c++boost.gif Matrix Expressions

Matrix Expression

Description

The templated class matrix_expression<E> forms the base for all static derived matrix expression classes including class matrix itself.

Definition

Defined in the header matrix_expression.hpp.

Template parameters

Parameter Description Default
E The type of the matrix expression.  

Model of

None.

Type requirements

None.

Public base classes

None.

Members

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

Interface

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

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

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

Matrix References

Constant Reference

Description

The templated class matrix_const_reference<E> contains a constant reference to a matrix expression.

Definition

Defined in the header matrix_expression.hpp.

Template parameters

Parameter Description Default
E The type of the matrix expression.  

Model of

Matrix Expression .

Type requirements

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

Public base classes

matrix_expression<matrix_const_reference<E> >

Members

Member Description
matrix_const_reference (const expression_type &e) Constructs a constant reference of the expression.
size_type size1 () const Returns the number of rows.
size_type size2 () const Returns the number of columns.
const_reference operator () (size_type i, size_type j) const Returns the value of the j-th element in the i-th row.
const_iterator1 begin1 () const Returns a const_iterator1 pointing to the beginning of the expression.
const_iterator1 end1 () const Returns a const_iterator1 pointing to the end of the expression.
const_iterator2 begin2 () const Returns a const_iterator2 pointing to the beginning of the expression.
const_iterator2 end2 () const Returns a const_iterator2 pointing to the end of the expression.
const_reverse_iterator1 rbegin1 () const Returns a const_reverse_iterator1 pointing to the beginning of the reversed expression.
const_reverse_iterator1 rend1 () const Returns a const_reverse_iterator1 pointing to the end of the reversed expression.
const_reverse_iterator2 rbegin2 () const Returns a const_reverse_iterator2 pointing to the beginning of the reversed expression.
const_reverse_iterator2 rend2 () const Returns a const_reverse_iterator2 pointing to the end of the reversed expression.

Interface

    template<class E>
class matrix_const_reference:
public matrix_expression<matrix_const_reference<E> > {
public:
typedef E expression_type;
typedef typename E::size_type size_type;
typedef typename E::difference_type difference_type;
typedef typename E::value_type value_type;
typedef typename E::const_reference const_reference;
typedef const_reference reference;
typedef typename E::const_pointer const_pointer;
typedef const_pointer pointer;
typedef typename E::orientation_category orientation_category;
typedef typename E::const_iterator1 const_iterator1_type;
typedef typename E::const_iterator2 const_iterator2_type;
typedef unknown_storage_tag storage_category;

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

// Accessors
size_type size1 () const;
size_type size2 () const;
const expression_type &expression () const;

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

typedef const_iterator1_type const_iterator1;
typedef const_iterator1 iterator1;
typedef const_iterator2_type const_iterator2;
typedef const_iterator2 iterator2;

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

// Iterators are the iterators of the referenced expression.

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

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

// Reverse iterators

typedef reverse_iterator_base1<const_iterator1> const_reverse_iterator1;

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

typedef reverse_iterator_base2<const_iterator2> const_reverse_iterator2;

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

Reference

Description

The templated class matrix_reference<E> contains a reference to a matrix expression.

Definition

Defined in the header matrix_expression.hpp.

Template parameters

Parameter Description Default
E The type of the matrix expression.  

Model of

Matrix Expression .

Type requirements

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

Public base classes

matrix_expression<matrix_reference<E> >

Members

Member Description
matrix_reference (expression_type &e) Constructs a constant reference of the expression.
void resize (size_type size1, size2) Resizes the expression to hold at most size1 rows of size2 elements.
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.
const_iterator1 begin1 () const Returns a const_iterator1 pointing to the beginning of the expression.
const_iterator1 end1 () const Returns a const_iterator1 pointing to the end of the expression.
iterator1 begin1 () Returns a iterator1 pointing to the beginning of the expression.
iterator1 end1 () Returns a iterator1 pointing to the end of the expression.
const_iterator2 begin2 () const Returns a const_iterator2 pointing to the beginning of the expression.
const_iterator2 end2 () const Returns a const_iterator2 pointing to the end of the expression.
iterator2 begin2 () Returns a iterator2 pointing to the beginning of the expression.
iterator2 end2 () Returns a iterator2 pointing to the end of the expression.
const_reverse_iterator1 rbegin1 () const Returns a const_reverse_iterator1 pointing to the beginning of the reversed expression.
const_reverse_iterator1 rend1 () const Returns a const_reverse_iterator1 pointing to the end of the reversed expression.
reverse_iterator1 rbegin1 () Returns a reverse_iterator1 pointing to the beginning of the reversed expression.
reverse_iterator1 rend1 () Returns a reverse_iterator1 pointing to the end of the reversed expression.
const_reverse_iterator2 rbegin2 () const Returns a const_reverse_iterator2 pointing to the beginning of the reversed expression.
const_reverse_iterator2 rend2 () const Returns a const_reverse_iterator2 pointing to the end of the reversed expression.
reverse_iterator2 rbegin2 () Returns a reverse_iterator2 pointing to the beginning of the reversed expression.
reverse_iterator2 rend2 () Returns a reverse_iterator2 pointing to the end of the reversed expression.

Interface

    template<class E>
class matrix_reference:
public matrix_expression<matrix_reference<E> > {
public:
typedef E expression_type;
typedef typename E::size_type size_type;
typedef typename E::difference_type difference_type;
typedef typename E::value_type value_type;
typedef typename E::const_reference const_reference;
typedef typename E::reference reference;
typedef typename E::const_pointer const_pointer;
typedef typename E::pointer pointer;
typedef typename E::orientation_category orientation_category;
typedef typename E::const_iterator1 const_iterator1_type;
typedef typename E::iterator1 iterator1_type;
typedef typename E::const_iterator2 const_iterator2_type;
typedef typename E::iterator2 iterator2_type;
typedef unknown_storage_tag storage_category;

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

// Accessors
size_type size1 () const;
size_type size2 () const;
const expression_type &expression () const;
expression_type &expression ();

// Resizing
void resize (size_type size1, size_type size2);

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

typedef const_iterator1_type const_iterator1;
typedef iterator1_type iterator1;
typedef const_iterator2_type const_iterator2;
typedef iterator2_type 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 are the iterators of the referenced expression.

