# Vector Expressions

## Vector Expression

#### Description

The templated class `vector_expression<E>` forms the base for all static derived vector expression classes including class `vector` itself.

#### Definition

Defined in the header vector_expression.hpp.

#### Template parameters

Parameter | Description | Default
--- | --- | ---
`E` | The type of the vector expression. |

#### Model of

None.

#### Type requirements

None.

#### Public base classes

None.

#### Members

Member | Description
--- | ---
`const expression_type &operator () () const` | Returns a `const` reference of the expression.
`expression_type &operator () ()` | Returns a reference of the expression.

#### Interface

        // Base class for the Barton Nackman trick
        template<class E>
        struct vector_expression {
            typedef E expression_type;
            typedef vector_tag type_category;

            // This class could define an common interface for all
            // statically derived expression type classes.
            // Due to a compiler deficiency - one can not reference class typedefs of E
            // on MSVC 6.0 (error C2027) - we only implement the casts.

            const expression_type &operator () () const;
            expression_type &operator () ();
        };

## Vector References

### Constant Reference

#### Description

The templated class `vector_const_reference<E>` contains a constant reference to a vector expression.

#### Definition

Defined in the header vector_expression.hpp.

#### Template parameters

Parameter | Description | Default
--- | --- | ---
`E` | The type of the vector expression. |

#### Model of

Vector Expression.

#### Type requirements

None, except for those imposed by the requirements of Vector Expression.

#### Public base classes

`vector_expression<vector_const_reference<E> >`

#### Members

Member | Description
--- | ---
`vector_const_reference (const expression_type &e)` | Constructs a constant reference of the expression.
`size_type size () const` | Returns the size of the expression.
`const_reference operator () (size_type i) const` | Returns the value of the `i`-th element.
`const_iterator begin () const` | Returns a `const_iterator` pointing to the beginning of the expression.
`const_iterator end () const` | Returns a `const_iterator` pointing to the end of the expression.
`const_reverse_iterator rbegin () const` | Returns a `const_reverse_iterator` pointing to the beginning of the reversed expression.
`const_reverse_iterator rend () const` | Returns a `const_reverse_iterator` pointing to the end of the reversed expression.

#### Interface

        template<class E>
        class vector_const_reference:
            public vector_expression<vector_const_reference<E> > {
        public:
            typedef E expression_type;
            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 const_reference reference;
            typedef typename E::const_pointer const_pointer;
            typedef const_pointer pointer;
            typedef typename E::const_iterator const_iterator_type;
            typedef unknown_storage_tag storage_category;

            // Construction and destruction
            vector_const_reference ();
            vector_const_reference (const expression_type &e);

            // Accessors
            size_type size () const;
            const expression_type &expression () const;

            // Element access
            const_reference operator () (size_type i) const;

            const_reference operator [] (size_type i) const;

            typedef const_iterator_type const_iterator;
            typedef const_iterator iterator;

            // Element lookup
            const_iterator find_first (size_type i) const;
            const_iterator find_last (size_type i) const;

            // Iterator is the iterator of the referenced expression.

            const_iterator begin () const;
            const_iterator end () const;

            // Reverse iterator

            typedef reverse_iterator_base<const_iterator> const_reverse_iterator;

            const_reverse_iterator rbegin () const;
            const_reverse_iterator rend () const;
        };

### Reference

#### Description

The templated class `vector_reference<E>` contains a reference to a vector expression.

#### Definition

Defined in the header vector_expression.hpp.

#### Template parameters

Parameter | Description | Default
--- | --- | ---
`E` | The type of the vector expression. |

#### Model of

Vector Expression.

#### Type requirements

None, except for those imposed by the requirements of Vector Expression.

#### Public base classes

`vector_expression<vector_reference<E> >`

#### Members

Member | Description
--- | ---
`vector_reference (expression_type &e)` | Constructs a reference of the expression.
`void resize (size_type size)` | Resizes the expression to hold at most `size` elements.
`size_type size () const` | Returns the size of the expression.
`const_reference operator () (size_type i) const` | Returns the value of the `i`-th element.
`reference operator () (size_type i)` | Returns a reference of the `i`-th element.
`const_iterator begin () const` | Returns a `const_iterator` pointing to the beginning of the expression.
`const_iterator end () const` | Returns a `const_iterator` pointing to the end of the expression.
`iterator begin ()` | Returns a `iterator` pointing to the beginning of the expression.
`iterator end ()` | Returns a `iterator` pointing to the end of the expression.
`const_reverse_iterator rbegin () const` | Returns a `const_reverse_iterator` pointing to the beginning of the reversed expression.
`const_reverse_iterator rend () const` | Returns a `const_reverse_iterator` pointing to the end of the reversed expression.
`reverse_iterator rbegin ()` | Returns a `reverse_iterator` pointing to the beginning of the reversed expression.
`reverse_iterator rend ()` | Returns a `reverse_iterator` pointing to the end of the reversed expression.

