vector_expression.hpp - Hosted on DriveHQ Cloud IT Platform

//
//  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 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 boost::mpl::if_<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 boost::mpl::if_<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 end () const {
            return expression ().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 rend () const {
            return const_reverse_iterator (begin ());
        }
        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 boost::mpl::if_<boost::is_same<F, scalar_identity<typename E::value_type> >,
                                          E,
                                          const E>::type expression_type;
        typedef typename boost::mpl::if_<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 boost::mpl::if_<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 ((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 ((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 end () const {
            return find (size ());
        }

        // 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 rend () const {
            return const_reverse_iterator (begin ());
        }

    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 end () const {
            return find (size ());
        }

        // 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 rend () const {
            return const_reverse_iterator (begin ());
        }

    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 end () const {
            return find (size ()); 
        }

        // 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 rend () const {
            return const_reverse_iterator (begin ());
        }

    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 vector_binary_scalar1_traits<const T1, E2, scalar_multiplies<T1, typename E2::value_type> >::result_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 end () const {
            return find (size ());
        }

        // 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 rend () const {
            return const_reverse_iterator (begin ());
        }

    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 vector_binary_scalar2_traits<E1, const T2, scalar_multiplies<typename E1::value_type, T2> >::result_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 vector_binary_scalar2_traits<E1, const T2, scalar_divides<typename E1::value_type, T2> >::result_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 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 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
vector_expression.hpp
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