# Matrix Proxies

## Matrix Row

#### Description

The templated class `matrix_row<M>` allows addressing a row of a matrix.

#### Example

    #include <boost/numeric/ublas/matrix.hpp>
    #include <boost/numeric/ublas/io.hpp>

    int main () {
        using namespace boost::numeric::ublas;
        matrix<double> m (3, 3);
        for (unsigned i = 0; i < m.size1 (); ++ i) {
            matrix_row<matrix<double> > mr (m, i);
            for (unsigned j = 0; j < mr.size (); ++ j)
                mr (j) = 3 * i + j;
            std::cout << mr << std::endl;
        }
    }

#### Definition

Defined in the header matrix_proxy.hpp.

#### Template parameters

Parameter | Description | Default
--- | --- | ---
`M` | The type of matrix referenced. |

#### Model of

Vector Expression .

#### Type requirements

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

#### Public base classes

`vector_expression<matrix_row<M> >`

#### Members

Member | Description
--- | ---
`matrix_row (matrix_type &data, size_type i)` | Constructs a sub vector.
`size_type size () const` | Returns the size of the sub vector.
`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.
`matrix_row &operator = (const matrix_row &mr)` | The assignment operator.
`matrix_row &assign_temporary (matrix_row &mr)` | Assigns a temporary. May change the matrix row `mr` .
`template<class AE> matrix_row &operator = (const vector_expression<AE> &ae)` | The extended assignment operator.
`template<class AE> matrix_row &assign (const vector_expression<AE> &ae)` | Assigns a vector expression to the sub vector. Left and right hand side of the assignment should be independent.
`template<class AE> matrix_row &operator += (const vector_expression<AE> &ae)` | A computed assignment operator. Adds the vector expression to the sub vector.
`template<class AE> matrix_row &plus_assign (const vector_expression<AE> &ae)` | Adds a vector expression to the sub vector. Left and right hand side of the assignment should be independent.
`template<class AE> matrix_row &operator -= (const vector_expression<AE> &ae)` | A computed assignment operator. Subtracts the vector expression from the sub vector.
`template<class AE> matrix_row &minus_assign (const vector_expression<AE> &ae)` | Subtracts a vector expression from the sub vector. Left and right hand side of the assignment should be independent.
`template<class AT> matrix_row &operator *= (const AT &at)` | A computed assignment operator. Multiplies the sub vector with a scalar.
`template<class AT> matrix_row &operator /= (const AT &at)` | A computed assignment operator. Divides the sub vector through a scalar.
`void swap (matrix_row &mr)` | Swaps the contents of the sub vectors.
`const_iterator begin () const` | Returns a `const_iterator` pointing to the beginning of the `matrix_row`.
`const_iterator end () const` | Returns a `const_iterator` pointing to the end of the `matrix_row`.
`iterator begin ()` | Returns a `iterator` pointing to the beginning of the `matrix_row`.
`iterator end ()` | Returns a `iterator` pointing to the end of the `matrix_row`.
`const_reverse_iterator rbegin () const` | Returns a `const_reverse_iterator` pointing to the beginning of the reversed `matrix_row`.
`const_reverse_iterator rend () const` | Returns a `const_reverse_iterator` pointing to the end of the reversed `matrix_row`.
`reverse_iterator rbegin ()` | Returns a `reverse_iterator` pointing to the beginning of the reversed `matrix_row`.
`reverse_iterator rend ()` | Returns a `reverse_iterator` pointing to the end of the reversed `matrix_row`.

#### Interface

        // Matrix based row vector class
        template<class M>
        class matrix_row:
            public vector_expression<matrix_row<M> > {
        public:
            typedef const M const_matrix_type;
            typedef M matrix_type;
            typedef typename M::size_type size_type;
            typedef typename M::difference_type difference_type;
            typedef typename M::value_type value_type;
            typedef typename M::const_reference const_reference;
            typedef typename M::reference reference;
            typedef typename M::const_pointer const_pointer;
            typedef typename M::pointer pointer;
            typedef const vector_const_reference<const matrix_row<matrix_type> > const_closure_type;
            typedef vector_reference<matrix_row<matrix_type> > closure_type;
            typedef typename M::const_iterator2 const_iterator_type;
            typedef typename M::iterator2 iterator_type;
            typedef typename storage_restrict_traits<typename M::storage_category,
                                                     dense_proxy_tag>::storage_category storage_category;

            // Construction and destruction
            matrix_row ();
            matrix_row (matrix_type &data, size_type i);

            // Accessors
            size_type size () const;
            const_matrix_type &data () const;
            matrix_type &data ();

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

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

            // Assignment
            matrix_row &operator = (const matrix_row &mr);
            matrix_row &assign_temporary (matrix_row &mr);
            template<class AE>
            matrix_row &operator = (const vector_expression<AE> &ae);
            template<class AE>
            matrix_row &assign (const vector_expression<AE> &ae);
            template<class AE>
            matrix_row &operator += (const vector_expression<AE> &ae);
            template<class AE>
            matrix_row &plus_assign (const vector_expression<AE> &ae);
            template<class AE>
            matrix_row &operator -= (const vector_expression<AE> &ae);
            template<class AE>
            matrix_row &minus_assign (const vector_expression<AE> &ae);
            template<class AT>
            matrix_row &operator *= (const AT &at);
            template<class AT>
            matrix_row &operator /= (const AT &at);

            // Swapping
            void swap (matrix_row &mr);
            friend void swap (matrix_row &mr1, matrix_row &mr2);

            class const_iterator;
            class iterator;

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

            // Iterators simply are pointers.

            class const_iterator:
                public container_const_reference<matrix_row>,
                public random_access_iterator_base<const_iterator, value_type> {
            public:
                typedef typename const_iterator_type::iterator_category iterator_category;
                typedef typename const_iterator_type::difference_type difference_type;
                typedef typename const_iterator_type::value_type value_type;
                typedef typename const_iterator_type::reference reference;
                typedef typename const_iterator_type::pointer pointer;

                // Construction and destruction
                const_iterator ();
                const_iterator (const matrix_row &mr, const const_iterator_type &it);
                const_iterator (const iterator &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;

            class iterator:
                public container_reference<matrix_row>,
                public random_access_iterator_base<iterator, value_type> {
            public:
                typedef typename iterator_type::iterator_category iterator_category;
                typedef typename iterator_type::difference_type difference_type;
                typedef typename iterator_type::value_type value_type;
                typedef typename iterator_type::reference reference;
                typedef typename iterator_type::pointer pointer;

                // Construction and destruction
                iterator ();
                iterator (matrix_row &mr, const iterator_type &it);

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

                // Dereference
                reference operator * () const;

                // Index
                size_type index () const;

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

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

### Projections

#### Prototypes

        template<class M>
        matrix_row<M> row (M &data, std::size_t i);
        template<class M>
        const matrix_row<const M> row (const M &data, std::size_t i);

#### Description

The free `row` functions support the construction of matrix rows.

#### Definition

Defined in the header matrix_proxy.hpp.

#### Type requirements

- `M` is a model of Matrix Expression .

#### Preconditions

- `i < data.size1 ()`

#### Complexity

Linear depending from the size of the row.

