# Sparse Matrix

#### Description

The templated class `sparse_matrix<T, F, A>` is the base container adaptor for sparse matrices. For a (m x n)-dimensional sparse matrix and 0 <= i < m, 0 <= j < n the non-zero elements mi, j are mapped via (i x n + j) for row major orientation or via (i + j x m) for column major orientation to consecutive elements of the associative container, i.e. for elements k = mi1 ,j1 and k + 1 = mi2 ,j2 of the container holds i1 < i2 or (i1 = i2 and j1 < j2) with row major orientation or j1 < j 2 or (j1 = j2 and i1 < i2 ) with column major orientation.

#### Example

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

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

#### Definition

Defined in the header matrix_sparse.hpp.

#### Template parameters

Parameter | Description | Default
--- | --- | ---
`T` | The type of object stored in the sparse matrix. |
`F` | Functor describing the storage organization. [1] | `row_major`
`A` | The type of the adapted array. [2] | `map_array<std::size_t, T>`

#### Model of

Matrix .

#### Type requirements

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

#### Public base classes

`matrix_expression<sparse_matrix<T, F, A> >`

#### Members

Member | Description
--- | ---
`sparse_matrix ()` | Allocates a `sparse_matrix` that holds at most zero rows of zero elements.
`sparse_matrix (size_type size1, size_type2, size_type non_zeros)` | Allocates a `sparse_matrix` that holds at most `size1` rows of `size2` elements.
`sparse_matrix (const sparse_matrix &m)` | The copy constructor.
`template<class AE> sparse_matrix (size_type non_zeros, const matrix_expression<AE> &ae)` | The extended copy constructor.
`void resize (size_type size1, size_type size2, size_type non_zeros)` | Reallocates a `sparse_matrix` to hold at most `size1` rows of `size2` elements. The content of the `sparse_matrix` is preserved.
`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.
`sparse_matrix &operator = (const sparse_matrix &m)` | The assignment operator.
`sparse_matrix &assign_temporary (sparse_matrix &m)` | Assigns a temporary. May change the sparse matrix `m` .
`template<class AE> sparse_matrix &operator = (const matrix_expression<AE> &ae)` | The extended assignment operator.
`template<class AE> sparse_matrix &assign (const matrix_expression<AE> &ae)` | Assigns a matrix expression to the sparse matrix. Left and right hand side of the assignment should be independent.
`template<class AE> sparse_matrix &operator += (const matrix_expression<AE> &ae)` | A computed assignment operator. Adds the matrix expression to the sparse matrix.
`template<class AE> sparse_matrix &plus_assign (const matrix_expression<AE> &ae)` | Adds a matrix expression to the sparse matrix. Left and right hand side of the assignment should be independent.
`template<class AE> sparse_matrix &operator -= (const matrix_expression<AE> &ae)` | A computed assignment operator. Subtracts the matrix expression from the sparse matrix.
`template<class AE> sparse_matrix &minus_assign (const matrix_expression<AE> &ae)` | Subtracts a matrix expression from the sparse matrix. Left and right hand side of the assignment should be independent.
`template<class AT> sparse_matrix &operator *= (const AT &at)` | A computed assignment operator. Multiplies the sparse matrix with a scalar.
`template<class AT> sparse_matrix &operator /= (const AT &at)` | A computed assignment operator. Divides the sparse matrix through a scalar.
`void swap (sparse_matrix &m)` | Swaps the contents of the sparse matrices.
`void insert (size_type i, size_type j, const_reference t)` | Inserts the value `t` at the `j`-th element of the `i`-th row.
`void erase (size_type i, size_type j)` | Erases the value at the `j`-th element of the `i`-th row.
`void clear ()` | Clears the sparse matrix.
`const_iterator1 begin1 () const` | Returns a `const_iterator1` pointing to the beginning of the `sparse_matrix`.
`const_iterator1 end1 () const` | Returns a `const_iterator1` pointing to the end of the `sparse_matrix`.
`iterator1 begin1 ()` | Returns a `iterator1` pointing to the beginning of the `sparse_matrix`.
`iterator1 end1 ()` | Returns a `iterator1` pointing to the end of the `sparse_matrix`.
`const_iterator2 begin2 () const` | Returns a `const_iterator2` pointing to the beginning of the `sparse_matrix`.
`const_iterator2 end2 () const` | Returns a `const_iterator2` pointing to the end of the `sparse_matrix`.
`iterator2 begin2 ()` | Returns a `iterator2` pointing to the beginning of the `sparse_matrix`.
`iterator2 end2 ()` | Returns a `iterator2` pointing to the end of the `sparse_matrix`.
`const_reverse_iterator1 rbegin1 () const` | Returns a `const_reverse_iterator1` pointing to the beginning of the reversed `sparse_matrix`.
`const_reverse_iterator1 rend1 () const` | Returns a `const_reverse_iterator1` pointing to the end of the reversed `sparse_matrix`.
`reverse_iterator1 rbegin1 ()` | Returns a `reverse_iterator1` pointing to the beginning of the reversed `sparse_matrix`.
`reverse_iterator1 rend1 ()` | Returns a `reverse_iterator1` pointing to the end of the reversed `sparse_matrix`.
`const_reverse_iterator2 rbegin2 () const` | Returns a `const_reverse_iterator2` pointing to the beginning of the reversed `sparse_matrix`.
`const_reverse_iterator2 rend2 () const` | Returns a `const_reverse_iterator2` pointing to the end of the reversed `sparse_matrix`.
`reverse_iterator2 rbegin2 ()` | Returns a `reverse_iterator2` pointing to the beginning of the reversed `sparse_matrix`.
`reverse_iterator2 rend2 ()` | Returns a `reverse_iterator2` pointing to the end of the reversed `sparse_matrix`.