#### Interface

        template<class E>
        class vector_reference:
            public vector_expression<vector_reference<E> > {
        public:
            typedef E expression_type;
            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 E::reference reference;
            typedef typename E::const_pointer const_pointer;
            typedef typename E::pointer pointer;
            typedef typename E::const_iterator const_iterator_type;
            typedef typename E::iterator iterator_type;
            typedef unknown_storage_tag storage_category;

            // Construction and destruction
            vector_reference ();
            vector_reference (expression_type &e);

            // Accessors
            size_type size () const;
            const expression_type &expression () const;
            expression_type &expression ();

            // Resizing
            void resize (size_type size);

            // Element access
            const_reference operator () (size_type i) const;
            reference operator () (size_type i);

            const_reference operator [] (size_type i) const;
            reference operator [] (size_type i);

            typedef const_iterator_type const_iterator;
            typedef iterator_type iterator;

            // Element lookup
            const_iterator find_first (size_type i) const;
            iterator find_first (size_type i);
            const_iterator find_last (size_type i) const;
            iterator find_last (size_type i);

            // Iterator is the iterator of the referenced expression.

            const_iterator begin () const;
            const_iterator end () const;

            iterator begin ();
            iterator end ();

            // Reverse iterator

            typedef reverse_iterator_base<const_iterator> const_reverse_iterator;

            const_reverse_iterator rbegin () const;
            const_reverse_iterator rend () const;

            typedef reverse_iterator_base<iterator> reverse_iterator;

            reverse_iterator rbegin ();
            reverse_iterator rend ();
        };

## Vector Operations

### Unary Operation Description

#### Description

The templated class `vector_unary<E, F>` describes a unary vector operation.

#### Definition

Defined in the header vector_expression.hpp.

#### Template parameters

Parameter | Description | Default
--- | --- | ---
`E` | The type of the vector expression. |
`F` | The type of the operation. |

#### Model of

Vector Expression.

#### Type requirements

None, except for those imposed by the requirements of Vector Expression.

#### Public base classes

`vector_expression<vector_unary<E, F> >`

#### Members

Member | Description
--- | ---
`vector_unary (const expression_type &e)` | Constructs a description of the expression.
`size_type size () const` | Returns the size of the expression.
`const_reference operator () (size_type i) const` | Returns the value of the `i`-th element.
`const_iterator begin () const` | Returns a `const_iterator` pointing to the beginning of the expression.
`const_iterator end () const` | Returns a `const_iterator` pointing to the end of the expression.
`const_reverse_iterator rbegin () const` | Returns a `const_reverse_iterator` pointing to the beginning of the reversed expression.
`const_reverse_iterator rend () const` | Returns a `const_reverse_iterator` pointing to the end of the reversed expression.

#### Interface

        template<class E, class F>
        class vector_unary:
            public vector_expression<vector_unary<E, F> > {
        public:
            typedef E expression_type;
            typedef F functor_type;
            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 const_reference reference;
            typedef const value_type *const_pointer;
            typedef const_pointer pointer;
            typedef const vector_unary<E, F> const_closure_type;
            typedef typename E::const_iterator const_iterator_type;
            typedef unknown_storage_tag storage_category;

            // Construction and destruction
            vector_unary ();
            vector_unary (const expression_type &e);

            // Accessors
            size_type size () const;
            const expression_type &expression () const;

            // Element access
            const_reference operator () (size_type i) const;

            const_reference operator [] (size_type i) const;

            class const_iterator;
            typedef const_iterator iterator;

            // Element lookup
            const_iterator find_first (size_type i) const;
            const_iterator find_last (size_type i) const;

            // Iterator enhances the iterator of the referenced expression
            // with the unary functor.

            class const_iterator:
                public container_const_reference<vector_unary>,
                public random_access_iterator_base<const_iterator, value_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
                const_iterator ();
                const_iterator (const vector_unary &vu, const const_iterator_type &it);

                // Arithmetic
                const_iterator &operator ++ ();
                const_iterator &operator -- ();
                const_iterator &operator += (difference_type n);
                const_iterator &operator -= (difference_type n);
                difference_type operator - (const const_iterator &it) const;

                // Dereference
                reference operator * () const;

                // Index
                size_type index () const;

                // Assignment
                const_iterator &operator = (const const_iterator &it);

                // Comparison
                bool operator == (const const_iterator &it) const;
                bool operator <(const const_iterator &it) const;
            };

            const_iterator begin () const;
            const_iterator end () const;