#### Examples

    #include <boost/numeric/ublas/matrix.hpp>
    #include <boost/numeric/ublas/io.hpp>

    int main () {
        using namespace boost::numeric::ublas;
        matrix<double> m (3, 3);
        for (unsigned i = 0; i < m.size1 (); ++ i) {
            for (unsigned j = 0; j < m.size2 (); ++ j)
                row (m, i) (j) = 3 * i + j;
            std::cout << row (m, i) << std::endl;
        }
    }

## Matrix Column

#### Description

The templated class `matrix_column<M>` allows addressing a column of a matrix.

#### Example

    #include <boost/numeric/ublas/matrix.hpp>
    #include <boost/numeric/ublas/io.hpp>

    int main () {
        using namespace boost::numeric::ublas;
        matrix<double> m (3, 3);
        for (unsigned j = 0; j < m.size2 (); ++ j) {
            matrix_column<matrix<double> > mc (m, j);
            for (unsigned i = 0; i < mc.size (); ++ i)
                mc (i) = 3 * i + j;
            std::cout << mc << std::endl;
        }
    }

#### Definition

Defined in the header matrix_proxy.hpp.

#### Template parameters

Parameter | Description | Default
--- | --- | ---
`M` | The type of matrix referenced. |

#### Model of

Vector Expression .

#### Type requirements

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

#### Public base classes

`vector_expression<matrix_column<M> >`

#### Members

Member | Description
--- | ---
`matrix_column (matrix_type &data, size_type j)` | Constructs a sub vector.
`size_type size () const` | Returns the size of the sub vector.
`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.
`matrix_column &operator = (const matrix_column &mc)` | The assignment operator.
`matrix_column &assign_temporary (matrix_column &mc)` | Assigns a temporary. May change the matrix column `mc` .
`template<class AE> matrix_column &operator = (const vector_expression<AE> &ae)` | The extended assignment operator.
`template<class AE> matrix_column &assign (const vector_expression<AE> &ae)` | Assigns a vector expression to the sub vector. Left and right hand side of the assignment should be independent.
`template<class AE> matrix_column &operator += (const vector_expression<AE> &ae)` | A computed assignment operator. Adds the vector expression to the sub vector.
`template<class AE> matrix_column &plus_assign (const vector_expression<AE> &ae)` | Adds a vector expression to the sub vector. Left and right hand side of the assignment should be independent.
`template<class AE> matrix_column &operator -= (const vector_expression<AE> &ae)` | A computed assignment operator. Subtracts the vector expression from the sub vector.
`template<class AE> matrix_column &minus_assign (const vector_expression<AE> &ae)` | Subtracts a vector expression from the sub vector. Left and right hand side of the assignment should be independent.
`template<class AT> matrix_column &operator *= (const AT &at)` | A computed assignment operator. Multiplies the sub vector with a scalar.
`template<class AT> matrix_column &operator /= (const AT &at)` | A computed assignment operator. Divides the sub vector through a scalar.
`void swap (matrix_column &mc)` | Swaps the contents of the sub vectors.
`const_iterator begin () const` | Returns a `const_iterator` pointing to the beginning of the `matrix_column`.
`const_iterator end () const` | Returns a `const_iterator` pointing to the end of the `matrix_column`.
`iterator begin ()` | Returns a `iterator` pointing to the beginning of the `matrix_column`.
`iterator end ()` | Returns a `iterator` pointing to the end of the `matrix_column`.
`const_reverse_iterator rbegin () const` | Returns a `const_reverse_iterator` pointing to the beginning of the reversed `matrix_column`.
`const_reverse_iterator rend () const` | Returns a `const_reverse_iterator` pointing to the end of the reversed `matrix_column`.
`reverse_iterator rbegin ()` | Returns a `reverse_iterator` pointing to the beginning of the reversed `matrix_column`.
`reverse_iterator rend ()` | Returns a `reverse_iterator` pointing to the end of the reversed `matrix_column`.

#### Interface

        // Matrix based column vector class
        template<class M>
        class matrix_column:
            public vector_expression<matrix_column<M> > {
        public:
            typedef const M const_matrix_type;
            typedef M matrix_type;
            typedef typename M::size_type size_type;
            typedef typename M::difference_type difference_type;
            typedef typename M::value_type value_type;
            typedef typename M::const_reference const_reference;
            typedef typename M::reference reference;
            typedef typename M::const_pointer const_pointer;
            typedef typename M::pointer pointer;
            typedef const vector_const_reference<const matrix_column<matrix_type> > const_closure_type;
            typedef vector_reference<matrix_column<matrix_type> > closure_type;
            typedef typename M::const_iterator1 const_iterator_type;
            typedef typename M::iterator1 iterator_type;
            typedef typename storage_restrict_traits<typename M::storage_category,
                                                     dense_proxy_tag>::storage_category storage_category;

            // Construction and destruction
            matrix_column ();
            matrix_column (matrix_type &data, size_type j);

            // Accessors
            size_type size () const;
            const_matrix_type &data () const;
            matrix_type &data ();

            // 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);

            // Assignment
            matrix_column &operator = (const matrix_column &mc);
            matrix_column &assign_temporary (matrix_column &mc);
            template<class AE>
            matrix_column &operator = (const vector_expression<AE> &ae);
            template<class AE>
            matrix_column &assign (const vector_expression<AE> &ae);
            template<class AE>
            matrix_column &operator += (const vector_expression<AE> &ae);
            template<class AE>
            matrix_column &plus_assign (const vector_expression<AE> &ae);
            template<class AE>
            matrix_column &operator -= (const vector_expression<AE> &ae);
            template<class AE>
            matrix_column &minus_assign (const vector_expression<AE> &ae);
            template<class AT>
            matrix_column &operator *= (const AT &at);
            template<class AT>
            matrix_column &operator /= (const AT &at);

            // Swapping
            void swap (matrix_column &mc);
            friend void swap (matrix_column &mc1, matrix_column &mc2);

            class const_iterator;
            class 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);

            // Iterators simply are pointers.

            class const_iterator:
                public container_const_reference<matrix_column>,
                public random_access_iterator_base<const_iterator, value_type> {
            public:
                typedef typename const_iterator_type::iterator_category iterator_category;
                typedef typename const_iterator_type::difference_type difference_type;
                typedef typename const_iterator_type::value_type value_type;
                typedef typename const_iterator_type::reference reference;
                typedef typename const_iterator_type::pointer pointer;

                // Construction and destruction
                const_iterator ();
                const_iterator (const matrix_column &mc, const const_iterator_type &it);
                const_iterator (const iterator &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;

            class iterator:
                public container_reference<matrix_column>,
                public random_access_iterator_base<iterator, value_type> {
            public:
                typedef typename iterator_type::iterator_category iterator_category;
                typedef typename iterator_type::difference_type difference_type;
                typedef typename iterator_type::value_type value_type;
                typedef typename iterator_type::reference reference;
                typedef typename iterator_type::pointer pointer;

                // Construction and destruction
                iterator ();
                iterator (matrix_column &mc, const iterator_type &it);

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

                // Dereference
                reference operator * () const;

                // Index
                size_type index () const;

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

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

### Projections

#### Prototypes

        template<class M>
        matrix_column<M> column (M &data, std::size_t j);
        template<class M>
        const matrix_column<const M> column (const M &data, std::size_t j);

#### Description

The free `column` functions support the construction of matrix columns.

#### Definition

Defined in the header matrix_proxy.hpp.

#### Type requirements

- `M` is a model of Matrix Expression .