#### Notes

[1] Supported parameters for the storage organization are `row_major` and `column_major`.

[2] Supported parameters for the adapted array are `map_array<std::size_t, T>` and `std::map<std::size_t, T>`.

#### Interface

        // Array based sparse matrix class
        template<class T, class F, class A>
        class sparse_matrix:
            public matrix_expression<sparse_matrix<T, F, A> > {
        public:
            typedef std::size_t size_type;
            typedef std::ptrdiff_t difference_type;
            typedef T value_type;
            typedef const T &const_reference;
            typedef T &reference;
            typedef const T *const_pointer;
            typedef T *pointer;
            typedef F functor_type;
            typedef A array_type;
            typedef const A const_array_type;
            typedef const sparse_matrix<T, F, A> const_self_type;
            typedef sparse_matrix<T, F, A> self_type;
            typedef const matrix_const_reference<const_self_type> const_closure_type;
            typedef matrix_reference<self_type> closure_type;
            typedef typename A::const_iterator const_iterator_type;
            typedef typename A::iterator iterator_type;
            typedef sparse_tag storage_category;
            typedef typename F::orientation_category orientation_category;

            // Construction and destruction
            sparse_matrix ();
            sparse_matrix (size_type size1, size_type size2, size_type non_zeros = 0);
            sparse_matrix (const sparse_matrix &m);
            template<class AE>
            sparse_matrix (const matrix_expression<AE> &ae, size_type non_zeros = 0);

            // Accessors
            size_type size1 () const;
            size_type size2 () const;
            size_type non_zeros () const;
            const_array_type &data () const;
            array_type &data ();

            // Resizing
            void resize (size_type size1, size_type size2, size_type non_zeros = 0);

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

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

            // Swapping
            void swap (sparse_matrix &m);
            friend void swap (sparse_matrix &m1, sparse_matrix &m2);

            // Element insertion and erasure
            void insert (size_type i, size_type j, const_reference t);
            void erase (size_type i, size_type j);
            void clear ();

            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 find1 (int rank, size_type i, size_type j) const;
            iterator1 find1 (int rank, size_type i, size_type j);
            const_iterator2 find2 (int rank, size_type i, size_type j) const;
            iterator2 find2 (int rank, size_type i, size_type j);
            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<sparse_matrix>,
                public bidirectional_iterator_base<const_iterator1, value_type> {
            public:
                typedef sparse_bidirectional_iterator_tag iterator_category;
                typedef typename sparse_matrix::difference_type difference_type;
                typedef typename sparse_matrix::value_type value_type;
                typedef typename sparse_matrix::const_reference reference;
                typedef typename sparse_matrix::const_pointer pointer;
                typedef const_iterator2 dual_iterator_type;
                typedef const_reverse_iterator2 dual_reverse_iterator_type;
                typedef typename functor_type::functor1_type functor1_type;

                // Construction and destruction
                const_iterator1 ();
                const_iterator1 (const sparse_matrix &m, int rank, size_type i, size_type j, const const_iterator_type &it);
                const_iterator1 (const iterator1 &it);

                // Arithmetic
                const_iterator1 &operator ++ ();
                const_iterator1 &operator -- ();

                // 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;
            };