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

iterator1 begin1 ();
iterator1 end1 ();

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

iterator2 begin2 ();
iterator2 end2 ();

// Reverse iterators

typedef reverse_iterator_base1<const_iterator1> const_reverse_iterator1;

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

typedef reverse_iterator_base1<iterator1> reverse_iterator1;

reverse_iterator1 rbegin1 ();
reverse_iterator1 rend1 ();

typedef reverse_iterator_base2<const_iterator2> const_reverse_iterator2;

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

typedef reverse_iterator_base2<iterator2> reverse_iterator2;

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

Matrix Operations

Unary Operation Description

Description

The templated classes matrix_unary1<E, F> and matrix_unary2<E, F> describe unary matrix operations.

Definition

Defined in the header matrix_expression.hpp.

Template parameters

Parameter Description Default
E The type of the matrix expression.  
F The type of the operation.  

Model of

Matrix Expression .

Type requirements

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

Public base classes

matrix_expression<matrix_unary1<E, F> > and matrix_expression<matrix_unary2<E, F> > resp.

Members

Member Description
matrix_unary1 (const expression_type &e) Constructs a description of the expression.
matrix_unary2 (const expression_type &e) Constructs a description of the expression.
size_type size1 () const Returns the number of rows.
size_type size2 () const Returns the number of columns.
const_reference operator () (size_type i, size_type j) const Returns the value of the j-th element in the i-th row.
const_iterator1 begin1 () const Returns a const_iterator1 pointing to the beginning of the expression.
const_iterator1 end1 () const Returns a const_iterator1 pointing to the end of the expression.
const_iterator2 begin2 () const Returns a const_iterator2 pointing to the beginning of the expression.
const_iterator2 end2 () const Returns a const_iterator2 pointing to the end of the expression.
const_reverse_iterator1 rbegin1 () const Returns a const_reverse_iterator1 pointing to the beginning of the reversed expression.
const_reverse_iterator1 rend1 () const Returns a const_reverse_iterator1 pointing to the end of the reversed expression.
const_reverse_iterator2 rbegin2 () const Returns a const_reverse_iterator2 pointing to the beginning of the reversed expression.
const_reverse_iterator2 rend2 () const Returns a const_reverse_iterator2 pointing to the end of the reversed expression.

Interface

    template<class E, class F>
class matrix_unary1:
public matrix_expression<matrix_unary1<E, F> > {
public:
typedef E expression_type;
typedef F functor_type;
typedef typename E::size_type size_type;
typedef typename E::difference_type difference_type;
typedef typename F::result_type value_type;
typedef value_type const_reference;
typedef const_reference reference;
typedef const value_type *const_pointer;
typedef const_pointer pointer;
typedef const matrix_unary1<E, F> const_closure_type;
typedef typename E::orientation_category orientation_category;
typedef typename E::const_iterator1 const_iterator1_type;
typedef typename E::const_iterator2 const_iterator2_type;
typedef unknown_storage_tag storage_category;

// Construction and destruction
matrix_unary1 ();
matrix_unary1 (const expression_type &e);

// Accessors
size_type size1 () const;
size_type size2 () const;
const expression_type &expression () const;

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

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

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

// Iterators enhance the iterators of the referenced expression
// with the unary functor.

class const_iterator1:
public container_const_reference<matrix_unary1>,
public random_access_iterator_base<const_iterator1, value_type> {
public:
typedef typename E::const_iterator1::iterator_category iterator_category;
typedef typename matrix_unary1::difference_type difference_type;
typedef typename matrix_unary1::value_type value_type;
typedef typename matrix_unary1::const_reference reference;
typedef typename matrix_unary1::const_pointer pointer;
typedef const_iterator2 dual_iterator_type;
typedef const_reverse_iterator2 dual_reverse_iterator_type;

// Construction and destruction
const_iterator1 ();
const_iterator1 (const matrix_unary1 &mu, const const_iterator1_type &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 const_iterator2:
public container_const_reference<matrix_unary1>,
public random_access_iterator_base<const_iterator2, value_type> {
public:
typedef typename E::const_iterator2::iterator_category iterator_category;
typedef typename matrix_unary1::difference_type difference_type;
typedef typename matrix_unary1::value_type value_type;
typedef typename matrix_unary1::const_reference reference;
typedef typename matrix_unary1::const_pointer pointer;
typedef const_iterator1 dual_iterator_type;
typedef const_reverse_iterator1 dual_reverse_iterator_type;

// Construction and destruction
const_iterator2 ();
const_iterator2 (const matrix_unary1 &mu, const const_iterator2_type &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;

// Reverse iterators

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

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

template<class E, class F>
class matrix_unary2:
public matrix_expression<matrix_unary2<E, F> > {
public:
typedef E expression_type;
typedef F functor_type;
typedef typename E::size_type size_type;
typedef typename E::difference_type difference_type;
typedef typename F::result_type value_type;
typedef value_type const_reference;
typedef const_reference reference;
typedef const value_type *const_pointer;
typedef const_pointer pointer;
typedef const matrix_unary2<E, F> const_closure_type;
typedef typename E::orientation_category orientation_category;
typedef typename E::const_iterator1 const_iterator2_type;
typedef typename E::const_iterator2 const_iterator1_type;
typedef unknown_storage_tag storage_category;

// Construction and destruction
matrix_unary2 ();
matrix_unary2 (const expression_type &e);

// Accessors
size_type size1 () const;
size_type size2 () const;
const expression_type &expression () const;

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

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

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

// Iterators enhance the iterators of the referenced expression
// with the unary functor.