#### Preconditions

- `j < data.size2 ()`

#### Complexity

Linear depending from the size of the column.

#### Examples

    #include <boost/numeric/ublas/matrix.hpp>
    #include <boost/numeric/ublas/io.hpp>

    int main () {
        using namespace boost::numeric::ublas;
        matrix<double> m (3, 3);
        for (unsigned j = 0; j < m.size2 (); ++ j) {
            for (unsigned i = 0; i < m.size1 (); ++ i)
                column (m, j) (i) = 3 * i + j;
            std::cout << column (m, j) << std::endl;
        }
    }

## Vector Range

#### Description

The templated class `matrix_vector_range<M>` allows addressing a sub vector of a matrix.

#### Example

    #include <boost/numeric/ublas/matrix.hpp>
    #include <boost/numeric/ublas/io.hpp>

    int main () {
        using namespace boost::numeric::ublas;
        matrix<double> m (3, 3);
        for (unsigned i = 0; i < m.size1 (); ++ i)
            for (unsigned j = 0; j < m.size2 (); ++ j)
                m (i, j) = 3 * i + j;

        matrix_vector_range<matrix<double> > mvr (m, range (0, 3), range (0, 3));
        std::cout << mvr << std::endl;
    }

#### Definition

Defined in the header matrix_proxy.hpp.

#### Template parameters

Parameter | Description | Default
--- | --- | ---
`M` | The type of matrix referenced. |

#### Model of

Vector Expression .

#### Type requirements

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

#### Public base classes

`vector_expression<matrix_vector_range<M> >`

#### Members

Member | Description
--- | ---
`matrix_vector_range (matrix_type &data, const range &r1, const range &r2)` | Constructs a sub vector.
`size_type size () const` | Returns the size of the sub vector.
`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.
`matrix_vector_range &operator = (const matrix_vector_range &mvr)` | The assignment operator.
`matrix_vector_range &assign_temporary (matrix_vector_range &mvr)` | Assigns a temporary. May change the matrix vector range `mvr`.
`template<class AE> matrix_vector_range &operator = (const vector_expression<AE> &ae)` | The extended assignment operator.
`template<class AE> matrix_vector_range &assign (const vector_expression<AE> &ae)` | Assigns a vector expression to the sub vector. Left and right hand side of the assignment should be independent.
`template<class AE> matrix_vector_range &operator += (const vector_expression<AE> &ae)` | A computed assignment operator. Adds the vector expression to the sub vector.
`template<class AE> matrix_vector_range &plus_assign (const vector_expression<AE> &ae)` | Adds a vector expression to the sub vector. Left and right hand side of the assignment should be independent.
`template<class AE> matrix_vector_range &operator -= (const vector_expression<AE> &ae)` | A computed assignment operator. Subtracts the vector expression from the sub vector.
`template<class AE> matrix_vector_range &minus_assign (const vector_expression<AE> &ae)` | Subtracts a vector expression from the sub vector. Left and right hand side of the assignment should be independent.
`template<class AT> matrix_vector_range &operator *= (const AT &at)` | A computed assignment operator. Multiplies the sub vector with a scalar.
`template<class AT> matrix_vector_range &operator /= (const AT &at)` | A computed assignment operator. Divides the sub vector through a scalar.
`void swap (matrix_vector_range &mvr)` | Swaps the contents of the sub vectors.
`const_iterator begin () const` | Returns a `const_iterator` pointing to the beginning of the `matrix_vector_range`.
`const_iterator end () const` | Returns a `const_iterator` pointing to the end of the `matrix_vector_range`.
`iterator begin ()` | Returns a `iterator` pointing to the beginning of the `matrix_vector_range`.
`iterator end ()` | Returns a `iterator` pointing to the end of the `matrix_vector_range`.
`const_reverse_iterator rbegin () const` | Returns a `const_reverse_iterator` pointing to the beginning of the `matrix_vector_range`.
`const_reverse_iterator rend () const` | Returns a `const_reverse_iterator` pointing to the end of the reversed `matrix_vector_range`.
`reverse_iterator rbegin ()` | Returns a `reverse_iterator` pointing to the beginning of the reversed `matrix_vector_range`.
`reverse_iterator rend ()` | Returns a `reverse_iterator` pointing to the end of the reversed `matrix_vector_range`.

#### Interface

        // Matrix based vector range class
        template<class M>
        class matrix_vector_range:
            public vector_expression<matrix_vector_range<M> > {
        public:
            typedef const M const_matrix_type;
            typedef M matrix_type;
            typedef typename M::size_type size_type;
            typedef typename M::difference_type difference_type;
            typedef typename M::value_type value_type;
            typedef typename M::const_reference const_reference;
            typedef typename M::reference reference;
            typedef typename M::const_pointer const_pointer;
            typedef typename M::pointer pointer;
            typedef const vector_const_reference<const matrix_vector_range<matrix_type> > const_closure_type;
            typedef vector_reference<matrix_vector_range<matrix_type> > closure_type;
            typedef range::const_iterator const_iterator1_type;
            typedef range::const_iterator iterator1_type;
            typedef range::const_iterator const_iterator2_type;
            typedef range::const_iterator iterator2_type;
            typedef typename storage_restrict_traits<typename M::storage_category,
                                                     dense_proxy_tag>::storage_category storage_category;

            // Construction and destruction
            matrix_vector_range ();
            matrix_vector_range (matrix_type &data, const range &r1, const range &r2);

            // Accessors
            size_type size () const;
            const_matrix_type &data () const;
            matrix_type &data ();

            // 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);

            // Assignment
            matrix_vector_range &operator = (const matrix_vector_range &mvr);
            matrix_vector_range &assign_temporary (matrix_vector_range &mvr);
            template<class AE>
            matrix_vector_range &operator = (const vector_expression<AE> &ae);
            template<class AE>
            matrix_vector_range &assign (const vector_expression<AE> &ae);
            template<class AE>
            matrix_vector_range &operator += (const vector_expression<AE> &ae);
            template<class AE>
            matrix_vector_range &plus_assign (const vector_expression<AE> &ae);
            template<class AE>
            matrix_vector_range &operator -= (const vector_expression<AE> &ae);
            template<class AE>
            matrix_vector_range &minus_assign (const vector_expression<AE> &ae);
            template<class AT>
            matrix_vector_range &operator *= (const AT &at);
            template<class AT>
            matrix_vector_range &operator /= (const AT &at);

            // Swapping
            void swap (matrix_vector_range &mvr);
            friend void swap (matrix_vector_range &mvr1, matrix_vector_range &mvr2);

            class const_iterator;
            class 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);

            // Iterators simply are indices.

            class const_iterator:
                public container_const_reference<matrix_vector_range>,
                public random_access_iterator_base<const_iterator, value_type> {
            public:
                typedef typename restrict_traits<typename M::const_iterator1::iterator_category,
                                                 typename M::const_iterator2::iterator_category>::iterator_category iterator_category;
                typedef typename matrix_vector_range::difference_type difference_type;
                typedef typename matrix_vector_range::value_type value_type;
                typedef typename matrix_vector_range::const_reference reference;
                typedef typename matrix_vector_range::const_pointer pointer;

                // Construction and destruction
                const_iterator ();
                const_iterator (const matrix_vector_range &mvr, const const_iterator1_type &it1, const const_iterator2_type &it2);
                const_iterator (const iterator &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;

            class iterator:
                public container_reference<matrix_vector_range>,
                public random_access_iterator_base<iterator, value_type> {
            public:
                typedef typename restrict_traits<typename M::iterator1::iterator_category,
                                                 typename M::iterator2::iterator_category>::iterator_category iterator_category;
                typedef typename matrix_vector_range::difference_type difference_type;
                typedef typename matrix_vector_range::value_type value_type;
                typedef typename matrix_vector_range::reference reference;
                typedef typename matrix_vector_range::pointer pointer;

                // Construction and destruction
                iterator ();
                iterator (matrix_vector_range &mvr, const iterator1_type &it1, const iterator2_type &it2);

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

                // Dereference
                reference operator * () const;

                // Index
                size_type index () const;

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

                // Comparison
                bool operator == (const iterator &it) const;
                bool operator <(const iterator &it) 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 Slice

#### Description

The templated class `matrix_vector_slice<M>` allows addressing a sliced sub vector of a matrix.