            // Reverse iterator

            typedef reverse_iterator_base<const_iterator> const_reverse_iterator;

            const_reverse_iterator rbegin () const;
            const_reverse_iterator rend () const;
        };

### Unary Operations

#### Prototypes

        template<class E, class F>
        struct vector_unary_traits {
            typedef vector_unary<typename E::const_closure_type, F> expression_type;
            typedef expression_type result_type;
        };

        // (- v) [i] = - v [i]
        template<class E>
        typename vector_unary_traits<E, scalar_negate<typename E::value_type> >::result_type
        operator - (const vector_expression<E> &e);

        // (conj v) [i] = conj (v [i])
        template<class E>
        typename vector_unary_traits<E, scalar_conj<typename E::value_type> >::result_type
        conj (const vector_expression<E> &e);

        // (real v) [i] = real (v [i])
        template<class E>
        typename vector_unary_traits<E, scalar_real<typename E::value_type> >::result_type
        real (const vector_expression<E> &e);

        // (imag v) [i] = imag (v [i])
        template<class E>
        typename vector_unary_traits<E, scalar_imag<typename E::value_type> >::result_type
        imag (const vector_expression<E> &e);

        // (trans v) [i] = v [i]
        template<class E>
        typename vector_unary_traits<E, scalar_identity<typename E::value_type> >::result_type
        trans (const vector_expression<E> &e);

        // (herm v) [i] = conj (v [i])
        template<class E>
        typename vector_unary_traits<E, scalar_conj<typename E::value_type> >::result_type
        herm (const vector_expression<E> &e);

#### Description

`operator -` computes the additive inverse of a vector expression. `conj` computes the complex conjugate of a vector expression. `real` and `imag` compute the real and imaginary parts of a vector expression. `trans` computes the transpose of a vector expression. `herm` computes the hermitian, i.e. the complex conjugate of the transpose of a vector expression.

#### Definition

Defined in the header vector_expression.hpp.

#### Type requirements

- `E` is a model of Vector Expression.

#### Preconditions

None.

#### Complexity

Linear depending from the size of the vector expression.

#### Examples

    int main () {
        using namespace boost::numeric::ublas;
        vector<std::complex<double> > v (3);
        for (int i = 0; i < v.size (); ++ i)
            v (i) = std::complex (i, i);

        std::cout << - v << std::endl;
        std::cout << conj (v) << std::endl;
        std::cout << real (v) << std::endl;
        std::cout << imag (v) << std::endl;
        std::cout << trans (v) << std::endl;
        std::cout << herm (v) << std::endl;
    }

### Binary Operation Description

#### Description

The templated class `vector_binary<E1, E2, F>` describes a binary vector operation.

#### Definition

Defined in the header vector_expression.hpp.

#### Template parameters

Parameter | Description | Default
--- | --- | ---
`E1` | The type of the first vector expression. |
`E2` | The type of the second vector expression. |
`F` | The type of the operation. |

#### Model of

Vector Expression.

#### Type requirements

None, except for those imposed by the requirements of Vector Expression.

#### Public base classes

`vector_expression<vector_binary<E1, E2, F> >`

#### Members

Member | Description
--- | ---
`vector_binary (const expression1_type &e1, const expression2_type &e2)` | Constructs a description of the expression.
`size_type size () const` | Returns the size of the expression.
`const_reference operator () (size_type i) const` | Returns the value of the `i`-th element.
`const_iterator begin () const` | Returns a `const_iterator` pointing to the beginning of the expression.
`const_iterator end () const` | Returns a `const_iterator` pointing to the end of the expression.
`const_reverse_iterator rbegin () const` | Returns a `const_reverse_iterator` pointing to the beginning of the reversed expression.
`const_reverse_iterator rend () const` | Returns a `const_reverse_iterator` pointing to the end of the reversed expression.

#### Interface

        template<class E1, class E2, class F>
        class vector_binary:
            public vector_expression<vector_binary<E1, E2, F> > {
        public:
            typedef E1 expression1_type;
            typedef E2 expression2_type;
            typedef F functor_type;
            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 value_type *const_pointer;
            typedef const_pointer pointer;
            typedef const vector_binary<E1, E2, F> const_closure_type;
            typedef typename E1::const_iterator const_iterator1_type;
            typedef typename E2::const_iterator const_iterator2_type;
            typedef unknown_storage_tag storage_category;

            // Construction and destruction
            vector_binary ();
            vector_binary (const expression1_type &e1, const expression2_type &e2);

            // Accessors
            size_type size () const;
            const expression1_type &expression1 () const;
            const expression2_type &expression2 () const;

            // Element access
            const_reference operator () (size_type i) const;

            const_reference operator [] (size_type i) const;

            class const_iterator;
            typedef const_iterator iterator;