class const_iterator1:
public container_const_reference<matrix_unary2>,
public random_access_iterator_base<const_iterator1, value_type> {
public:
typedef typename E::const_iterator2::iterator_category iterator_category;
typedef typename matrix_unary2::difference_type difference_type;
typedef typename matrix_unary2::value_type value_type;
typedef typename matrix_unary2::const_reference reference;
typedef typename matrix_unary2::const_pointer pointer;
typedef const_iterator2 dual_iterator_type;
typedef const_reverse_iterator2 dual_reverse_iterator_type;

// Construction and destruction
const_iterator1 ();
const_iterator1 (const matrix_unary2 &mu, const const_iterator1_type &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 const_iterator2:
public container_const_reference<matrix_unary2>,
public random_access_iterator_base<const_iterator2, value_type> {
public:
typedef typename E::const_iterator1::iterator_catego ry iterator_category;
typedef typename matrix_unary2::difference_type difference_type;
typedef typename matrix_unary2::value_type value_type;
typedef typename matrix_unary2::const_reference reference;
typedef typename matrix_unary2::const_pointer pointer;
typedef const_iterator1 dual_iterator_type;
typedef const_reverse_iterator1 dual_reverse_iterator_type;

// Construction and destruction
const_iterator2 ();
const_iterator2 (const matrix_unary2 &mu, const const_iterator2_type &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;

// Reverse iterators

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

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

Unary Operations

Prototypes

    template<class E, class F>
struct matrix_unary1_traits {
typedef matrix_unary1<typename E::const_closure_type, F> expression_type;
typedef expression_type result_type;
};

// (- m) [i] [j] = - m [i] [j]
template<class E>
typename matrix_unary1_traits<E, scalar_negate<typename E::value_type> >::result_type
operator - (const matrix_expression<E> &e);

// (conj m) [i] [j] = conj (m [i] [j])
template<class E>
typename matrix_unary1_traits<E, scalar_conj<typename E::value_type> >::result_type
conj (const matrix_expression<E> &e);

// (real m) [i] [j] = real (m [i] [j])
template<class E>
typename matrix_unary1_traits<E, scalar_real<typename E::value_type> >::result_type
real (const matrix_expression<E> &e);

// (imag m) [i] [j] = imag (m [i] [j])
template<class E>
typename matrix_unary1_traits<E, scalar_imag<typename E::value_type> >::result_type
imag (const matrix_expression<E> &e);

template<class E, class F>
struct matrix_unary2_traits {
typedef matrix_unary2<typename E::const_closure_type, F> expression_type;
typedef expression_type result_type;
};

// (trans m) [i] [j] = m [j] [i]
template<class E>
typename matrix_unary2_traits<E, scalar_identity<typename E::value_type> >::result_type
trans (const matrix_expression<E> &e);

// (herm m) [i] [j] = conj (m [j] [i])
template<class E>
typename matrix_unary2_traits<E, scalar_conj<typename E::value_type> >::result_type
herm (const matrix_expression<E> &e);

Description

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

Definition

Defined in the header matrix_expression.hpp.

Type requirements

  • E is a model of Matrix Expression .
  • Preconditions

    None.

    Complexity

    Quadratic depending from the size of the matrix expression.

    Examples

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

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

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

    Binary Operation Description

    Description

    The templated class matrix_binary<E1, E2, F> describes a binary matrix operation.

    Definition

    Defined in the header matrix_expression.hpp.

    Template parameters

    Parameter Description Default
    E1 The type of the first matrix expression.  
    E2 The type of the second matrix expression.  
    F The type of the operation.  

    Model of

    Matrix Expression .

    Type requirements

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

    Public base classes

    matrix_expression<matrix_binary<E1, E2, F> >.

    Members

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

    Interface

        template<class E1, class E2, class F>
    class matrix_binary:
    public matrix_expression<matrix_binary<E1, E2, F> > {
    public:
    typedef E1 expression1_type;
    typedef E2 expression2_type;
    typedef F functor_type;
    typedef typename promote_traits<typename E1::size_type, typename E2::size_type>::promote_type size_type;
    typedef typename promote_traits<typename E1::difference_type, typename E2::difference_type>::promote_type difference_type;
    typedef typename F::result_type value_type;
    typedef value_type const_reference;
    typedef const_reference reference;
    typedef const value_type *const_pointer;
    typedef const_pointer pointer;
    typedef const matrix_binary<E1, E2, F> const_closure_type;
    typedef unknown_orientation_tag orientation_category;
    typedef typename E1::const_iterator1 const_iterator11_type;
    typedef typename E1::const_iterator2 const_iterator12_type;
    typedef typename E2::const_iterator1 const_iterator21_type;
    typedef typename E2::const_iterator2 const_iterator22_type;
    typedef unknown_storage_tag storage_category;

    // Construction and destruction
    matrix_binary ();
    matrix_binary (const E1 &e1, const E2 &e2);

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

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

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

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

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

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

    // Construction and destruction
    const_iterator1 ();
    const_iterator1 (const matrix_binary &mb, size_type i, size_type j,
    const const_iterator11_type &it1, const const_iterator11_type &it1_end,
    const const_iterator21_type &it2, const const_iterator21_type &it2_end);

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

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

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

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

    // Dereference
    reference operator * () const;

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

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

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

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

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

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

    // Construction and destruction
    const_iterator2 ();
    const_iterator2 (const matrix_binary &mb, size_type i, size_type j,
    const const_iterator12_type &it1, const const_iterator12_type &it1_end,
    const const_iterator22_type &it2, const const_iterator22_type &it2_end);

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

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

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

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

    // Dereference
    reference operator * () const;

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

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

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

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

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

    // Reverse iterators

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

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

    Binary Operations

    Prototypes

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

    // (m1 + m2) [i] [j] = m1 [i] [j] + m2 [i] [j]
    template<class E1, class E2>
    typename matrix_binary_traits<E1, E2, scalar_plus<typename E1::value_type,
    typename E2::value_type> >::result_type
    operator + (const matrix_expression<E1> &e1,
    const matrix_expression<E2> &e2);

    // (m1 - m2) [i] [j] = m1 [i] [j] - m2 [i] [j]
    template<class E1, class E2>
    typename matrix_binary_traits<E1, E2, scalar_minus<typename E1::value_type,
    typename E2::value_type> >::result_type
    operator - (const matrix_expression<E1> &e1,
    const matrix_expression<E2> &e2);

    Description

    operator + computes the sum of two matrix expressions. operator - computes the difference of two matrix expressions.

