::type
      operator * (const vector_expression<E1> &e1,
                  const T2 &e2) {
          typedef typename vector_binary_scalar2_traits<E1, const T2, scalar_multiplies<typename E1::value_type, T2> >::expression_type expression_type;
          return expression_type (e1 (), e2);
      }
  
      // (v / t) [i] = v [i] / t
      template<class E1, class T2>
      BOOST_UBLAS_INLINE
      typename enable_if< is_convertible<T2, typename E1::value_type >,    
      typename vector_binary_scalar2_traits<E1, const T2, scalar_divides<typename E1::value_type, T2> >::result_type
      >::type
      operator / (const vector_expression<E1> &e1,
                  const T2 &e2) {
          typedef typename vector_binary_scalar2_traits<E1, const T2, scalar_divides<typename E1::value_type, T2> >::expression_type expression_type;
          return expression_type (e1 (), e2);
      }
  
  
      template<class E, class F>
      class vector_scalar_unary:
          public scalar_expression<vector_scalar_unary<E, F> > {
  
          typedef E expression_type;
          typedef F functor_type;
          typedef typename E::const_closure_type expression_closure_type;
          typedef typename E::const_iterator::iterator_category iterator_category;
          typedef vector_scalar_unary<E, F> self_type;
      public:
          typedef typename F::result_type value_type;
          typedef typename E::difference_type difference_type;
          typedef const self_type const_closure_type;
          typedef const_closure_type closure_type;
          typedef unknown_storage_tag storage_category;
  
          // Construction and destruction
          BOOST_UBLAS_INLINE
          explicit vector_scalar_unary (const expression_type &e):
              e_ (e) {}
  
      private:
          // Expression accessors
          BOOST_UBLAS_INLINE
          const expression_closure_type &expression () const {
              return e_;
          }
  
      public:
          BOOST_UBLAS_INLINE
          operator value_type () const {
              return evaluate (iterator_category ());
          }
  
      private:
          // Dense random access specialization
          BOOST_UBLAS_INLINE
          value_type evaluate (dense_random_access_iterator_tag) const {
  #ifdef BOOST_UBLAS_USE_INDEXING
              return functor_type::apply (e_);
  #elif BOOST_UBLAS_USE_ITERATING
              difference_type size = e_.size ();
              return functor_type::apply (size, e_.begin ());
  #else
              difference_type size = e_.size ();
              if (size >= BOOST_UBLAS_ITERATOR_THRESHOLD)
                  return functor_type::apply (size, e_.begin ());
              else
                  return functor_type::apply (e_);
  #endif
          }
  
          // Packed bidirectional specialization
          BOOST_UBLAS_INLINE
          value_type evaluate (packed_random_access_iterator_tag) const {
              return functor_type::apply (e_.begin (), e_.end ());
          }
  
          // Sparse bidirectional specialization
          BOOST_UBLAS_INLINE
          value_type evaluate (sparse_bidirectional_iterator_tag) const {
              return functor_type::apply (e_.begin (), e_.end ());
          }
  
      private:
          expression_closure_type e_;
      };
  
      template<class E, class F>
      struct vector_scalar_unary_traits {
          typedef vector_scalar_unary<E, F> expression_type;
  #if !defined (BOOST_UBLAS_SIMPLE_ET_DEBUG) && defined (BOOST_UBLAS_USE_SCALAR_ET)
  // FIXME don't define USE_SCALAR_ET other then for testing
  // They do not work for complex types
           typedef expression_type result_type;
  #else
           typedef typename F::result_type result_type;
  #endif
      };
  
      // sum v = sum (v [i])
      template<class E>
      BOOST_UBLAS_INLINE
      typename vector_scalar_unary_traits<E, vector_sum<E> >::result_type
      sum (const vector_expression<E> &e) {
          typedef typename vector_scalar_unary_traits<E, vector_sum<E> >::expression_type expression_type;
          return expression_type (e ());
      }
  
      // real: norm_1 v = sum (abs (v [i]))
      // complex: norm_1 v = sum (abs (real (v [i])) + abs (imag (v [i])))
      template<class E>
      BOOST_UBLAS_INLINE
      typename vector_scalar_unary_traits<E, vector_norm_1<E> >::result_type
      norm_1 (const vector_expression<E> &e) {
          typedef typename vector_scalar_unary_traits<E, vector_norm_1<E> >::expression_type expression_type;
          return expression_type (e ());
      }
  
      // real: norm_2 v = sqrt (sum (v [i] * v [i]))
      // complex: norm_2 v = sqrt (sum (v [i] * conj (v [i])))
      template<class E>
      BOOST_UBLAS_INLINE
      typename vector_scalar_unary_traits<E, vector_norm_2<E> >::result_type
      norm_2 (const vector_expression<E> &e) {
          typedef typename vector_scalar_unary_traits<E, vector_norm_2<E> >::expression_type expression_type;
          return expression_type (e ());
      }
  
      // real: norm_inf v = maximum (abs (v [i]))
      // complex: norm_inf v = maximum (maximum (abs (real (v [i])), abs (imag (v [i]))))
      template<class E>
      BOOST_UBLAS_INLINE
      typename vector_scalar_unary_traits<E, vector_norm_inf<E> >::result_type
      norm_inf (const vector_expression<E> &e) {
          typedef typename vector_scalar_unary_traits<E, vector_norm_inf<E> >::expression_type expression_type;
          return expression_type (e ());
      }
  