            // Element lookup
            const_iterator find_first (size_type i) const;
            const_iterator find_last (size_type i) const;

            // Iterator merges the iterators of the referenced expressions and
            // enhances them with the binary functor.

            class const_iterator:
                public container_const_reference<vector_binary>,
                public random_access_iterator_base<const_iterator, value_type> {
            public:
                typedef typename 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
                const_iterator ();
                const_iterator (const vector_binary &vb, size_type i,
                                const const_iterator1_type &it1, const const_iterator1_type &it1_end,
                                const const_iterator2_type &it2, const const_iterator2_type &it2_end);

                // Dense specializations
                void increment (dense_random_access_iterator_tag);
                void decrement (dense_random_access_iterator_tag);
                value_type dereference (dense_random_access_iterator_tag) const;

                // Packed specializations
                void increment (packed_random_access_iterator_tag);
                void decrement (packed_random_access_iterator_tag);
                value_type dereference (packed_random_access_iterator_tag) const;

                // Sparse specializations
                void increment (sparse_bidirectional_iterator_tag);
                void decrement (sparse_bidirectional_iterator_tag);
                value_type dereference (sparse_bidirectional_iterator_tag) const;

                // Arithmetic
                const_iterator &operator ++ ();
                const_iterator &operator -- ();
                const_iterator &operator += (difference_type n);
                const_iterator &operator -= (difference_type n);
                difference_type operator - (const const_iterator &it) const;

                // Dereference
                reference operator * () const;

                // Index
                size_type index () const;

                // Assignment
                const_iterator &operator = (const const_iterator &it);

                // Comparison
                bool operator == (const const_iterator &it) const;
                bool operator <(const const_iterator &it) const;
            };

            const_iterator begin () const;
            const_iterator end () const;

            // Reverse iterator

            typedef reverse_iterator_base<const_iterator> const_reverse_iterator;

            const_reverse_iterator rbegin () const;
            const_reverse_iterator rend () const;
        };

### Binary Operations

#### Prototypes

        template<class E1, class E2, class F>
        struct vector_binary_traits {
            typedef vector_binary<typename E1::const_closure_type,
                                  typename E2::const_closure_type, F> expression_type;
            typedef expression_type result_type;
        };

        // (v1 + v2) [i] = v1 [i] + v2 [i]
        template<class E1, class E2>
        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);

        // (v1 - v2) [i] = v1 [i] - v2 [i]
        template<class E1, class E2>
        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);

#### Description

`operator +` computes the sum of two vector expressions. `operator -` computes the difference of two vector expressions.

#### Definition

Defined in the header vector_expression.hpp.

#### Type requirements

- `E1` is a model of Vector Expression.
- `E2` is a model of Vector Expression.

#### Preconditions

- `e1 ().size () == e2 ().size ()`

#### Complexity

Linear depending from the size of the vector expressions.

#### Examples

    int main () {
        using namespace boost::numeric::ublas;
        vector<double> v1 (3), v2 (3);
        for (int i = 0; i < std::min (v1.size (), v2.size ()); ++ i)
            v1 (i) = v2 (i) = i;

        std::cout << v1 + v2 << std::endl;
        std::cout << v1 - v2 << std::endl;
    }

### Binary Outer Operation Description

#### Description

The templated class `vector_matrix_binary<E1, E2, F>` describes a binary outer vector operation.

#### Definition

Defined in the header matrix_expression.hpp.

#### Template parameters

Parameter | Description | Default
--- | --- | ---
`E1` | The type of the first vector expression. |
`E2` | The type of the second vector expression. |
`F` | The type of the operation. |

#### Model of

Matrix Expression.

#### Type requirements

None, except for those imposed by the requirements of Matrix Expression.

#### Public base classes

`matrix_expression<vector_matrix_binary<E1, E2, F> >`

#### Members

Member | Description
--- | ---
`vector_matrix_binary (const expression1_type &e1, const expression2_type &e2)` | Constructs a description of the expression.
`size_type size1 () const` | Returns the number of rows.
`size_type size2 () const` | Returns the number of columns.
`const_reference operator () (size_type i, size_type j) const` | Returns the value of the `j`-th element in the`i`-th row.
`const_iterator1 begin1 () const` | Returns a `const_iterator1` pointing to the beginning of the expression.
`const_iterator1 end1 () const` | Returns a `const_iterator1` pointing to the end of the expression.
`const_iterator2 begin2 () const` | Returns a `const_iterator2` pointing to the beginning of the expression.
`const_iterator2 end2 () const` | Returns a `const_iterator2` pointing to the end of the expression.
`const_reverse_iterator1 rbegin1 () const` | Returns a `const_reverse_iterator1` pointing to the beginning of the reversed expression.
`const_reverse_iterator1 rend1 () const` | Returns a `const_reverse_iterator1` pointing to the end of the reversed expression.
`const_reverse_iterator2 rbegin2 () const` | Returns a `const_reverse_iterator2` pointing to the beginning of the reversed expression.
`const_reverse_iterator2 rend2 () const` | Returns a `const_reverse_iterator2` pointing to the end of the reversed expression.