    Definition

    Defined in the header matrix_expression.hpp.

    Type requirements

  • E1 is a model of Matrix Expression .
  • E2 is a model of Matrix Expression .
  • Preconditions

  • e1 ().size1 () == e2 ().size1 ()
  • e1 ().size2 () == e2 ().size2 ()
  • Complexity

    Quadratic depending from the size of the matrix expressions.

    Examples

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

    int main () {
    using namespace boost::numeric::ublas;
    matrix<double> m1 (3, 3), m2 (3, 3);
    for (unsigned i = 0; i < std::min (m1.size1 (), m2.size1 ()); ++ i)
    for (unsigned j = 0; j < std::min (m1.size2 (), m2.size2 ()); ++ j)
    m1 (i, j) = m2 (i, j) = 3 * i + j;

    std::cout << m1 + m2 << std::endl;
    std::cout << m1 - m2 << std::endl;
    }

    Scalar Matrix Operation Description

    Description

    The templated classes matrix_binary_scalar1<E1, E2, F> and matrix_binary_scalar2<E1, E2, F> describe binary operations between a scalar and a matrix.

    Definition

    Defined in the header matrix_expression.hpp.

    Template parameters

    Parameter Description Default
    E1/E2 The type of the scalar expression.  
    E2/E1 The type of the matrix expression.  
    F The type of the operation.  

    Model of

    Matrix Expression .

    Type requirements

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

    Public base classes

    matrix_expression<matrix_binary_scalar1<E1, E2, F> > and matrix_expression<matrix_binary_scalar2<E1, E2, F> > resp.

    Members

    Member Description
    matrix_binary_scalar1 (const expression1_type &e1, const expression2_type &e2) Constructs a description of the expression.
    matrix_binary_scalar1 (const expression1_type &e1, const expression2_type &e2) Constructs a description of the expression.
    size_type size1 () const Returns the number of rows.
    size_type size2 () const Returns the number of columns.
    const_reference operator () (size_type i, size_type j) const Returns the value of the j-th element in the i-th row.
    const_iterator1 begin1 () const Returns a const_iterator1 pointing to the beginning of the expression.
    const_iterator1 end1 () const Returns a const_iterator1 pointing to the end of the expression.
    const_iterator2 begin2 () const Returns a const_iterator2 pointing to the beginning of the expression.
    const_iterator2 end2 () const Returns a const_iterator2 pointing to the end of the expression.
    const_reverse_iterator1 rbegin1 () const Returns a const_reverse_iterator1 pointing to the beginning of the reversed expression.
    const_reverse_iterator1 rend1 () const Returns a const_reverse_iterator1 pointing to the end of the reversed expression.
    const_reverse_iterator2 rbegin2 () const Returns a const_reverse_iterator2 pointing to the beginning of the reversed expression.
    const_reverse_iterator2 rend2 () const Returns a const_reverse_iterator2 pointing to the end of the reversed expression.

    Interface

        template<class E1, class E2, class F>
    class matrix_binary_scalar1:
    public matrix_expression<matrix_binary_scalar1<E1, E2, F> > {
    public:
    typedef E1 expression1_type;
    typedef E2 expression2_type;
    typedef F functor_type;
    typedef typename E2::size_type size_type;
    typedef typename E2::difference_type difference_type;
    typedef typename F::result_type value_type;
    typedef value_type const_reference;
    typedef const_reference reference;
    typedef const value_type *const_pointer;
    typedef const_pointer pointer;
    typedef const matrix_binary_scalar1<E1, E2, F> const_closure_type;
    typedef typename E2::orientation_category orientation_category;
    typedef typename E1::value_type const_iterator1_type;
    typedef typename E2::const_iterator1 const_iterator21_type;
    typedef typename E2::const_iterator2 const_iterator22_type;
    typedef unknown_storage_tag storage_category;

    // Construction and destruction
    matrix_binary_scalar1 ();
    matrix_binary_scalar1 (const expression1_type &e1, const expression2_type &e2);

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

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

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

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

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

    class const_iterator1:
    public container_const_reference<matrix_binary_scalar1>,
    public random_access_iterator_base<const_iterator1, value_type> {
    public:
    typedef typename E2::const_iterator1::iterator_category iterator_category;
    typedef typename matrix_binary_scalar1::difference_type difference_type;
    typedef typename matrix_binary_scalar1::value_type value_type;
    typedef typename matrix_binary_scalar1::const_reference reference;
    typedef typename matrix_binary_scalar1::const_pointer pointer;
    typedef const_iterator2 dual_iterator_type;
    typedef const_reverse_iterator2 dual_reverse_iterator_type;

    // Construction and destruction
    const_iterator1 ();
    const_iterator1 (const matrix_binary_scalar1 &mbs, const const_iterator1_type &it1, const const_iterator21_type &it2);

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

    // Dereference
    reference operator * () const;