#### Example

    #include <boost/numeric/ublas/matrix.hpp>
    #include <boost/numeric/ublas/io.hpp>

    int main () {
        using namespace boost::numeric::ublas;
        matrix<double> m (3, 3);
        for (unsigned i = 0; i < m.size1 (); ++ i)
            for (unsigned j = 0; j < m.size2 (); ++ j)
                m (i, j) = 3 * i + j;

        matrix_vector_slice<matrix<double> > mvs (m, slice (0, 1, 3), slice (0, 1, 3));
        std::cout << mvs << std::endl;
    }

#### Definition

Defined in the header matrix_proxy.hpp.

#### Template parameters

Parameter | Description | Default
--- | --- | ---
`M` | The type of matrix referenced. |

#### Model of

Vector Expression .

#### Type requirements

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

#### Public base classes

`vector_expression<matrix_vector_slice<M> >`

#### Members

Member | Description
--- | ---
`matrix_vector_slice (matrix_type &data, const slice &s1, const slice &s2)` | Constructs a sub vector.
`size_type size () const` | Returns the size of the sub vector.
`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.
`matrix_vector_slice &operator = (const matrix_vector_slice &mvs)` | The assignment operator.
`matrix_vector_slice &assign_temporary (matrix_vector_slice &mvs)` | Assigns a temporary. May change the matrix vector slice `vs`.
`template<class AE> matrix_vector_slice &operator = (const vector_expression<AE> &ae)` | The extended assignment operator.
`template<class AE> matrix_vector_slice &assign (const vector_expression<AE> &ae)` | Assigns a vector expression to the sub vector. Left and right hand side of the assignment should be independent.
`template<class AE> matrix_vector_slice &operator += (const vector_expression<AE> &ae)` | A computed assignment operator. Adds the vector expression to the sub vector.
`template<class AE> matrix_vector_slice &plus_assign (const vector_expression<AE> &ae)` | Adds a vector expression to the sub vector. Left and right hand side of the assignment should be independent.
`template<class AE> matrix_vector_slice &operator -= (const vector_expression<AE> &ae)` | A computed assignment operator. Subtracts the vector expression from the sub vector.
`template<class AE> matrix_vector_slice &minus_assign (const vector_expression<AE> &ae)` | Subtracts a vector expression from the sub vector. Left and right hand side of the assignment should be independent.
`template<class AT> matrix_vector_slice &operator *= (const AT &at)` | A computed assignment operator. Multiplies the sub vector with a scalar.
`template<class AT> matrix_vector_slice &operator /= (const AT &at)` | A computed assignment operator. Divides the sub vector through a scalar.
`void swap (matrix_vector_slice &mvs)` | Swaps the contents of the sub vectors.
`const_iterator begin () const` | Returns a `const_iterator` pointing to the beginning of the `matrix_vector_slice`.
`const_iterator end () const` | Returns a `const_iterator` pointing to the end of the `matrix_vector_slice`.
`iterator begin ()` | Returns a `iterator` pointing to the beginning of the `matrix_vector_slice`.
`iterator end ()` | Returns a `iterator` pointing to the end of the `matrix_vector_slice`.
`const_reverse_iterator rbegin () const` | Returns a `const_reverse_iterator` pointing to the beginning of the reversed `matrix_vector_slice`.
`const_reverse_iterator rend () const` | Returns a `const_reverse_iterator` pointing to the end of the reversed `matrix_vector_slice`.
`reverse_iterator rbegin ()` | Returns a `reverse_iterator` pointing to the beginning of the reversed `matrix_vector_slice`.
`reverse_iterator rend ()` | Returns a `reverse_iterator` pointing to the end of the reversed `matrix_vector_slice`.

#### Interface

        // Matrix based vector slice class
        template<class M>
        class matrix_vector_slice:
            public vector_expression<matrix_vector_slice<M> > {
        public:
            typedef const M const_matrix_type;
            typedef M matrix_type;
            typedef typename M::size_type size_type;
            typedef typename M::difference_type difference_type;
            typedef typename M::value_type value_type;
            typedef typename M::const_reference const_reference;
            typedef typename M::reference reference;
            typedef typename M::const_pointer const_pointer;
            typedef typename M::pointer pointer;
            typedef const vector_const_reference<const matrix_vector_slice<matrix_type> > const_closure_type;
            typedef vector_reference<matrix_vector_slice<matrix_type> > closure_type;
            typedef slice::const_iterator const_iterator1_type;
            typedef slice::const_iterator iterator1_type;
            typedef slice::const_iterator const_iterator2_type;
            typedef slice::const_iterator iterator2_type;
            typedef typename storage_restrict_traits<typename M::storage_category,
                                                     dense_proxy_tag>::storage_category storage_category;

            // Construction and destruction
            matrix_vector_slice ();
            matrix_vector_slice (matrix_type &data, const slice &s1, const slice &s2);

            // Accessors
            size_type size () const;
            const_matrix_type &data () const;
            matrix_type &data ();

            // 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);

            // Assignment
            matrix_vector_slice &operator = (const matrix_vector_slice &mvs);
            matrix_vector_slice &assign_temporary (matrix_vector_slice &mvs);
            template<class AE>
            matrix_vector_slice &operator = (const vector_expression<AE> &ae);
            template<class AE>
            matrix_vector_slice &assign (const vector_expression<AE> &ae);
            template<class AE>
            matrix_vector_slice &operator += (const vector_expression<AE> &ae);
            template<class AE>
            matrix_vector_slice &plus_assign (const vector_expression<AE> &ae);
            template<class AE>
            matrix_vector_slice &operator -= (const vector_expression<AE> &ae);
            template<class AE>
            matrix_vector_slice &minus_assign (const vector_expression<AE> &ae);
            template<class AT>
            matrix_vector_slice &operator *= (const AT &at);
            template<class AT>
            matrix_vector_slice &operator /= (const AT &at);

            // Swapping
            void swap (matrix_vector_slice &mvs);
            friend void swap (matrix_vector_slice &mvs1, matrix_vector_slice &mvs2);

            class const_iterator;
            class 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);

            // Iterators simply are indices.