#### Interface

        template<class E1, class E2, class F>
        class vector_matrix_binary:
            public matrix_expression<vector_matrix_binary<E1, E2, F> > {
        public:
            typedef E1 expression1_type;
            typedef E2 expression2_type;
            typedef F functor_type;
            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 value_type *const_pointer;
            typedef const_pointer pointer;
            typedef const vector_matrix_binary<E1, E2, F> const_closure_type;
            typedef unknown_orientation_tag orientation_category;
            typedef typename E1::const_iterator const_iterator1_type;
            typedef typename E2::const_iterator const_iterator2_type;
            typedef unknown_storage_tag storage_category;

            // Construction and destruction
            vector_matrix_binary ();
            vector_matrix_binary (const expression1_type &e1, const expression2_type &e2);

            // Accessors
            size_type size1 () const;
            size_type size2 () const;
            const expression1_type &expression1 () const;
            const expression2_type &expression2 () const;

            // Element access
            const_reference operator () (size_type i, size_type j) const;

            class const_iterator1;
            typedef const_iterator1 iterator1;
            class const_iterator2;
            typedef const_iterator2 iterator2;

            // Element lookup
            const_iterator1 find_first1 (int rank, size_type i, size_type j) const;
            const_iterator1 find_last1 (int rank, size_type i, size_type j) const;
            const_iterator2 find_first2 (int rank, size_type i, size_type j) const;
            const_iterator2 find_last2 (int rank, size_type i, size_type j) const;

            // Iterators enhance the iterators of the referenced expressions
            // with the binary functor.

            class const_iterator1:
                public container_const_reference<vector_matrix_binary>,
                public random_access_iterator_base<const_iterator1, value_type> {
            public:
                typedef typename restrict_traits<typename E1::const_iterator::iterator_category,
                                                 typename E2::const_iterator::iterator_category>::iterator_category iterator_category;
                typedef typename vector_matrix_binary::difference_type difference_type;
                typedef typename vector_matrix_binary::value_type value_type;
                typedef typename vector_matrix_binary::const_reference reference;
                typedef typename vector_matrix_binary::const_pointer pointer;
                typedef const_iterator2 dual_iterator_type;
                typedef const_reverse_iterator2 dual_reverse_iterator_type;

                // Construction and destruction
                const_iterator1 ();
                const_iterator1 (const vector_matrix_binary &vmb, const const_iterator1_type &it1, const const_iterator2_type &it2);

                // Arithmetic
                const_iterator1 &operator ++ ();
                const_iterator1 &operator -- ();
                const_iterator1 &operator += (difference_type n);
                const_iterator1 &operator -= (difference_type n);
                difference_type operator - (const const_iterator1 &it) const;

                // Dereference
                reference operator * () const;

                const_iterator2 begin () const;
                const_iterator2 end () const;
                const_reverse_iterator2 rbegin () const;
                const_reverse_iterator2 rend () const;

                // Indices
                size_type index1 () const;
                size_type  index2 () const;

                // Assignment
                const_iterator1 &operator = (const const_iterator1 &it);

                // Comparison
                bool operator == (const const_iterator1 &it) const;
                bool operator <(const const_iterator1 &it) const;
            };

            const_iterator1 begin1 () const;
            const_iterator1 end1 () const;

            class const_iterator2:
                public container_const_reference<vector_matrix_binary>,
                public random_access_iterator_base<const_iterator2, value_type> {
            public:
                typedef typename restrict_traits<typename E1::const_iterator::iterator_category,
                                                 typename E2::const_iterator::iterator_category>::iterator_category iterator_category;
                typedef typename vector_matrix_binary::difference_type difference_type;
                typedef typename vector_matrix_binary::value_type value_type;
                typedef typename vector_matrix_binary::const_reference reference;
                typedef typename vector_matrix_binary::const_pointer pointer;
                typedef const_iterator1 dual_iterator_type;
                typedef const_reverse_iterator1 dual_reverse_iterator_type;

                // Construction and destruction
                const_iterator2 ();
                const_iterator2 (const vector_matrix_binary &vmb, const const_iterator1_type &it1, const const_iterator2_type &it2);

                // Arithmetic
                const_iterator2 &operator ++ ();
                const_iterator2 &operator -- ();
                const_iterator2 &operator += (difference_type n);
                const_iterator2 &operator -= (difference_type n);
                difference_type operator - (const const_iterator2 &it) const;