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

            class iterator1:
                public container_reference<sparse_matrix>,
                public bidirectional_iterator_base<iterator1, value_type> {
            public:
                typedef sparse_bidirectional_iterator_tag iterator_category;
                typedef typename sparse_matrix::difference_type difference_type;
                typedef typename sparse_matrix::value_type value_type;
                typedef typename sparse_matrix::reference reference;
                typedef typename sparse_matrix::pointer pointer;
                typedef iterator2 dual_iterator_type;
                typedef reverse_iterator2 dual_reverse_iterator_type;
                typedef typename functor_type::functor1_type functor1_type;

                // Construction and destruction
                iterator1 ();
                iterator1 (sparse_matrix &m, int rank, size_type i, size_type j, const iterator_type &it);

                // Arithmetic
                iterator1 &operator ++ ();
                iterator1 &operator -- ();

                // 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;
            };

            iterator1 begin1 ();
            iterator1 end1 ();

            class const_iterator2:
                public container_const_reference<sparse_matrix>,
                public bidirectional_iterator_base<const_iterator2, value_type> {
            public:
                typedef sparse_bidirectional_iterator_tag iterator_category;
                typedef typename sparse_matrix::difference_type difference_type;
                typedef typename sparse_matrix::value_type value_type;
                typedef typename sparse_matrix::const_reference reference;
                typedef typename sparse_matrix::const_pointer pointer;
                typedef const_iterator1 dual_iterator_type;
                typedef const_reverse_iterator1 dual_reverse_iterator_type;
                typedef typename functor_type::functor2_type functor2_type;

                // Construction and destruction
                const_iterator2 ();
                const_iterator2 (const sparse_matrix &m, int rank, size_type i, size_type j, const const_iterator_type &it);
                const_iterator2 (const iterator2 &it);

                // Arithmetic
                const_iterator2 &operator ++ ();
                const_iterator2 &operator -- ();

                // 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;
            };

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

            class iterator2:
                public container_reference<sparse_matrix>,
                public bidirectional_iterator_base<iterator2, value_type> {
            public:
                typedef sparse_bidirectional_iterator_tag iterator_category;
                typedef typename sparse_matrix::difference_type difference_type;
                typedef typename sparse_matrix::value_type value_type;
                typedef typename sparse_matrix::reference reference;
                typedef typename sparse_matrix::pointer pointer;
                typedef iterator1 dual_iterator_type;
                typedef reverse_iterator1 dual_reverse_iterator_type;
                typedef typename functor_type::functor2_type functor2_type;

                // Construction and destruction
                iterator2 ();
                iterator2 (sparse_matrix &m, int rank, size_type i, size_type j, const iterator_type &it);

                // Arithmetic
                iterator2 &operator ++ ();
                iterator2 &operator -- ();

                // 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;
            };

            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 ();
        };

## Compressed Matrix

#### Description

The templated class `compressed_matrix<T, F, IB, IA, TA>` is the base container adaptor for compressed matrices. For a (m x n )-dimensional compressed matrix and 0 <= i < m, 0 <= j < n the non-zero elements mi, j are mapped via (i x n + j) for row major orientation or via (i + j x m) for column major orientation to consecutive elements of the index and value containers, i.e. for elements k = mi 1,j1 and k + 1 = mi2 ,j2 of the container holds i1 < i2 or (i1 = i2 and j1 < j2) with row major orientation or j1 < j 2 or (j1 = j2 and i1 < i2 ) with column major orientation.

#### Example

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

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

#### Definition

Defined in the header matrix_sparse.hpp.

#### Template parameters

Parameter | Description | Default
--- | --- | ---
`T` | The type of object stored in the compressed matrix. |
`F` | Functor describing the storage organization. [1] | `row_major`
`IB` | The index base of the compressed vector. [2] | `0`
`IA` | The type of the adapted array for indices. [3] | `unbounded_array<std::size_t>`
`TA` | The type of the adapted array for values. [3] | `unbounded_array<T>`

#### Model of

Matrix .