            class const_iterator:
                public container_const_reference<matrix_vector_slice>,
                public random_access_iterator_base<const_iterator, value_type> {
            public:
                typedef typename restrict_traits<typename M::const_iterator1::iterator_category,
                                                 typename M::const_iterator2::iterator_category>::iterator_category iterator_category;
                typedef typename matrix_vector_slice::difference_type difference_type;
                typedef typename matrix_vector_slice::value_type value_type;
                typedef typename matrix_vector_slice::const_reference reference;
                typedef typename matrix_vector_slice::const_pointer pointer;

                // Construction and destruction
                const_iterator ();
                const_iterator (const matrix_vector_slice &mvs, const const_iterator1_type &it1, const const_iterator2_type &it2);
                const_iterator (const iterator &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;

            class iterator:
                public container_reference<matrix_vector_slice>,
                public random_access_iterator_base<iterator, value_type> {
            public:
                typedef typename restrict_traits<typename M::iterator1::iterator_category,
                                                 typename M::iterator2::iterator_category>::iterator_category iterator_category;
                typedef typename matrix_vector_slice::difference_type difference_type;
                typedef typename matrix_vector_slice::value_type value_type;
                typedef typename matrix_vector_slice::reference reference;
                typedef typename matrix_vector_slice::pointer pointer;

                // Construction and destruction
                iterator ();
                iterator (matrix_vector_slice &mvs, const iterator1_type &it1, const iterator2_type &it2);

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

                // Dereference
                reference operator * () const;

                // Index
                size_type index () const;

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

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

## Matrix Range

#### Description

The templated class `matrix_range<M>` allows addressing a sub matrix of a matrix.

#### Example

    #include <boost/numeric/ublas/matrix.hpp>
    #include <boost/numeric/ublas/io.hpp>

    int main () {
        using namespace boost::numeric::ublas;
        matrix<double> m (3, 3);
        matrix_range<matrix<double> > mr (m, range (0, 3), range (0, 3));
        for (unsigned i = 0; i < mr.size1 (); ++ i)
            for (unsigned j = 0; j < mr.size2 (); ++ j)
                mr (i, j) = 3 * i + j;
        std::cout << mr << std::endl;
    }

#### Definition

Defined in the header matrix_proxy.hpp.

#### Template parameters

Parameter | Description | Default
--- | --- | ---
`M` | The type of matrix referenced. |

#### Model of

Matrix Expression .

#### Type requirements

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

#### Public base classes

`matrix_expression<matrix_range<M> >`

#### Members

Member | Description
--- | ---
`matrix_range (matrix_type &data, const range &r1, const range &r2)` | Constructs a sub matrix.
`size_type start1 () const` | Returns the index of the first row.
`size_type size1 () const` | Returns the number of rows.
`size_type start2 () const` | Returns the index of the first column.
`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.
`reference operator () (size_type i, size_type j)` | Returns a reference of the `j`-th element in the`i`-th row.
`matrix_range &operator = (const matrix_range &mr)` | The assignment operator.
`matrix_range &assign_temporary (matrix_range &mr)` | Assigns a temporary. May change the matrix range `mr` .
`template<class AE> matrix_range &operator = (const matrix_expression<AE> &ae)` | The extended assignment operator.
`template<class AE> matrix_range &assign (const matrix_expression<AE> &ae)` | Assigns a matrix expression to the sub matrix. Left and right hand side of the assignment should be independent.
`template<class AE> matrix_range &operator += (const matrix_expression<AE> &ae)` | A computed assignment operator. Adds the matrix expression to the sub matrix.
`template<class AE> matrix_range &plus_assign (const matrix_expression<AE> &ae)` | Adds a matrix expression to the sub matrix. Left and right hand side of the assignment should be independent.
`template<class AE> matrix_range &operator -= (const matrix_expression<AE> &ae)` | A computed assignment operator. Subtracts the matrix expression from the sub matrix.
`template<class AE> matrix_range &minus_assign (const matrix_expression<AE> &ae)` | Subtracts a matrix expression from the sub matrix. Left and right hand side of the assignment should be independent.
`template<class AT> matrix_range &operator *= (const AT &at)` | A computed assignment operator. Multiplies the sub matrix with a scalar.
`template<class AT> matrix_range &operator /= (const AT &at)` | A computed assignment operator. Divides the sub matrix through a scalar.
`void swap (matrix_range &mr)` | Swaps the contents of the sub matrices.
`const_iterator1 begin1 () const` | Returns a `const_iterator1` pointing to the beginning of the `matrix_range`.
`const_iterator1 end1 () const` | Returns a `const_iterator1` pointing to the end of the `matrix_range`.
`iterator1 begin1 ()` | Returns a `iterator1` pointing to the beginning of the `matrix_range`.
`iterator1 end1 ()` | Returns a `iterator1` pointing to the end of the `matrix_range`.
`const_iterator2 begin2 () const` | Returns a `const_iterator2` pointing to the beginning of the `matrix_range`.
`const_iterator2 end2 () const` | Returns a `const_iterator2` pointing to the end of the `matrix_range`.
`iterator2 begin2 ()` | Returns a `iterator2` pointing to the beginning of the `matrix_range`.
`iterator2 end2 ()` | Returns a `iterator2` pointing to the end of the `matrix_range`.
`const_reverse_iterator1 rbegin1 () const` | Returns a `const_reverse_iterator1` pointing to the beginning of the reversed `matrix_range`.
`const_reverse_iterator1 rend1 () const` | Returns a `const_reverse_iterator1` pointing to the end of the reversed `matrix_range`.
`reverse_iterator1 rbegin1 ()` | Returns a `reverse_iterator1` pointing to the beginning of the reversed `matrix_range`.
`reverse_iterator1 rend1 ()` | Returns a `reverse_iterator1` pointing to the end of the reversed `matrix_range`.
`const_reverse_iterator2 rbegin2 () const` | Returns a `const_reverse_iterator2` pointing to the beginning of the reversed `matrix_range`.
`const_reverse_iterator2 rend2 () const` | Returns a `const_reverse_iterator2` pointing to the end of the reversed `matrix_range`.
`reverse_iterator2 rbegin2 ()` | Returns a `reverse_iterator2` pointing to the beginning of the reversed `matrix_range`.
`reverse_iterator2 rend2 ()` | Returns a `reverse_iterator2` pointing to the end of reversed the `matrix_range`.

#### Interface

        // Matrix based range class
        template<class M>
        class matrix_range:
            public matrix_expression<matrix_range<M> > {
        public:
            typedef const M const_matrix_type;
            typedef M matrix_type;
            typedef typename M::size_type size_type;
            typedef typename M::difference_type difference_type;
            typedef typename M::value_type value_type;
            typedef typename M::const_reference const_reference;
            typedef typename M::reference reference;
            typedef typename M::const_pointer const_pointer;
            typedef typename M::pointer pointer;
            typedef const matrix_const_reference<const matrix_range<matrix_type> > const_closure_type;
            typedef matrix_reference<matrix_range<matrix_type> > closure_type;
            typedef typename M::const_iterator1 const_iterator1_type;
            typedef typename M::iterator1 iterator1_type;
            typedef typename M::const_iterator2 const_iterator2_type;
            typedef typename M::iterator2 iterator2_type;
            typedef typename storage_restrict_traits<typename M::storage_category,
                                                     dense_proxy_tag>::storage_category storage_category;
            typedef typename M::orientation_category orientation_category;

            // Construction and destruction
            matrix_range ();
            matrix_range (matrix_type &data, const range &r1, const range &r2);