                // Dereference
                reference operator * () const;

                const_iterator1 begin () const;
                const_iterator1 end () const;
                const_reverse_iterator1 rbegin () const;
                const_reverse_iterator1 rend () const;

                // Indices
                size_type index1 () const;
                size_type  index2 () const;

                // Assignment
                const_iterator2 &operator = (const const_iterator2 &it);

                // Comparison
                bool operator == (const const_iterator2 &it) const;
                bool operator <(const const_iterator2 &it) const;
            };

            const_iterator2 begin2 () const;
            const_iterator2 end2 () const;

            // Reverse iterators

            const_reverse_iterator1 rbegin1 () const;
            const_reverse_iterator1 rend1 () const;

            const_reverse_iterator2 rbegin2 () const;
            const_reverse_iterator2 rend2 () const;
        };

### Binary Outer Operations

#### Prototypes

        template<class E1, class E2, class F>
        struct vector_matrix_binary_traits {
            typedef vector_matrix_binary<typename E1::const_closure_type,
                                         typename E2::const_closure_type, F> expression_type;
            typedef expression_type result_type;
        };

        // (outer_prod (v1, v2)) [i] [j] = v1 [i] * v2 [j]
        template<class E1, class E2>
        typename vector_matrix_binary_traits<E1, E2, scalar_multiplies<typename E1::value_type, typename E2::value_type> >::result_type
        outer_prod (const vector_expression<E1> &e1,
                    const vector_expression<E2> &e2);

#### Description

`outer_prod` computes the outer product of two vector expressions.

#### Definition

Defined in the header matrix_expression.hpp.

#### Type requirements

- `E1` is a model of Vector Expression.
- `E2` is a model of Vector Expression.

#### Preconditions

None.

#### Complexity

Quadratic depending from the size of the vector expressions.

#### Examples

    int main () {
        using namespace boost::numeric::ublas;
        vector<double> v1 (3), v2 (3);
        for (int i = 0; i < std::min (v1.size (), v2.size ()); ++ i)
            v1 (i) = v2 (i) = i;

        std::cout << outer_prod (v1, v2) << std::endl;
    }

### Scalar Vector Operation Description

#### Description

The templated classes `vector_binary_scalar1<E1, E2, F>` and `vector_binary_scalar2<E1, E2, F>` describe binary operations between a scalar and a vector.

#### Definition

Defined in the header vector_expression.hpp.

#### Template parameters

Parameter | Description | Default
--- | --- | ---
`E1/E2` | The type of the scalar expression. |
`E2/E1` | The type of the vector expression. |
`F` | The type of the operation. |

#### Model of

Vector Expression.

#### Type requirements

None, except for those imposed by the requirements of Vector Expression.

#### Public base classes

`vector_expression<vector_binary_scalar1<E1, E2, F> >` and`vector_expression<vector_binary_scalar2<E1, E2, F> >` resp.

#### Members

Member | Description
--- | ---
`vector_binary_scalar1 (const expression1_type &e1, const expression2_type &e2)` | Constructs a description of the expression.
`vector_binary_scalar2 (const expression1_type &e1, const expression2_type &e2)` | Constructs a description of the expression.
`size_type size () const` | Returns the size of the expression.
`const_reference operator () (size_type i) const` | Returns the value of the `i`-th element.
`const_iterator begin () const` | Returns a `const_iterator` pointing to the beginning of the expression.
`const_iterator end () const` | Returns a `const_iterator` pointing to the end of the expression.
`const_reverse_iterator rbegin () const` | Returns a `const_reverse_iterator` pointing to the beginning of the reversed expression.
`const_reverse_iterator rend () const` | Returns a `const_reverse_iterator` pointing to the end of the reversed expression.

#### Interface

        template<class E1, class E2, class F>
        class vector_binary_scalar1:
            public vector_expression<vector_binary_scalar1<E1, E2, F> > {
        public:
            typedef E1 expression1_type;
            typedef E2 expression2_type;
            typedef F functor_type;
            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 value_type *const_pointer;
            typedef const_pointer pointer;
            typedef const vector_binary_scalar1<E1, E2, F> const_closure_type;
            typedef typename E1::value_type const_iterator1_type;
            typedef typename E2::const_iterator const_iterator2_type;
            typedef unknown_storage_tag storage_category;

            // Construction and destruction
            vector_binary_scalar1 ();
            vector_binary_scalar1 (const expression1_type &e1, const expression2_type &e2);

            // Accessors
            size_type size () const;
            const expression1_type &expression1 () const;
            const expression2_type &expression2 () const;

            // Element access
            const_reference operator () (size_type i) const;

            const_reference operator [] (size_type i) const;

            class const_iterator;
            typedef const_iterator iterator;