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

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

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

    // Comparison
    bool operator == (const const_iterator1 &it) const;
    bool operator <(const const_iterator1 &it) const;
    };
    const_iterator1 begin1 () const; const_iterator1 end1 () const; class const_iterator2: public container_const_reference<matrix_binary_scalar1>, public random_access_iterator_base<const_iterator2, value_type> { public: typedef typename E2::const_iterator2::iterator_category iterator_category; typedef typename matrix_binary_scalar1::difference_type difference_type; typedef typename matrix_binary_scalar1::value_type value_type; typedef typename matrix_binary_scalar1::const_reference reference; typedef typename matrix_binary_scalar1::const_pointer pointer; typedef const_iterator1 dual_iterator_type; typedef const_reverse_iterator1 dual_reverse_iterator_type; // Construction and destruction const_iterator2 (); const_iterator2 (const matrix_binary_scalar1 &mbs, const const_iterator1_type &it1, const const_iterator22_type &it2); // Arithmetic const_iterator2 &operator ++ (); const_iterator2 &operator -- (); const_iterator2 &operator += (difference_type n); const_iterator2 &operator -= (difference_type n); difference_type operator - (const const_iterator2 &it) const; // Dereference reference operator * () const; const_iterator1 begin () const; const_iterator1 end () const; const_reverse_iterator1 rbegin () const; const_reverse_iterator1 rend () const; // Indices size_type index1 () const; size_type index2 () const; // Assignment const_iterator2 &operator = (const const_iterator2 &it); // Comparison bool operator == (const const_iterator2 &it) const; bool operator <(const const_iterator2 &it) const; }; const_iterator2 begin2 () const; const_iterator2 end2 () const; // Reverse iterators const_reverse_iterator1 rbegin1 () const; const_reverse_iterator1 rend1 () const; const_reverse_iterator2 rbegin2 () const; const_reverse_iterator2 rend2 () const; }; template<class E1, class E2, class F> class matrix_binary_scalar2: public matrix_expression<matrix_binary_scalar2<E1, E2, F> > { public: typedef E1 expression1_type; typedef E2 expression2_type; typedef F functor_type; typedef typename E1::size_type size_type; typedef typename E1::difference_type difference_type; typedef typename F::result_type value_type; typedef value_type const_reference; typedef const_reference reference; typedef const value_type *const_pointer; typedef const_pointer pointer; typedef const matrix_binary_scalar2<E1, E2, F> const_closure_type; typedef typename E1::orientation_category orientation_category; typedef typename E1::const_iterator1 const_iterator11_type; typedef typename E1::const_iterator2 const_iterator12_type; typedef typename E2::value_type const_iterator2_type; typedef unknown_storage_tag storage_category; // Construction and destruction matrix_binary_scalar2 (); matrix_binary_scalar2 (const expression1_type &e1, const expression2_type &e2); // Accessors size_type size1 () const; size_type size2 () const; const expression1_type &expression1 () const; const expression2_type &expression2 () const; // Element access const_reference operator () (size_type i, size_type j) const; class const_iterator1; typedef const_iterator1 iterator1; class const_iterator2; typedef const_iterator2 iterator2; typedef reverse_iterator_base1<const_iterator1> const_reverse_iterator1; typedef reverse_iterator_base2<const_iterator2> const_reverse_iterator2; // Element lookup const_iterator1 find_first1 (int rank, size_type i, size_type j) const;
    const_iterator1 find_last1 (int rank, size_type i, size_type j) const;
    const_iterator2 find_first2 (int rank, size_type i, size_type j) const;
    const_iterator2 find_last2 (int rank, size_type i, size_type j) const;

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

    class const_iterator1:
    public container_const_reference<matrix_binary_scalar2>,
    public random_access_iterator_base<const_iterator1, value_type> {
    public:
    typedef typename E1::const_iterator1::iterator_category iterator_category;
    typedef typename matrix_binary_scalar2::difference_type difference_type;
    typedef typename matrix_binary_scalar2::value_type value_type;
    typedef typename matrix_binary_scalar2::const_reference reference;
    typedef typename matrix_binary_scalar2::const_pointer pointer;
    typedef const_iterator2 dual_iterator_type;
    typedef const_reverse_iterator2 dual_reverse_iterator_type;

    // Construction and destruction
    const_iterator1 ();
    const_iterator1 (const matrix_binary_scalar2 &mbs, const const_iterator11_type &it1, const const_iterator2_type &it2);

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

    // Dereference
    reference operator * () const;

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

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

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

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

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

    class const_iterator2:
    public container_const_reference<matrix_binary_scalar2>,
    public random_access_iterator_base<const_iterator2, value_type> {
    public:
    typedef typename E1::const_iterator2::iterator_category iterator_category;
    typedef typename matrix_binary_scalar2::difference_type difference_type;
    typedef typename matrix_binary_scalar2::value_type value_type;
    typedef typename matrix_binary_scalar2::const_reference reference;
    typedef typename matrix_binary_scalar2::const_pointer pointer;
    typedef const_iterator1 dual_iterator_type;
    typedef const_reverse_iterator1 dual_reverse_iterator_type;

    // Construction and destruction
    const_iterator2 ();
    const_iterator2 (const matrix_binary_scalar2 &mbs, const const_iterator12_type &it1, const const_iterator2_type &it2);

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

    // Dereference
    reference operator * () const;

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

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

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

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

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

    // Reverse iterators

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

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

    Scalar Matrix Operations

    Prototypes

        template<class T1, class E2, class F>
    struct matrix_binary_scalar1_traits {
    typedef matrix_binary_scalar1<scalar_const_reference<T1>,
    typename E2::const_closure_type, F> expression_type;
    typedef expression_type result_type;
    };

    // (t * m) [i] [j] = t * m [i] [j]
    template<class T1, class E2>
    typename matrix_binary_scalar1_traits<T1, E2, scalar_multiplies<T1, typename E2::value_type> >::result_type
    operator * (const T1 &e1,
    const matrix_expression<E2> &e2);

    template<class E1, class T2, class F>
    struct matrix_binary_scalar2_traits {
    typedef matrix_binary_scalar2<typename E1::const_closure_type,
    scalar_const_reference<T2>, F> expression_type;
    typedef expression_type result_type;
    };

    // (m * t) [i] [j] = m [i] [j] * t
    template<class E1, class T2>
    typename matrix_binary_scalar2_traits<E1, T2, scalar_multiplies<typename E1::value_type, T2> >::result_type
    operator * (const matrix_expression<E1> &e1,
    const T2 &e2);

    // (m / t) [i] [j] = m [i] [j] / t
    template<class E1, class T2>
    typename matrix_binary_scalar2_traits<E1, T2, scalar_divides<typename E1::value_type, T2> >::result_type
    operator / (const matrix_expression<E1> &e1,
    const T2 &e2);

    Description

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

    Definition

    Defined in the header matrix_expression.hpp.