            // Accessors
            size_type start1 () const;
            size_type size1 () const;
            size_type start2() const;
            size_type size2 () const;
            const_matrix_type &data () const;
            matrix_type &data ();

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

            matrix_range<matrix_type> project (const range &r1, const range &r2) const;

            // Assignment
            matrix_range &operator = (const matrix_range &mr);
            matrix_range &assign_temporary (matrix_range &mr);
            template<class AE>
            matrix_range &operator = (const matrix_expression<AE> &ae);
            template<class AE>
            matrix_range &assign (const matrix_expression<AE> &ae);
            template<class AE>
            matrix_range& operator += (const matrix_expression<AE> &ae);
            template<class AE>
            matrix_range &plus_assign (const matrix_expression<AE> &ae);
            template<class AE>
            matrix_range& operator -= (const matrix_expression<AE> &ae);
            template<class AE>
            matrix_range &minus_assign (const matrix_expression<AE> &ae);
            template<class AT>
            matrix_range& operator *= (const AT &at);
            template<class AT>
            matrix_range& operator /= (const AT &at);

            // Swapping
            void swap (matrix_range &mr);
            friend void swap (matrix_range &mr1, matrix_range &mr2);

            class const_iterator1;
            class iterator1;
            class const_iterator2;
            class iterator2;
            typedef reverse_iterator_base1<const_iterator1> const_reverse_iterator1;
            typedef reverse_iterator_base1<iterator1> reverse_iterator1;
            typedef reverse_iterator_base2<const_iterator2> const_reverse_iterator2;
            typedef reverse_iterator_base2<iterator2> reverse_iterator2;

            // Element lookup
            const_iterator1 find_first1 (int rank, size_type i, size_type j) const;
            iterator1 find_first1 (int rank, size_type i, size_type j);
            const_iterator1 find_last1 (int rank, size_type i, size_type j) const;
            iterator1 find_last1 (int rank, size_type i, size_type j);
            const_iterator2 find_first2 (int rank, size_type i, size_type j) const;
            iterator2 find_first2 (int rank, size_type i, size_type j);
            const_iterator2 find_last2 (int rank, size_type i, size_type j) const;
            iterator2 find_last2 (int rank, size_type i, size_type j);

            // Iterators simply are pointers.

            class const_iterator1:
                public container_const_reference<matrix_range>,
                public random_access_iterator_base<const_iterator1, value_type> {
            public:
                typedef typename const_iterator1_type::iterator_category iterator_category;
                typedef typename const_iterator1_type::difference_type difference_type;
                typedef typename const_iterator1_type::value_type value_type;
                typedef typename const_iterator1_type::reference reference;
                typedef typename const_iterator1_type::pointer pointer;
                typedef const_iterator2 dual_iterator_type;
                typedef const_reverse_iterator2 dual_reverse_iterator_type;

                // Construction and destruction
                const_iterator1 ();
                const_iterator1 (const matrix_range &mr, const const_iterator1_type &it);
                const_iterator1 (const iterator1 &it);

                // 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 iterator1:
                public container_reference<matrix_range>,
                public random_access_iterator_base<iterator1, value_type> {
            public:
                typedef typename iterator1_type::iterator_category iterator_category;
                typedef typename iterator1_type::difference_type difference_type;
                typedef typename iterator1_type::value_type value_type;
                typedef typename iterator1_type::reference reference;
                typedef typename iterator1_type::pointer pointer;
                typedef iterator2 dual_iterator_type;
                typedef reverse_iterator2 dual_reverse_iterator_type;

                // Construction and destruction
                iterator1 ();
                iterator1 (matrix_range &mr, const iterator1_type &it);

                // Arithmetic
                iterator1 &operator ++ ();
                iterator1 &operator -- ();
                iterator1 &operator += (difference_type n);
                iterator1 &operator -= (difference_type n);
                difference_type operator - (const iterator1 &it) const;

                // Dereference
                reference operator * () const;

                iterator2 begin () const;
                iterator2 end () const;
                reverse_iterator2 rbegin () const;
                reverse_iterator2 rend () const;

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

                // Assignment
                iterator1 &operator = (const iterator1 &it);

                // Comparison
                bool operator == (const iterator1 &it) const;
                bool operator <(const iterator1 &it) const;
            };

            iterator1 begin1 ();
            iterator1 end1 ();

            class const_iterator2:
                public container_const_reference<matrix_range>,
                public random_access_iterator_base<const_iterator2, value_type> {
            public:
                typedef typename const_iterator2_type::iterator_category iterator_category;
                typedef typename const_iterator2_type::difference_type difference_type;
                typedef typename const_iterator2_type::value_type value_type;
                typedef typename const_iterator2_type::reference reference;
                typedef typename const_iterator2_type::pointer pointer;
                typedef const_iterator1 dual_iterator_type;
                typedef const_reverse_iterator1 dual_reverse_iterator_type;

                // Construction and destruction
                const_iterator2 ();
                const_iterator2 (const matrix_range &mr, const const_iterator2_type &it);
                const_iterator2 (const iterator2 &it);

                // 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;

            class iterator2:
                public container_reference<matrix_range>,
                public random_access_iterator_base<iterator2, value_type> {
            public:
                typedef typename iterator2_type::iterator_category iterator_category;
                typedef typename iterator2_type::difference_type difference_type;
                typedef typename iterator2_type::value_type value_type;
                typedef typename iterator2_type::reference reference;
                typedef typename iterator2_type::pointer pointer;
                typedef iterator1 dual_iterator_type;
                typedef reverse_iterator1 dual_reverse_iterator_type;

                // Construction and destruction
                iterator2 ();
                iterator2 (matrix_range &mr, const iterator2_type &it);

                // Arithmetic
                iterator2 &operator ++ ();
                iterator2 &operator -- ();
                iterator2 &operator += (difference_type n);
                iterator2 &operator -= (difference_type n);
                difference_type operator - (const iterator2 &it) const;

                // Dereference
                reference operator * () const;

                iterator1 begin () const;
                iterator1 end () const;
                reverse_iterator1 rbegin () const;
                reverse_iterator1 rend () const;

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

                // Assignment
                iterator2 &operator = (const iterator2 &it);

                // Comparison
                bool operator == (const iterator2 &it) const;
                bool operator <(const iterator2 &it) const;
            };

            iterator2 begin2 ();
            iterator2 end2 ();

            // Reverse iterators

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

            reverse_iterator1 rbegin1 ();
            reverse_iterator1 rend1 ();

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

            reverse_iterator2 rbegin2 ();
            reverse_iterator2 rend2 ();
        };

### Projections

#### Prototypes

        template<class M>
        matrix_range<M> project (M &data, const range &r1, const range &r2);
        template<class M>
        const matrix_range<const M> project (const M &data, const range &r1, const range &r2);
        template<class M>
        matrix_range<M> project (const matrix_range<M> &data, const range &r1, const range &r2);

#### Description

The free `project` functions support the construction of matrix ranges.

#### Definition

Defined in the header matrix_proxy.hpp.

#### Type requirements

- `M` is a model of Matrix Expression .

#### Preconditions

- `r1.start () + r1.size () <= data.size1 ()`
- `r2.start () + r2.size () <= data.size2 ()`

#### Complexity

Quadratic depending from the size of the ranges.