            // Element lookup
            const_iterator find_first (size_type i) const;
            const_iterator find_last (size_type i) const;

            // Iterator enhances the iterator of the referenced vector expression
            // with the binary functor.

            class const_iterator:
                public container_const_reference<vector_binary_scalar1>,
                public random_access_iterator_base<const_iterator, value_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
                const_iterator ();
                const_iterator (const vector_binary_scalar1 &vbs, const const_iterator1_type &it1, const const_iterator2_type &it2);

                // Arithmetic
                const_iterator &operator ++ ();
                const_iterator &operator -- ();
                const_iterator &operator += (difference_type n);
                const_iterator &operator -= (difference_type n);
                difference_type operator - (const const_iterator &it) const;

                // Dereference
                reference operator * () const;

                // Index
                size_type index () const;

                // Assignment
                const_iterator &operator = (const const_iterator &it);

                // Comparison
                bool operator == (const const_iterator &it) const;
                bool operator <(const const_iterator &it) const;
            };

            const_iterator begin () const;
            const_iterator end () const;

            // Reverse iterator

            typedef reverse_iterator_base<const_iterator> const_reverse_iterator;

            const_reverse_iterator rbegin () const;
            const_reverse_iterator rend () const;
        };

        template<class E1, class E2, class F>
        class vector_binary_scalar2:
            public vector_expression<vector_binary_scalar2<E1, E2, F> > {
        public:
            typedef E1 expression1_type;
            typedef E2 expression2_type;
            typedef F functor_type;
            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 value_type *const_pointer;
            typedef const_pointer pointer;
            typedef const vector_binary_scalar2<E1, E2, F> const_closure_type;
            typedef typename E1::const_iterator const_iterator1_type;
            typedef typename E2::value_type const_iterator2_type;
            typedef unknown_storage_tag storage_category;

            // Construction and destruction
            vector_binary_scalar2 ();
            vector_binary_scalar2 (const expression1_type &e1, const expression2_type &e2);

            // Accessors
            size_type size () const;
            const expression1_type &expression1 () const;
            const expression2_type &expression2 () const;

            // Element access
            const_reference operator () (size_type i) const;

            const_reference operator [] (size_type i) const ;

            class const_iterator;
            typedef const_iterator iterator;

            // Element lookup
            const_iterator find_first (size_type i) const;
            const_iterator find_last (size_type i) const;

            // Iterator enhances the iterator of the referenced vector expression
            // with the binary functor.

            class const_iterator:
                public container_const_reference<vector_binary_scalar2>,
                public random_access_iterator_base<const_iterator, value_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
                const_iterator ();
                const_iterator (const vector_binary_scalar2 &vbs, const const_iterator1_type &it1, const const_iterator2_type &it2);

                // Arithmetic
                const_iterator &operator ++ ();
                const_iterator &operator -- ();
                const_iterator &operator += (difference_type n);
                const_iterator &operator -= (difference_type n);
                difference_type operator - (const const_iterator &it) const;

                // Dereference
                reference operator * () const;

                // Index
                size_type index () const;

                // Assignment
                const_iterator &operator = (const const_iterator &it);

                // Comparison
                bool operator == (const const_iterator &it) const;
                bool operator <(const const_iterator &it) const;
            };

            const_iterator begin () const;
            const_iterator end () const;

            // Reverse iterator

            typedef reverse_iterator_base<const_iterator> const_reverse_iterator;

            const_reverse_iterator rbegin () const;
            const_reverse_iterator rend () const;
        };

### Scalar Vector Operations

#### Prototypes

        template<class T1, class E2, class F>
        struct vector_binary_scalar1_traits {
            typedef vector_binary_scalar1<scalar_const_reference<T1>,
                                          typename E2::const_closure_type, F> expression_type;
            typedef expression_type result_type;
        };

        // (t * v) [i] = t * v [i]
        template<class T1, class E2>
        typename vector_binary_scalar1_traits<T1, E2, scalar_multiplies<T1, typename E2::value_type> >::result_type
        operator * (const T1 &e1,
                    const vector_expression<E2> &e2);

        template<class E1, class T2, class F>
        struct vector_binary_scalar2_traits {
            typedef vector_binary_scalar2<typename E1::const_closure_type,
                                          scalar_const_reference<T2>, F> expression_type;
            typedef expression_type result_type;
        };

        // (v * t) [i] = v [i] * t
        template<class E1, class T2>
        typename vector_binary_scalar2_traits<E1, T2, scalar_multiplies<typename E1::value_type, T2> >::result_type
        operator * (const vector_expression<E1> &e1,
                    const T2 &e2);

        // (v / t) [i] = v [i] / t
        template<class E1, class T2>
        typename vector_binary_scalar2_traits<E1, T2, scalar_divides<typename E1::value_type, T2> >::result_type
        operator / (const vector_expression<E1> &e1,
                    const T2 &e2);

#### Description

`operator *` computes the product of a scalar and a vector expression. `operator /` multiplies the vector with the reciprocal of the scalar.