#### Type requirements

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

#### Public base classes

`matrix_expression<compressed_matrix<T, F, IB, IA, TA> >`

#### Members

Member | Description
--- | ---
`compressed_matrix ()` | Allocates a `compressed_matrix` that holds at most zero rows of zero elements.
`compressed_matrix (size_type size1, size_type2, size_type non_zeros)` | Allocates a `compressed_matrix` that holds at most `size1` rows of `size2` elements.
`compressed_matrix (const compressed_matrix &m)` | The copy constructor.
`template<class AE> compressed_matrix (size_type non_zeros, const matrix_expression<AE> &ae)` | The extended copy constructor.
`void resize (size_type size1, size_type size2, size_type non_zeros)` | Reallocates a `compressed_matrix` to hold at most `size1` rows of `size2` elements. The content of the `compressed_matrix` is preserved.
`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.
`compressed_matrix &operator = (const compressed_matrix &m)` | The assignment operator.
`compressed_matrix &assign_temporary (compressed_matrix &m)` | Assigns a temporary. May change the compressed matrix `m`.
`template<class AE> compressed_matrix &operator = (const matrix_expression<AE> &ae)` | The extended assignment operator.
`template<class AE> compressed_matrix &assign (const matrix_expression<AE> &ae)` | Assigns a matrix expression to the compressed matrix. Left and right hand side of the assignment should be independent.
`template<class AE> compressed_matrix &operator += (const matrix_expression<AE> &ae)` | A computed assignment operator. Adds the matrix expression to the compressed matrix.
`template<class AE> compressed_matrix &plus_assign (const matrix_expression<AE> &ae)` | Adds a matrix expression to the compressed matrix. Left and right hand side of the assignment should be independent.
`template<class AE> compressed_matrix &operator -= (const matrix_expression<AE> &ae)` | A computed assignment operator. Subtracts the matrix expression from the compressed matrix.
`template<class AE> compressed_matrix &minus_assign (const matrix_expression<AE> &ae)` | Subtracts a matrix expression from the compressed matrix. Left and right hand side of the assignment should be independent.
`template<class AT> compressed_matrix &operator *= (const AT &at)` | A computed assignment operator. Multiplies the compressed matrix with a scalar.
`template<class AT> compressed_matrix &operator /= (const AT &at)` | A computed assignment operator. Divides the compressed matrix through a scalar.
`void swap (compressed_matrix &m)` | Swaps the contents of the compressed matrices.
`void insert (size_type i, size_type j, const_reference t)` | Inserts the value `t` at the `j`-th element of the `i`-th row.
`void erase (size_type i, size_type j)` | Erases the value at the `j`-th element of the `i`-th row.
`void clear ()` | Clears the compressed matrix.
`const_iterator1 begin1 () const` | Returns a `const_iterator1` pointing to the beginning of the `compressed_matrix`.
`const_iterator1 end1 () const` | Returns a `const_iterator1` pointing to the end of the `compressed_matrix`.
`iterator1 begin1 ()` | Returns a `iterator1` pointing to the beginning of the `compressed_matrix`.
`iterator1 end1 ()` | Returns a `iterator1` pointing to the end of the `compressed_matrix`.
`const_iterator2 begin2 () const` | Returns a `const_iterator2` pointing to the beginning of the `compressed_matrix`.
`const_iterator2 end2 () const` | Returns a `const_iterator2` pointing to the end of the `compressed_matrix`.
`iterator2 begin2 ()` | Returns a `iterator2` pointing to the beginning of the `compressed_matrix`.
`iterator2 end2 ()` | Returns a `iterator2` pointing to the end of the `compressed_matrix`.
`const_reverse_iterator1 rbegin1 () const` | Returns a `const_reverse_iterator1` pointing to the beginning of the reversed `compressed_matrix`.
`const_reverse_iterator1 rend1 () const` | Returns a `const_reverse_iterator1` pointing to the end of the reversed `compressed_matrix`.
`reverse_iterator1 rbegin1 ()` | Returns a `reverse_iterator1` pointing to the beginning of the reversed `compressed_matrix`.
`reverse_iterator1 rend1 ()` | Returns a `reverse_iterator1` pointing to the end of the reversed `compressed_matrix`.
`const_reverse_iterator2 rbegin2 () const` | Returns a `const_reverse_iterator2` pointing to the beginning of the reversed `compressed_matrix`.
`const_reverse_iterator2 rend2 () const` | Returns a `const_reverse_iterator2` pointing to the end of the reversed `compressed_matrix`.
`reverse_iterator2 rbegin2 ()` | Returns a `reverse_iterator2` pointing to the beginning of the reversed `compressed_matrix`.
`reverse_iterator2 rend2 ()` | Returns a `reverse_iterator2` pointing to the end of the reversed `compressed_matrix`.

#### Notes

[1] Supported parameters for the storage organization are `row_major` and `column_major`.

[2] Supported parameters for the index base are `0` and `1` at least.

[3] Supported parameters for the adapted array are `unbounded_array<>` , `bounded_array<>` and `std::vector<>` .