#### Examples

    #include <boost/numeric/ublas/matrix.hpp>
    #include <boost/numeric/ublas/io.hpp>

    int main () {
        using namespace boost::numeric::ublas;
        matrix<double> m (3, 3);
        for (unsigned i = 0; i < m.size1 (); ++ i)
            for (unsigned j = 0; j < m.size2 (); ++ j)
                project (m, range (0, 3), range (0, 3)) (i, j) = 3 * i + j;
        std::cout << project (m, range (0, 3), range (0, 3)) << std::endl;
    }

## Matrix Slice

#### Description

The templated class `matrix_slice<M>` allows addressing a sliced sub matrix of a matrix.

#### Example

    #include <boost/numeric/ublas/matrix.hpp>
    #include <boost/numeric/ublas/io.hpp>

    int main () {
        using namespace boost::numeric::ublas;
        matrix<double> m (3, 3);
        matrix_slice<matrix<double> > ms (m, slice (0, 1, 3), slice (0, 1, 3));
        for (unsigned i = 0; i < ms.size1 (); ++ i)
            for (unsigned j = 0; j < ms.size2 (); ++ j)
                ms (i, j) = 3 * i + j;
        std::cout << ms << std::endl;
    }

#### Definition

Defined in the header matrix_proxy.hpp.

#### Template parameters

Parameter | Description | Default
--- | --- | ---
`M` | The type of matrix referenced. |

#### Model of

Matrix Expression .

#### Type requirements

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

#### Public base classes

`matrix_expression<matrix_slice<M> >`

#### Members

Member | Description
--- | ---
`matrix_slice (matrix_type &data, const slice &s1, const slice &s2)` | Constructs a sub matrix.
`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.
`reference operator () (size_type i, size_type j)` | Returns a reference of the `j`-th element in the`i`-th row.
`matrix_slice &operator = (const matrix_slice &ms)` | The assignment operator.
`matrix_slice &assign_temporary (matrix_slice &ms)` | Assigns a temporary. May change the matrix slice `ms` .
`template<class AE> matrix_slice &operator = (const matrix_expression<AE> &ae)` | The extended assignment operator.
`template<class AE> matrix_slice &assign (const matrix_expression<AE> &ae)` | Assigns a matrix expression to the sub matrix. Left and right hand side of the assignment should be independent.
`template<class AE> matrix_slice &operator += (const matrix_expression<AE> &ae)` | A computed assignment operator. Adds the matrix expression to the sub matrix.
`template<class AE> matrix_slice &plus_assign (const matrix_expression<AE> &ae)` | Adds a matrix expression to the sub matrix. Left and right hand side of the assignment should be independent.
`template<class AE> matrix_slice &operator -= (const matrix_expression<AE> &ae)` | A computed assignment operator. Subtracts the matrix expression from the sub matrix.
`template<class AE> matrix_slice &minus_assign (const matrix_expression<AE> &ae)` | Subtracts a matrix expression from the sub matrix. Left and right hand side of the assignment should be independent.
`template<class AT> matrix_slice &operator *= (const AT &at)` | A computed assignment operator. Multiplies the sub matrix with a scalar.
`template<class AT> matrix_slice &operator /= (const AT &at)` | A computed assignment operator. Multiplies the sub matrix through a scalar.
`void swap (matrix_slice &ms)` | Swaps the contents of the sub matrices.
`const_iterator1 begin1 () const` | Returns a `const_iterator1` pointing to the beginning of the `matrix_slice`.
`const_iterator1 end1 () const` | Returns a `const_iterator1` pointing to the end of the `matrix_slice`.
`iterator1 begin1 ()` | Returns a `iterator1` pointing to the beginning of the `matrix_slice`.
`iterator1 end1 ()` | Returns a `iterator1` pointing to the end of the `matrix_slice`.
`const_iterator2 begin2 () const` | Returns a `const_iterator2` pointing to the beginning of the `matrix_slice`.
`const_iterator2 end2 () const` | Returns a `const_iterator2` pointing to the end of the `matrix_slice`.
`iterator2 begin2 ()` | Returns a `iterator2` pointing to the beginning of the `matrix_slice`.
`iterator2 end2 ()` | Returns a `iterator2` pointing to the end of the `matrix_slice`.
`const_reverse_iterator1 rbegin1 () const` | Returns a `const_reverse_iterator1` pointing to the beginning of the reversed `matrix_slice`.
`const_reverse_iterator1 rend1 () const` | Returns a `const_reverse_iterator1` pointing to the end of the reversed `matrix_slice`.
`reverse_iterator1 rbegin1 ()` | Returns a `reverse_iterator1` pointing to the beginning of the reversed `matrix_slice`.
`reverse_iterator1 rend1 ()` | Returns a `reverse_iterator1` pointing to the end of the reversed `matrix_slice`.
`const_reverse_iterator2 rbegin2 () const` | Returns a `const_reverse_iterator2` pointing to the beginning of the reversed `matrix_slice`.
`const_reverse_iterator2 rend2 () const` | Returns a `const_reverse_iterator2` pointing to the end of the reversed `matrix_slice`.
`reverse_iterator2 rbegin2 ()` | Returns a `reverse_iterator2` pointing to the beginning of the reversed `matrix_slice`.
`reverse_iterator2 rend2 ()` | Returns a `reverse_iterator2` pointing to the end of the reversed `matrix_slice`.

#### Interface

        // Matrix based slice class
        template<class M>
        class matrix_slice:
            public matrix_expression<matrix_slice<M> > {
        public:
            typedef const M const_matrix_type;
            typedef M matrix_type;
            typedef typename M::size_type size_type;
            typedef typename M::difference_type difference_type;
            typedef typename M::value_type value_type;
            typedef typename M::const_reference const_reference;
            typedef typename M::reference reference;
            typedef typename M::const_pointer const_pointer;
            typedef typename M::pointer pointer;
            typedef const matrix_const_reference<const matrix_slice<matrix_type> > const_closure_type;
            typedef matrix_reference<matrix_slice<matrix_type> > closure_type;
            typedef slice::const_iterator const_iterator1_type;
            typedef slice::const_iterator iterator1_type;
            typedef slice::const_iterator const_iterator2_type;
            typedef slice::const_iterator iterator2_type;
            typedef typename storage_restrict_traits<typename M::storage_category,
                                                     dense_proxy_tag>::storage_category storage_category;
            typedef typename M::orientation_category orientation_category;

            // Construction and destruction
            matrix_slice ();
            matrix_slice (matrix_type &data, const slice &s1, const slice &s2);

            // Accessors
            size_type size1 () const;
            size_type size2 () const;
            const_matrix_type &data () const;
            matrix_type &data ();

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

            matrix_slice<matrix_type> project (const range &r1, const range &r2) const;
            matrix_slice<matrix_type> project (const slice &s1, const slice &s2) const;

            // Assignment
            matrix_slice &operator = (const matrix_slice &ms);
            matrix_slice &assign_temporary (matrix_slice &ms);
            template<class AE>
            matrix_slice &operator = (const matrix_expression<AE> &ae);
            template<class AE>
            matrix_slice &assign (const matrix_expression<AE> &ae);
            template<class AE>
            matrix_slice& operator += (const matrix_expression<AE> &ae);
            template<class AE>
            matrix_slice &plus_assign (const matrix_expression<AE> &ae);
            template<class AE>
            matrix_slice& operator -= (const matrix_expression<AE> &ae);
            template<class AE>
            matrix_slice &minus_assign (const matrix_expression<AE> &ae);
            template<class AT>
            matrix_slice& operator *= (const AT &at);
            template<class AT>
            matrix_slice& operator /= (const AT &at);