    Type requirements

  • T1/T2 is a model of Scalar Expression .
  • E2/E1 is a model of Matrix Expression .
  • Preconditions

    None.

    Complexity

    Quadratic depending from the size of the matrix expression.

    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)
    m (i, j) = 3 * i + j;

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

    
        
    

    Matrix Vector Operations

    Binary Operation Description

    Description

    The templated classes matrix_vector_binary1<E1, E2, F> and matrix_vector_binary2<E1, E2, F> describe binary matrix vector operations.

    Definition

    Defined in the header matrix_expression.hpp.

    Template parameters

    Parameter Description Default
    E1 The type of the matrix or vector expression.  
    E2 The type of the vector or matrix expression.  
    F The type of the operation.  

    Model of

    Vector Expression .

    Type requirements

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

    Public base classes

    vector_expression<matrix_vector_binary1<E1, E2, F> > and vector_expression<matrix_vector_binary2<E1, E2, F> > resp.

    Members

    Member Description
    matrix_vector_binary1 (const expression1_type &e1, const expression2_type &e2) Constructs a description of the expression.
    matrix_vector_binary2 (const expression1_type &e1, const expression2_type &e2) Constructs a description of the expression.
    size_type size () const Returns the size of the expression.
    const_reference operator () (size_type i) const Returns the value of the i-th element.
    const_iterator begin () const Returns a const_iterator pointing to the beginning of the expression.
    const_iterator end () const Returns a const_iterator pointing to the end of the expression.
    const_reverse_iterator rbegin () const Returns a const_reverse_iterator pointing to the beginning of the reversed expression.
    const_reverse_iterator rend () const Returns a const_reverse_iterator pointing to the end of the reversed expression.

    Interface

        template<class E1, class E2, class F>
    class matrix_vector_binary1:
    public vector_expression<matrix_vector_binary1<E1, E2, F> > {
    public:
    typedef E1 expression1_type;
    typedef E2 expression2_type;
    typedef F functor_type;
    typedef typename promote_traits<typename E1::size_type, typename E2::size_type>::promote_type size_type;
    typedef typename promote_traits<typename E1::difference_type, typename E2::difference_type>::promote_type difference_type;
    typedef typename F::result_type value_type;
    typedef value_type const_reference;
    typedef const_reference reference;
    typedef const value_type *const_pointer;
    typedef const_pointer pointer;
    typedef const matrix_vector_binary1<E1, E2, F> const_closure_type;
    typedef typename E1::const_iterator1 const_iterator1_type;
    typedef typename E2::const_iterator const_iterator2_type;
    typedef unknown_storage_tag storage_category;

    // Construction and destruction
    matrix_vector_binary1 ();
    matrix_vector_binary1 (const expression1_type &e1, const expression2_type &e2);

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

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

    class const_iterator;
    typedef const_iterator iterator;

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

    // Iterator simply is a pointer.

    class const_iterator:
    public container_const_reference<matrix_vector_binary1>,
    public random_access_iterator_base<const_iterator, value_type> {
    public:
    typedef typename restrict_traits<typename E1::const_iterator1::iterator_category,
    typename E2::const_iterator::iterator_category>::iterator_category iterator_category;
    typedef typename matrix_vector_binary1::difference_type difference_type;
    typedef typename matrix_vector_binary1::value_type value_type;
    typedef typename matrix_vector_binary1::const_reference reference;
    typedef typename matrix_vector_binary1::const_pointer pointer;

    // Construction and destruction
    const_iterator ();
    const_iterator (const matrix_vector_binary1 &mvb, const const_iterator1_type &it1);

    // Dense random access specialization
    value_type dereference (dense_random_access_iterator_tag) const;

    // Packed bidirectional specialization
    value_type dereference (packed_bidirectional_iterator_tag) const;

    // Sparse bidirectional specialization
    value_type dereference (sparse_bidirectional_iterator_tag) const;

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

    // Dereference
    reference operator * () const;

    // Index
    size_type index () const;

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

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

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

    // Reverse iterator

    typedef reverse_iterator_base<const_iterator> const_reverse_iterator;

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

    template<class E1, class E2, class F>
    class matrix_vector_binary2:
    public vector_expression<matrix_vector_binary2<E1, E2, F> > {
    public:
    typedef E1 expression1_type;
    typedef E2 expression2_type;
    typedef F functor_type;
    typedef typename promote_traits<typename E1::size_type, typename E2::size_type>::promote_type size_type;
    typedef typename promote_traits<typename E1::difference_type, typename E2::difference_type>::promote_type difference_type;
    typedef typename F::result_type value_type;
    typedef value_type const_reference;
    typedef const_reference reference;
    typedef const value_type *const_pointer;
    typedef const_pointer pointer;
    typedef const matrix_vector_binary2<E1, E2, F> const_closure_type;
    typedef typename E1::const_iterator const_iterator1_type;
    typedef typename E2::const_iterator2 const_iterator2_type;
    typedef unknown_storage_tag storage_category;

    // Construction and destruction
    matrix_vector_binary2 ();
    matrix_vector_binary2 (const expression1_type &e1, const expression2_type &e2);

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

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

    class const_iterator;
    typedef const_iterator iterator;

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

    // Iterator simply is a pointer.

    class const_iterator:
    public container_const_reference<matrix_vector_binary2>,
    public random_access_iterator_base<const_iterator, value_type> {
    public:
    typedef typename restrict_traits<typename E1::const_iterator::iterator_category,
    typename E2::const_iterator2::iterator_category>::iterator_category iterator_category;
    typedef typename matrix_vector_binary2::difference_type difference_type;
    typedef typename matrix_vector_binary2::value_type value_type;
    typedef typename matrix_vector_binary2::const_reference reference;
    typedef typename matrix_vector_binary2::const_pointer pointer;