#### Interface

        // Array based sparse matrix class
        template<class T, class F, std::size_t IB, class IA, class TA>
        class compressed_matrix:
            public matrix_expression<compressed_matrix<T, F, IB, IA, TA> > {
        public:
            typedef std::size_t size_type;
            typedef std::ptrdiff_t difference_type;
            typedef T value_type;
            typedef const T &const_reference;
            typedef T &reference;
            typedef const T *const_pointer;
            typedef T *pointer;
            typedef F functor_type;
            typedef IA index_array_type;
            typedef TA value_array_type;
            typedef const compressed_matrix<T, F, IB, IA, TA> const_self_type;
            typedef compressed_matrix<T, F, IB, IA, TA> self_type;
            typedef const matrix_const_reference<const_self_type> const_closure_type;
            typedef matrix_reference<self_type> closure_type;
            typedef typename A::const_iterator const_iterator_type;
            typedef typename A::iterator iterator_type;
            typedef sparse_tag storage_category;
            typedef typename F::orientation_category orientation_category;

            // Construction and destruction
            compressed_matrix ();
            compressed_matrix (size_type size1, size_type size2, size_type non_zeros = 0);
            compressed_matrix (const compressed_matrix &m);
            template<class AE>
            compressed_matrix (const matrix_expression<AE> &ae, size_type non_zeros = 0);

            // Accessors
            size_type size1 () const;
            size_type size2 () const;
            size_type non_zeros () const;
            static size_type index_base ();
            const index_array_type &index1_data () const;
            index_array_type &index1_data ();
            const index_array_type &index2_data () const;
            index_array_type &index2_data ();
            const value_array_type &value_data () const;
            value_array_type &value_data ();

            // Resizing
            void resize (size_type size1, size_type size2, size_type non_zeros = 0);

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

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

            // Swapping
            void swap (compressed_matrix &m);
            friend void swap (compressed_matrix &m1, compressed_matrix &m2);

            // Element insertion and erasure
            void insert (size_type i, size_type j, const_reference t);
            void erase (size_type i, size_type j);
            void clear ();

            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 find1 (int rank, size_type i, size_type j) const;
            iterator1 find1 (int rank, size_type i, size_type j);
            const_iterator2 find2 (int rank, size_type i, size_type j) const;
            iterator2 find2 (int rank, size_type i, size_type j);
            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<compressed_matrix>,
                public bidirectional_iterator_base<const_iterator1, value_type> {
            public:
                typedef sparse_bidirectional_iterator_tag iterator_category;
                typedef typename compressed_matrix::difference_type difference_type;
                typedef typename compressed_matrix::value_type value_type;
                typedef typename compressed_matrix::const_reference reference;
                typedef typename compressed_matrix::const_pointer pointer;
                typedef const_iterator2 dual_iterator_type;
                typedef const_reverse_iterator2 dual_reverse_iterator_type;
                typedef typename functor_type::functor1_type functor1_type;

                // Construction and destruction
                const_iterator1 ();
                const_iterator1 (const compressed_matrix &m, int rank, size_type i, size_type j, const const_iterator_type &it);
                const_iterator1 (const iterator1 &it);

                // Arithmetic
                const_iterator1 &operator ++ ();
                const_iterator1 &operator -- ();

                // 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;
            };

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

            class iterator1:
                public container_reference<compressed_matrix>,
                public bidirectional_iterator_base<iterator1, value_type> {
            public:
                typedef sparse_bidirectional_iterator_tag iterator_category;
                typedef typename compressed_matrix::difference_type difference_type;
                typedef typename compressed_matrix::value_type value_type;
                typedef typename compressed_matrix::reference reference;
                typedef typename compressed_matrix::pointer pointer;
                typedef iterator2 dual_iterator_type;
                typedef reverse_iterator2 dual_reverse_iterator_type;
                typedef typename functor_type::functor1_type functor1_type;

                // Construction and destruction
                iterator1 ();
                iterator1 (compressed_matrix &m, int rank, size_type i, size_type j, const iterator_type &it);

                // Arithmetic
                iterator1 &operator ++ ();
                iterator1 &operator -- ();

                // 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;
            };

            iterator1 begin1 ();
            iterator1 end1 ();

            class const_iterator2:
                public container_const_reference<compressed_matrix>,
                public bidirectional_iterator_base<const_iterator2, value_type> {
            public:
                typedef sparse_bidirectional_iterator_tag iterator_category;
                typedef typename compressed_matrix::difference_type difference_type;
                typedef typename compressed_matrix::value_type value_type;
                typedef typename compressed_matrix::const_reference reference;
                typedef typename compressed_matrix::const_pointer pointer;
                typedef const_iterator1 dual_iterator_type;
                typedef const_reverse_iterator1 dual_reverse_iterator_type;
                typedef typename functor_type::functor2_type functor2_type;