            // Swapping
            void swap (matrix_slice &ms);
            friend void swap (matrix_slice &ms1, matrix_slice &ms2);

            class const_iterator1;
            class iterator1;
            class const_iterator2;
            class iterator2;
            typedef reverse_iterator_base1<const_iterator1> const_reverse_iterator1;
            typedef reverse_iterator_base1<iterator1> reverse_iterator1;
            typedef reverse_iterator_base2<const_iterator2> const_reverse_iterator2;
            typedef reverse_iterator_base2<iterator2> reverse_iterator2;

            // Element lookup
            const_iterator1 find_first1 (int rank, size_type i, size_type j) const;
            iterator1 find_first1 (int rank, size_type i, size_type j);
            const_iterator1 find_last1 (int rank, size_type i, size_type j) const;
            iterator1 find_last1 (int rank, size_type i, size_type j);
            const_iterator2 find_first2 (int rank, size_type i, size_type j) const;
            iterator2 find_first2 (int rank, size_type i, size_type j);
            const_iterator2 find_last2 (int rank, size_type i, size_type j) const;
            iterator2 find_last2 (int rank, size_type i, size_type j);

            // Iterators simply are indices.

            class const_iterator1:
                public container_const_reference<matrix_slice>,
                public random_access_iterator_base<const_iterator1, value_type> {
            public:
                typedef typename M::const_iterator1::iterator_category iterator_category;
                typedef typename M::const_iterator1::difference_type difference_type;
                typedef typename M::const_iterator1::value_type value_type;
                typedef typename M::const_iterator1::reference reference;
                typedef typename M::const_iterator1::pointer pointer;
                typedef const_iterator2 dual_iterator_type;
                typedef const_reverse_iterator2 dual_reverse_iterator_type;

                // Construction and destruction
                const_iterator1 ();
                const_iterator1 (const matrix_slice &ms, const const_iterator1_type &it1, const const_iterator2_type &it2);
                const_iterator1 (const iterator1 &it);

                // 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 iterator1:
                public container_reference<matrix_slice>,
                public random_access_iterator_base<iterator1, value_type> {
            public:
                typedef typename M::iterator1::iterator_category iterator_category;
                typedef typename M::iterator1::difference_type difference_type;
                typedef typename M::iterator1::value_type value_type;
                typedef typename M::iterator1::reference reference;
                typedef typename M::iterator1::pointer pointer;
                typedef iterator2 dual_iterator_type;
                typedef reverse_iterator2 dual_reverse_iterator_type;

                // Construction and destruction
                iterator1 ();
                iterator1 (matrix_slice &ms, const iterator1_type &it1, const iterator2_type &it2);

                // Arithmetic
                iterator1 &operator ++ ();
                iterator1 &operator -- ();
                iterator1 &operator += (difference_type n);
                iterator1 &operator -= (difference_type n);
                difference_type operator - (const iterator1 &it) const;

                // Dereference
                reference operator * () const;

                iterator2 begin () const;
                iterator2 end () const;
                reverse_iterator2 rbegin () const;
                reverse_iterator2 rend () const;

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

                // Assignment
                iterator1 &operator = (const iterator1 &it);

                // Comparison
                bool operator == (const iterator1 &it) const;
                bool operator <(const iterator1 &it) const;
            };

            iterator1 begin1 ();
            iterator1 end1 ();

            class const_iterator2:
                public container_const_reference<matrix_slice>,
                public random_access_iterator_base<const_iterator2, value_type> {
            public:
                typedef typename M::const_iterator2::iterator_category iterator_category;
                typedef typename M::const_iterator2::difference_type difference_type;
                typedef typename M::const_iterator2::value_type value_type;
                typedef typename M::const_iterator2::reference reference;
                typedef typename M::const_iterator2::pointer pointer;
                typedef const_iterator1 dual_iterator_type;
                typedef const_reverse_iterator1 dual_reverse_iterator_type;

                // Construction and destruction
                const_iterator2 ();
                const_iterator2 (const matrix_slice &ms, const const_iterator1_type &it1, const const_iterator2_type &it2);
                const_iterator2 (const iterator2 &it);

                // 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;

            class iterator2:
                public container_reference<matrix_slice>,
                public random_access_iterator_base<iterator2, value_type> {
            public:
                typedef typename M::iterator2::iterator_category iterator_category;
                typedef typename M::iterator2::difference_type difference_type;
                typedef typename M::iterator2::value_type value_type;
                typedef typename M::iterator2::reference reference;
                typedef typename M::iterator2::pointer pointer;
                typedef iterator1 dual_iterator_type;
                typedef reverse_iterator1 dual_reverse_iterator_type;

                // Construction and destruction
                iterator2 ();
                iterator2 (matrix_slice &ms, const iterator1_type &it1, const iterator2_type &it2);

                // Arithmetic
                iterator2 &operator ++ ();
                iterator2 &operator -- ();
                iterator2 &operator += (difference_type n);
                iterator2 &operator -= (difference_type n);
                difference_type operator - (const iterator2 &it) const;

                // Dereference
                reference operator * () const;

                iterator1 begin () const;
                iterator1 end () const;
                reverse_iterator1 rbegin () const;
                reverse_iterator1 rend () const;

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

                // Assignment
                iterator2 &operator = (const iterator2 &it);

                // Comparison
                bool operator == (const iterator2 &it) const;
                bool operator <(const iterator2 &it) const;
            };

            iterator2 begin2 ();
            iterator2 end2 ();

            // Reverse iterators

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

            reverse_iterator1 rbegin1 ();
            reverse_iterator1 rend1 ();

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

            reverse_iterator2 rbegin2 ();
            reverse_iterator2 rend2 ();
        };

### Projections

#### Prototypes

        template<class M>
        matrix_slice<M> project (const matrix_slice<M> &data, const range &r1, const range &r2);
        template<class M>
        matrix_slice<M> project (M &data, const slice &s1, const slice &s2);
        template<class M>
        const matrix_slice<const M> project (const M &data, const slice &s1, const slice &s2);
        template<class M>
        matrix_slice<M> project (const matrix_slice<M> &data, const slice &s1, const slice &s2);

#### Description

The free `project` functions support the construction of matrix slices.

#### Definition

Defined in the header matrix_proxy.hpp.

#### Type requirements

- `M` is a model of Matrix Expression .

#### Preconditions

- `s1.start () + s1.stride () * s1.size () <= data.size1 ()`
- `s2.start () + s2.stride () * s2.size () <= data.size2 ()`

#### Complexity

Quadratic depending from the size of the slices.

#### Examples

    #include <boost/numeric/ublas/matrix.hpp>
    #include <boost/numeric/ublas/io.hpp>

    int main () {
        using namespace boost::numeric::ublas;
        matrix<double> m (3, 3);
        for (unsigned i = 0; i < m.size1 (); ++ i)
            for (unsigned j = 0; j < m.size2 (); ++ j)
                project (m, slice (0, 1, 3), slice (0, 1, 3)) (i, j) = 3 * i + j;
        std::cout << project (m, slice (0, 1, 3), slice (0, 1, 3)) << 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: 1/15/2003