#### Definition

Defined in the header vector_expression.hpp.

#### Type requirements

- `T1/T2` is a model of Scalar Expression.
- `E2/E1` is a model of Vector Expression.

#### Preconditions

None.

#### Complexity

Linear depending from the size of the vector expression.

#### Examples

    int main () {
        using namespace boost::numeric::ublas;
        vector<double> v (3);
        for (int i = 0; i < v.size (); ++ i)
            v (i) = i;

        std::cout << 2.0 * v << std::endl;
        std::cout << v * 2.0 << std::endl;
    }

## Vector Reductions

### Unary Reductions

#### Prototypes

        template<class E, class F>
        struct vector_scalar_unary_traits {
             typedef typename F::result_type result_type;
        };

        // sum v = sum (v [i])
        template<class E>
        typename vector_scalar_unary_traits<E, vector_sum<typename E::value_type> >::result_type
        sum (const vector_expression<E> &e);

        // norm_1 v = sum (abs (v [i]))
        template<class E>
        typename vector_scalar_unary_traits<E, vector_norm_1<typename E::value_type> >::result_type
        norm_1 (const vector_expression<E> &e);

        // norm_2 v = sqrt (sum (v [i] * v [i]))
        template<class E>
        typename vector_scalar_unary_traits<E, vector_norm_2<typename E::value_type> >::result_type
        norm_2 (const vector_expression<E> &e);

        // norm_inf v = max (abs (v [i]))
        template<class E>
        typename vector_scalar_unary_traits<E, vector_norm_inf<typename E::value_type> >::result_type
        norm_inf (const vector_expression<E> &e);

        // index_norm_inf v = min (i: abs (v [i]) == max (abs (v [i])))
        template<class E>
        typename vector_scalar_unary_traits<E, vector_index_norm_inf<typename E::value_type> >::result_type
        index_norm_inf (const vector_expression<E> &e);

#### Description

`sum` computes the sum of the vector expression's elements. `norm_1`, `norm_2` and `norm_inf` compute the corresponding ||.||1, ||.||2 and ||.||inf vector norms.`index_norm_1` computes the index of the vector expression's first element having maximal absolute value.

#### Definition

Defined in the header vector_expression.hpp.

#### Type requirements

- `E` is a model of Vector Expression.

#### Preconditions

None.

#### Complexity

Linear depending from the size of the vector expression.

#### Examples

    int main () {
        using namespace boost::numeric::ublas;
        vector<double> v (3);
        for (int i = 0; i < v.size (); ++ i)
            v (i) = i;

        std::cout << sum (v) << std::endl;
        std::cout << norm_1 (v) << std::endl;
        std::cout << norm_2 (v) << std::endl;
        std::cout << norm_inf (v) << std::endl;
        std::cout << index_norm_inf (v) << std::endl;
    }

### Binary Reductions

#### Prototypes

        template<class E1, class E2, class F>
        struct vector_scalar_binary_traits {
            typedef typename F::result_type result_type;
        };

        // inner_prod (v1, v2) = sum (v1 [i] * v2 [i]
        template<class E1, class E2>
        typename vector_scalar_binary_traits<E1, E2, vector_inner_prod<typename E1::value_type,
                                                                       typename E2::value_type,
                                                                       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);

        template<class E1, class E2>
        typename vector_scalar_binary_traits<E1, E2, vector_inner_prod<typename E1::value_type,
                                                                       typename E2::value_type,
                                                                       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);

#### Description

`inner_prod` computes the inner product of the vector expressions. `prec_inner_prod` computes the double precision inner product of the vector expressions`.`

#### Definition

Defined in the header vector_expression.hpp.

#### Type requirements

- `E1` is a model of Vector Expression.
- `E2` is a model of Vector Expression.

#### Preconditions

- `e1 ().size () == e2 ().size ()`

#### Complexity

Linear depending from the size of the vector expressions.

#### Examples

    int main () {
        using namespace boost::numeric::ublas;
        vector<double> v1 (3), v2 (3);
        for (int i = 0; i < std::min (v1.size (), v2.size ()); ++ i)
            v1 (i) = v2 (i) = i;

        std::cout << inner_prod (v1, v2) << std::endl;
    }

Copyright (©) 2000-2002 Joerg Walter, Mathias Koch
Permission to copy, use, modify, sell and distribute this document is granted provided this copyright notice appears in all copies. This document is provided ``as is'' without express or implied warranty, and with no claim as to its suitability for any purpose.

Last revised: 8/3/2002