                // Construction and destruction
                const_iterator2 ();
                const_iterator2 (const compressed_matrix &m, int rank, size_type i, size_type j, const const_iterator_type &it);
                const_iterator2 (const iterator2 &it);

                // Arithmetic
                const_iterator2 &operator ++ ();
                const_iterator2 &operator -- ();

                // 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 &operato
    r = (const const_iterator2 &it);

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

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

            class iterator2:
                public container_reference<compressed_matrix>,
                public bidirectional_iterator_base<iterator2, value_type> {
            public:
                typedef sparse_bidirectional_iterator_tag iterator_category;
                typedef typename compressed_matrix::difference_type difference_type;
                typedef typename compressed_matrix::value_type value_type;
                typedef typename compressed_matrix::reference reference;
                typedef typename compressed_matrix::pointer pointer;
                typedef iterator1 dual_iterator_type;
                typedef reverse_iterator1 dual_reverse_iterator_type;
                typedef typename functor_type::functor2_type functor2_type;

                // Construction and destruction
                iterator2 ();
                iterator2 (compressed_matrix &m, int rank, size_type i, size_type j, const iterator_type &it);

                // Arithmetic
                iterator2 &operator ++ ();
                iterator2 &operator -- ();

                // 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;
            };

            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 ();
        };

## Coordinate Matrix

#### Description

The templated class `coordinate_matrix<T, F, IB, IA, TA>` is the base container adaptor for compressed matrices. For a (m x n )-dimensional sorted coordinate matrix and 0 <= i < m, 0 <= j < n the non-zero elements mi, j are mapped via (i x n + j) for row major orientation or via (i + j x m) for column major orientation to consecutive elements of the index and value containers, i.e. for elements k = m i1,j 1 and k + 1 = mi 2,j2 of the container holds i1 < i2 or (i1 = i 2 and j1 < j 2) with row major orientation or j1 < j2 or (j1 = j2 and i1 < i2) with column major orientation.

#### Example

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

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

#### Definition

Defined in the header matrix_sparse.hpp.

#### Template parameters

Parameter | Description | Default
--- | --- | ---
`T` | The type of object stored in the coordinate matrix. |
`F` | Functor describing the storage organization. [1] | `row_major`
`IB` | The index base of the coordinate vector. [2] | `0`
`IA` | The type of the adapted array for indices. [3] | `unbounded_array<std::size_t>`
`TA` | The type of the adapted array for values. [3] | `unbounded_array<T>`

#### Model of

Matrix .