      // real: index_norm_inf v = minimum (i: abs (v [i]) == maximum (abs (v [i])))
      template<class E>
      BOOST_UBLAS_INLINE
      typename vector_scalar_unary_traits<E, vector_index_norm_inf<E> >::result_type
      index_norm_inf (const vector_expression<E> &e) {
          typedef typename vector_scalar_unary_traits<E, vector_index_norm_inf<E> >::expression_type expression_type;
          return expression_type (e ());
      }
  
      template<class E1, class E2, class F>
      class vector_scalar_binary:
          public scalar_expression<vector_scalar_binary<E1, E2, F> > {
  
          typedef E1 expression1_type;
          typedef E2 expression2_type;
          typedef F functor_type;
          typedef typename E1::const_closure_type expression1_closure_type;
          typedef typename E2::const_closure_type expression2_closure_type;
          typedef typename iterator_restrict_traits<typename E1::const_iterator::iterator_category,
                                                    typename E2::const_iterator::iterator_category>::iterator_category iterator_category;
          typedef vector_scalar_binary<E1, E2, F> self_type;
      public:
          static const unsigned complexity = 1;
          typedef typename F::result_type value_type;
          typedef typename E1::difference_type difference_type;
          typedef const self_type const_closure_type;
          typedef const_closure_type closure_type;
          typedef unknown_storage_tag storage_category;
  
          // Construction and destruction
          BOOST_UBLAS_INLINE
          vector_scalar_binary (const expression1_type &e1, const expression2_type  &e2):
              e1_ (e1), e2_ (e2) {}
  
      private:
          // Accessors
          BOOST_UBLAS_INLINE
          const expression1_closure_type &expression1 () const {
              return e1_;
          }
          BOOST_UBLAS_INLINE
          const expression2_closure_type &expression2 () const {
              return e2_;
          }
  
      public:
          BOOST_UBLAS_INLINE
          operator value_type () const {
              return evaluate (iterator_category ());
          }
  
      private:
          // Dense random access specialization
          BOOST_UBLAS_INLINE
          value_type evaluate (dense_random_access_iterator_tag) const {
              BOOST_UBLAS_CHECK (e1_.size () == e2_.size (), external_logic());
  #ifdef BOOST_UBLAS_USE_INDEXING
              return functor_type::apply (e1_, e2_);
  #elif BOOST_UBLAS_USE_ITERATING
              difference_type size = BOOST_UBLAS_SAME (e1_.size (), e2_.size ());
              return functor_type::apply (size, e1_.begin (), e2_.begin ());
  #else
              difference_type size = BOOST_UBLAS_SAME (e1_.size (), e2_.size ());
              if (size >= BOOST_UBLAS_ITERATOR_THRESHOLD)
                  return functor_type::apply (size, e1_.begin (), e2_.begin ());
              else
                  return functor_type::apply (e1_, e2_);
  #endif
          }
  
          // Packed bidirectional specialization
          BOOST_UBLAS_INLINE
          value_type evaluate (packed_random_access_iterator_tag) const {
              BOOST_UBLAS_CHECK (e1_.size () == e2_.size (), external_logic());
              return functor_type::apply (e1_.begin (), e1_.end (), e2_.begin (), e2_.end ());
          }
  
          // Sparse bidirectional specialization
          BOOST_UBLAS_INLINE
          value_type evaluate (sparse_bidirectional_iterator_tag) const {
              BOOST_UBLAS_CHECK (e1_.size () == e2_.size (), external_logic());
              return functor_type::apply (e1_.begin (), e1_.end (), e2_.begin (), e2_.end (), sparse_bidirectional_iterator_tag ());
          }
  
      private:
          expression1_closure_type e1_;
          expression2_closure_type e2_;
      };
  
      template<class E1, class E2, class F>
      struct vector_scalar_binary_traits {
          typedef vector_scalar_binary<E1, E2, F> expression_type;
  #if !defined (BOOST_UBLAS_SIMPLE_ET_DEBUG) && defined (BOOST_UBLAS_USE_SCALAR_ET)
  // FIXME don't define USE_SCALAR_ET other then for testing
  // They do not work for complex types
          typedef expression_type result_type;
  #else
          typedef typename F::result_type result_type;
  #endif
      };
  
      // inner_prod (v1, v2) = sum (v1 [i] * v2 [i])
      template<class E1, class E2>
      BOOST_UBLAS_INLINE
      typename vector_scalar_binary_traits<E1, E2, vector_inner_prod<E1, E2,
                                                                     typename promote_traits<typename E1::value_type,
                                                                                             typename E2::value_type>::promote_type> >::result_type
      inner_prod (const vector_expression<E1> &e1,
                  const vector_expression<E2> &e2) {
          typedef typename vector_scalar_binary_traits<E1, E2, vector_inner_prod<E1, E2,
                                                                     typename promote_traits<typename E1::value_type,
                                                                                             typename E2::value_type>::promote_type> >::expression_type expression_type;
          return expression_type (e1 (), e2 ());
      }
  
      template<class E1, class E2>
      BOOST_UBLAS_INLINE
      typename vector_scalar_binary_traits<E1, E2, vector_inner_prod<E1, E2,
                                                                     typename type_traits<typename promote_traits<typename E1::value_type,
                                                                                                                  typename E2::value_type>::promote_type>::precision_type> >::result_type
      prec_inner_prod (const vector_expression<E1> &e1,
                       const vector_expression<E2> &e2) {
          typedef typename vector_scalar_binary_traits<E1, E2, vector_inner_prod<E1, E2,
                                                                     typename type_traits<typename promote_traits<typename E1::value_type,
                                                                                                                  typename E2::value_type>::promote_type>::precision_type> >::expression_type expression_type;
          return expression_type (e1 (), e2 ());
      }
  
  }}}
  
  #endif
  