    // Construction and destruction
    const_iterator ();
    const_iterator (const matrix_vector_binary2 &mvb, const const_iterator2_type &it2);

    // Dense random access specialization
    value_type dereference (dense_random_access_iterator_tag) const;

    // Packed bidirectional specialization
    value_type dereference (packed_bidirectional_iterator_tag) const;

    // Sparse bidirectional specialization
    value_type dereference (sparse_bidirectional_iterator_tag) const;

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

    // Dereference
    reference operator * () const;

    // Index
    size_type index () const;

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

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

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

    // Reverse iterator

    typedef reverse_iterator_base<const_iterator> const_reverse_iterator;

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

    Binary Operations

    Prototypes

        template<class T1, class E1, class T2, class E2>
    struct matrix_vector_binary1_traits {
    typedef row_major_tag dispatch_category;
    typedef typename promote_traits<T1, T2>::promote_type promote_type;
    typedef matrix_vector_binary1<typename E1::const_closure_type,
    typename E2::const_closure_type,
    matrix_vector_prod1<T1, T2, promote_type> > expression_type;
    typedef expression_type result_type;
    };

    template<class E1, class E2>
    typename matrix_vector_binary1_traits<typename E1::value_type, E1,
    typename E2::value_type, E2>::result_type
    prod (const matrix_expression<E1> &e1,
    const vector_expression<E2> &e2,
    row_major_tag);

    // Dispatcher
    template<class E1, class E2>
    typename matrix_vector_binary1_traits<typename E1::value_type, E1,
    typename E2::value_type, E2>::result_type
    prod (const matrix_expression<E1> &e1,
    const vector_expression<E2> &e2);

    template<class E1, class E2>
    typename matrix_vector_binary1_traits<typename type_traits<typename E1::value_type>::precision_type, E1,
    typename type_traits<typename E2::value_type>::precision_type, E2>::result_type
    prec_prod (const matrix_expression<E1> &e1,
    const vector_expression<E2> &e2,
    row_major_tag);

    // Dispatcher
    template<class E1, class E2>
    typename matrix_vector_binary1_traits<typename type_traits<typename E1::value_type>::precision_type, E1,
    typename type_traits<typename E2::value_type>::precision_type, E2>::result_type
    prec_prod (const matrix_expression<E1> &e1,
    const vector_expression<E2> &e2);

    template<class V, class E1, class E2>
    V
    prod (const matrix_expression<E1> &e1,
    const vector_expression<E2> &e2);

    template<class V, class E1, class E2>
    V
    prec_prod (const matrix_expression<E1> &e1,
    const vector_expression<E2> &e2);

    template<class T1, class E1, class T2, class E2>
    struct matrix_vector_binary2_traits {
    typedef column_major_tag dispatch_category;
    typedef typename promote_traits<T1, T2>::promote_type promote_type;
    typedef matrix_vector_binary2<typename E1::const_closure_type,
    typename E2::const_closure_type,
    matrix_vector_prod2<T1, T2, promote_type> > expression_type;
    typedef expression_type result_type;
    };

    template<class E1, class E2>
    typename matrix_vector_binary2_traits<typename E1::value_type, E1,
    typename E2::value_type, E2>::result_type
    prod (const vector_expression<E1> &e1,
    const matrix_expression<E2> &e2,
    column_major_tag);

    // Dispatcher
    template<class E1, class E2>
    typename matrix_vector_binary2_traits<typename E1::value_type, E1,
    typename E2::value_type, E2>::result_type
    prod (const vector_expression<E1> &e1,
    const matrix_expression<E2> &e2);

    template<class E1, class E2>
    typename matrix_vector_binary2_traits<typename type_traits<typename E1::value_type>::precision_type, E1,
    typename type_traits<typename E2::value_type>::precision_type, E2>::result_type
    prec_prod (const vector_expression<E1> &e1,
    const matrix_expression<E2> &e2,
    column_major_tag);

    // Dispatcher
    template<class E1, class E2>
    typename matrix_vector_binary2_traits<typename type_traits<typename E1::value_type>::precision_type, E1,
    typename type_traits<typename E2::value_type>::precision_type, E2>::result_type
    prec_prod (const vector_expression<E1> &e1,
    const matrix_expression<E2> &e2);

    template<class V, class E1, class E2>
    V
    prod (const vector_expression<E1> &e1,
    const matrix_expression<E2> &e2);

    template<class V, class E1, class E2>
    V
    prec_prod (const vector_expression<E1> &e1,
    const matrix_expression<E2> &e2);

    Description

    prod computes the product of the matrix and the vector expression. prec_prod computes the double precision product of the matrix and the vector expression.

    Definition

    Defined in the header matrix_expression.hpp.

    Type requirements

  • E1 is a model of Matrix Expression  or Vector Expression .
  • E2 is a model of Vector Expression or Matrix Expression .
  • Preconditions

  • e1 ().size2 () == e2 ().size ()
  • e1 ().size () == e2 ().size1 ()
  • Complexity

    Quadratic depending from the size of the matrix expression.

    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);
    vector<double> v (3);
    for (unsigned i = 0; i < std::min (m.size1 (), v.size ()); ++ i) {
    for (unsigned j = 0; j < m.size2 (); ++ j)
    m (i, j) = 3 * i + j;
    v (i) = i;
    }

    std::cout << prod (m, v) << std::endl;
    std::cout << prod (v, m) << std::endl;
    }