#### Type requirements

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

#### Public base classes

`matrix_expression<coordinate_matrix<T, F, IB, IA, TA> >`

#### Members

Member | Description
--- | ---
`coordinate_matrix ()` | Allocates a `coordinate_matrix` that holds at most zero rows of zero elements.
`coordinate_matrix (size_type size1, size_type2, size_type non_zeros)` | Allocates a `coordinate_matrix` that holds at most `size1` rows of `size2` elements.
`coordinate_matrix (const coordinate_matrix &m)` | The copy constructor.
`template<class AE> coordinate_matrix (size_type non_zeros, const matrix_expression<AE> &ae)` | The extended copy constructor.
`void resize (size_type size1, size_type size2, size_type non_zeros)` | Reallocates a `coordinate_matrix` to hold at most `size1` rows of `size2` elements. The content of the `coordinate_matrix` is preserved.
`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.
`coordinate_matrix &operator = (const coordinate_matrix &m)` | The assignment operator.
`coordinate_matrix &assign_temporary (coordinate_matrix &m)` | Assigns a temporary. May change the coordinate matrix `m`.
`template<class AE> coordinate_matrix &operator = (const matrix_expression<AE> &ae)` | The extended assignment operator.
`template<class AE> coordinate_matrix &assign (const matrix_expression<AE> &ae)` | Assigns a matrix expression to the coordinate matrix. Left and right hand side of the assignment should be independent.
`template<class AE> coordinate_matrix &operator += (const matrix_expression<AE> &ae)` | A computed assignment operator. Adds the matrix expression to the coordinate matrix.
`template<class AE> coordinate_matrix &plus_assign (const matrix_expression<AE> &ae)` | Adds a matrix expression to the coordinate matrix. Left and right hand side of the assignment should be independent.
`template<class AE> coordinate_matrix &operator -= (const matrix_expression<AE> &ae)` | A computed assignment operator. Subtracts the matrix expression from the coordinate matrix.
`template<class AE> coordinate_matrix &minus_assign (const matrix_expression<AE> &ae)` | Subtracts a matrix expression from the coordinate matrix. Left and right hand side of the assignment should be independent.
`template<class AT> coordinate_matrix &operator *= (const AT &at)` | A computed assignment operator. Multiplies the coordinate matrix with a scalar.
`template<class AT> coordinate_matrix &operator /= (const AT &at)` | A computed assignment operator. Divides the coordinate matrix through a scalar.
`void swap (coordinate_matrix &m)` | Swaps the contents of the coordinate matrices.
`void insert (size_type i, size_type j, const_reference t)` | Inserts the value `t` at the `j`-th element of the `i`-th row.
`void erase (size_type i, size_type j)` | Erases the value at the `j`-th element of the `i`-th row.
`void clear ()` | Clears the coordinate matrix.
`const_iterator1 begin1 () const` | Returns a `const_iterator1` pointing to the beginning of the `coordinate_matrix`.
`const_iterator1 end1 () const` | Returns a `const_iterator1` pointing to the end of the `coordinate_matrix`.
`iterator1 begin1 ()` | Returns a `iterator1` pointing to the beginning of the `coordinate_matrix`.
`iterator1 end1 ()` | Returns a `iterator1` pointing to the end of the `coordinate_matrix`.
`const_iterator2 begin2 () const` | Returns a `const_iterator2` pointing to the beginning of the `coordinate_matrix`.
`const_iterator2 end2 () const` | Returns a `const_iterator2` pointing to the end of the `coordinate_matrix`.
`iterator2 begin2 ()` | Returns a `iterator2` pointing to the beginning of the `coordinate_matrix`.
`iterator2 end2 ()` | Returns a `iterator2` pointing to the end of the `coordinate_matrix`.
`const_reverse_iterator1 rbegin1 () const` | Returns a `const_reverse_iterator1` pointing to the beginning of the reversed `coordinate_matrix`.
`const_reverse_iterator1 rend1 () const` | Returns a `const_reverse_iterator1` pointing to the end of the reversed `coordinate_matrix`.
`reverse_iterator1 rbegin1 ()` | Returns a `reverse_iterator1` pointing to the beginning of the reversed `coordinate_matrix`.
`reverse_iterator1 rend1 ()` | Returns a `reverse_iterator1` pointing to the end of the reversed `coordinate_matrix`.
`const_reverse_iterator2 rbegin2 () const` | Returns a `const_reverse_iterator2` pointing to the beginning of the reversed `coordinate_matrix`.
`const_reverse_iterator2 rend2 () const` | Returns a `const_reverse_iterator2` pointing to the end of the reversed `coordinate_matrix`.
`reverse_iterator2 rbegin2 ()` | Returns a `reverse_iterator2` pointing to the beginning of the reversed `coordinate_matrix`.
`reverse_iterator2 rend2 ()` | Returns a `reverse_iterator2` pointing to the end of the reversed `coordinate_matrix`.

#### Notes

[1] Supported parameters for the storage organization are `row_major` and `column_major`.

[2] Supported parameters for the index base are `0` and `1` at least.

[3] Supported parameters for the adapted array are `unbounded_array<>` , `bounded_array<>` and `std::vector<>` .

#### Interface

        // Array based sparse matrix class
        template<class T, class F, std::size_t IB, class IA, class TA>
        class coordinate_matrix:
            public matrix_expression<coordinate_matrix<T, F, IB, IA, TA> > {
        public:
            typedef std::size_t size_type;
            typedef std::ptrdiff_t difference_type;
            typedef T value_type;
            typedef const T &const_reference;
            typedef T &reference;
            typedef const T *const_pointer;
            typedef T *pointer;
            typedef F functor_type;
            typedef IA index_array_type;
            typedef TA value_array_type;
            typedef const coordinate_matrix<T, F, IB, IA, TA> const_self_type;
            typedef coordinate_matrix<T, F, IB, IA, TA> self_type;
            typedef const matrix_const_reference<const_self_type> const_closure_type;
            typedef matrix_reference<self_type> closure_type;
            typedef typename A::const_iterator const_iterator_type;
            typedef typename A::iterator iterator_type;
            typedef sparse_tag storage_category;
            typedef typename F::orientation_category orientation_category;

            // Construction and destruction
            coordinate_matrix ();
            coordinate_matrix (size_type size1, size_type size2, size_type non_zeros = 0);
            coordinate_matrix (const coordinate_matrix &m);
            template<class AE>
            coordinate_matrix (const matrix_expression<AE> &ae, size_type non_zeros = 0);

            // Accessors
            size_type size1 () const;
            size_type size2 () const;
            size_type non_zeros () const;
            static size_type index_base ();
            const index_array_type &index1_data () const;
            index_array_type &index1_data ();
            const index_array_type &index2_data () const;
            index_array_type &index2_data ();
            const value_array_type &value_data () const;
            value_array_type &value_data ();

            // Resizing
            void resize (size_type size1, size_type size2, size_type non_zeros = 0);

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

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

            // Swapping
            void swap (coordinate_matrix &m);
            friend void swap (coordinate_matrix &m1, coordinate_matrix &m2);

            // Element insertion and erasure
            void insert (size_type i, size_type j, const_reference t);
            void erase (size_type i, size_type j);
            void clear ();

            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 find1 (int rank, size_type i, size_type j) const;
            iterator1 find1 (int rank, size_type i, size_type j);
            const_iterator2 find2 (int rank, size_type i, size_type j) const;
            iterator2 find2 (int rank, size_type i, size_type j);
            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<coordinate_matrix>,
                public bidirectional_iterator_base<const_iterator1, value_type> {
            public:
                typedef sparse_bidirectional_iterator_tag iterator_category;
                typedef typename coordinate_matrix::difference_type difference_type;
                typedef typename coordinate_matrix::value_type value_type;
                typedef typename coordinate_matrix::const_reference reference;
                typedef typename coordinate_matrix::const_pointer pointer;
                typedef const_iterator2 dual_iterator_type;
                typedef const_reverse_iterator2 dual_reverse_iterator_type;
                typedef typename functor_type::functor1_type functor1_type;

                // Construction and destruction
                const_iterator1 ();
                const_iterator1 (const coordinate_matrix &m, int rank, size_type i, size_type j, const const_iterator_type &it);
                const_iterator1 (const iterator1 &it);

                // Arithmetic
                const_iterator1 &operator ++ ();
                const_iterator1 &operator -- ();

                // 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;
            };

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

            class iterator1:
                public container_reference<coordinate_matrix>,
                public bidirectional_iterator_base<iterator1, value_type> {
            public:
                typedef sparse_bidirectional_iterator_tag iterator_category;
                typedef typename coordinate_matrix::difference_type difference_type;
                typedef typename coordinate_matrix::value_type value_type;
                typedef typename coordinate_matrix::reference reference;
                typedef typename coordinate_matrix::pointer pointer;
                typedef iterator2 dual_iterator_type;
                typedef reverse_iterator2 dual_reverse_iterator_type;
                typedef typename functor_type::functor1_type functor1_type;

                // Construction and destruction
                iterator1 ();
                iterator1 (coordinate_matrix &m, int rank, size_type i, size_type j, const iterator_type &it);

                // Arithmetic
                iterator1 &operator ++ ();
                iterator1 &operator -- ();

                // 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;
            };

            iterator1 begin1 ();<
    br>        iterator1 end1 ();

            class const_iterator2:
                public container_const_reference<coordinate_matrix>,
                public bidirectional_iterator_base<const_iterator2, value_type> {
            public:
                typedef sparse_bidirectional_iterator_tag iterator_category;
                typedef typename coordinate_matrix::difference_type difference_type;
                typedef typename coordinate_matrix::value_type value_type;
                typedef typename coordinate_matrix::const_reference reference;
                typedef typename coordinate_matrix::const_pointer pointer;
                typedef const_iterator1 dual_iterator_type;
                typedef const_reverse_iterator1 dual_reverse_iterator_type;
                typedef typename functor_type::functor2_type functor2_type;

                // Construction and destruction
                const_iterator2 ();
                const_iterator2 (const coordinate_matrix &m, int rank, size_type i, size_type j, const const_iterator_type &it);
                const_iterator2 (const iterator2 &it);

                // Arithmetic
                const_iterator2 &operator ++ ();
                const_iterator2 &operator -- ();

                // 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;
            };

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

            class iterator2:
                public container_reference<coordinate_matrix>,
                public bidirectional_iterator_base<iterator2, value_type> {
            public:
                typedef sparse_bidirectional_iterator_tag iterator_category;
                typedef typename coordinate_matrix::difference_type difference_type;
                typedef typename coordinate_matrix::value_type value_type;
                typedef typename coordinate_matrix::reference reference;
                typedef typename coordinate_matrix::pointer pointer;
                typedef iterator1 dual_iterator_type;
                typedef reverse_iterator1 dual_reverse_iterator_type;
                typedef typename functor_type::functor2_type functor2_type;

                // Construction and destruction
                iterator2 ();
                iterator2 (coordinate_matrix &m, int rank, size_type i, size_type j, const iterator_type &it);

                // Arithmetic
                iterator2 &operator ++ ();
                iterator2 &operator -- ();

                // 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;
            };

            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 ();
        };

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