    Triangular Solver

    Prototypes

        template<class E1, class E2>
    struct matrix_vector_solve_traits {
    typedef typename promote_traits<typename E1::value_type, typename E2::value_type>::promote_type promote_type;
    typedef vector<promote_type> result_type;
    };

    template<class E1, class E2>
    void inplace_solve (const matrix_expression<E1> &e1,
    E2 &e2,
    lower_tag,
    vector_tag);
    template<class E1, class E2>
    void inplace_solve (const matrix_expression<E1> &e1,
    E2 &e2,
    upper_tag,
    vector_tag);
    template<class E1, class E2>
    void inplace_solve (const matrix_expression<E1> &e1,
    E2 &e2,
    unit_lower_tag,
    vector_tag);
    template<class E1, class E2>
    void inplace_solve (const matrix_expression<E1> &e1,
    E2 &e2,
    unit_upper_tag,
    vector_tag);

    template<class E1, class E2, class C>
    typename matrix_vector_solve_traits<E1, E2>::result_type
    solve (const matrix_expression<E1> &e1,
    const vector_expression<E2> &e2,
    C);

    template<class E1, class E2>
    void inplace_solve (E1 &e1,
    const matrix_expression<E2> &e2,
    vector_tag,
    lower_tag);
    template<class E1, class E2>
    void inplace_solve (E1 &e1,
    const matrix_expression<E2> &e2,
    vector_tag,
    upper_tag);
    template<class E1, class E2>
    void inplace_solve (E1 &e1,
    const matrix_expression<E2> &e2,
    vector_tag,
    unit_lower_tag);
    template<class E1, class E2>
    void inplace_solve (E1 &e1,
    const matrix_expression<E2> &e2,
    vector_tag,
    unit_upper_tag);

    template<class E1, class E2, class C>
    typename matrix_vector_solve_traits<E1, E2>::result_type
    solve (const vector_expression<E1> &e1,
    const matrix_expression<E2> &e2,
    C);

    Description

    solve solves a linear equation for lower or upper (unit) triangular matrices.

    Definition

    Defined in the header triangular.hpp.

    Type requirements

  • E1 is a model of Matrix Expression or Vector Expression .
  • E2 is a model of Vector Expression or Matrix Expression .
  • Preconditions

  • e1 ().size1 () == e1 ().size2 ()
  • e1 ().size2 () == e2 ().size ()
  • e1 ().size () == e2 ().size1 ()
  • e2 ().size1 () == e2 ().size2 ()
  • Complexity

    Quadratic depending from the size of the matrix expression.

    Examples

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

    int main () {
    using namespace boost::numeric::ublas;
    matrix<double> m (3, 3);
    vector<double> v (3);
    for (unsigned i = 0; i < std::min (m.size1 (), v.size ()); ++ i) {
    for (unsigned j = 0; j <= i; ++ j)
    m (i, j) = 3 * i + j + 1;
    v (i) = i;
    }

    std::cout << solve (m, v, lower_tag ()) << std::endl;
    std::cout << solve (v, m, lower_tag ()) << std::endl;
    }

    Matrix Matrix Operations

    Binary Operation Description

    Description

    The templated class matrix_matrix_binary<E1, E2, F> describes a binary matrix operation.

    Definition

    Defined in the header matrix_expression.hpp.

    Template parameters

    Parameter Description Default
    E1 The type of the first matrix expression.  
    E2 The type of the second matrix expression.  
    F The type of the operation.  

    Model of

    Matrix Expression .

    Type requirements

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

    Public base classes

    matrix_expression<matrix_matrix_binary<E1, E2, F> > .

    Members

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

    Interface

        template<class E1, class E2, class F>
    class matrix_matrix_binary:
    public matrix_expression<matrix_matrix_binary<E1, E2, F> > {
    public:
    typedef E1 expression1_type;
    typedef E2 expression2_type;
    typedef F functor_type;
    typedef typename promote_traits<typename E1::size_type, typename E2::size_type>::promote_type size_type;
    typedef typename promote_traits<typename E1::difference_type, typename E2::difference_type>::promote_type difference_type;
    typedef typename F::result_type value_type;
    typedef value_type const_reference;
    typedef const_reference reference;
    typedef const value_type *const_pointer;
    typedef const_pointer pointer;
    typedef const matrix_matrix_binary<E1, E2, F> const_closure_type;
    typedef unknown_orientation_tag orientation_category;
    typedef typename E1::const_iterator1 const_iterator11_type;
    typedef typename E1::const_iterator2 const_iterator12_type;
    typedef typename E2::const_iterator1 const_iterator21_type;
    typedef typename E2::const_iterator2 const_iterator22_type;
    typedef unknown_storage_tag storage_category;

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

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

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

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

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

    // Iterators simply are pointers.

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

    // Construction and destruction
    const_iterator1 ();
    const_iterator1 (const matrix_matrix_binary &mmb, const const_iterator11_type &it1, const const_iterator22_type &it2);

    // Random access specialization
    value_type dereference (dense_random_access_iterator_tag) const;

    // Packed bidirectional specialization
    value_type dereference (packed_bidirectional_iterator_tag) const;

    // Sparse bidirectional specialization
    value_type dereference (sparse_bidirectional_iterator_tag) const;

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

    // Dereference
    reference operator * () const;

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

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

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

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

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

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

    // Construction and destruction
    const_iterator2 ();
    const_iterator2 (const matrix_matrix_binary &mmb, const const_iterator11_type &it1, const const_iterator22_type &it2);

    // Random access specialization
    value_type dereference (dense_random_access_iterator_tag) const;

    // Packed bidirectional specialization
    value_type dereference (packed_bidirectional_iterator_tag) const;

    // Sparse bidirectional specialization
    value_type dereference (sparse_bidirectional_iterator_tag) const;

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

    // Dereference
    reference operator * () const;

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

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

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

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

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

    // Reverse iterators

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

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

    Binary Operations

    Prototypes

        template<class T1, class E1, class T2, class E2>
    struct matrix_matrix_binary_traits {
    typedef unknown_orientation_tag dispatch_category;
    typedef typename promote_traits<T1, T2>::promote_type promote_type;
    typedef matrix_matrix_binary<typename E1::const_closure_type,
    typename E2::const_closure_type,
    matrix_matrix_prod<T1, T2, promote_type> > expression_type;
    typedef expression_type result_type;
    };

    template<class E1, class E2>
    typename matrix_matrix_binary_traits<typename E1::value_type, E1,
    typename E2::value_type, E2>::result_type
    prod (const matrix_expression<E1> &e1,
    const matrix_expression<E2> &e2,
    unknown_orientation_tag);

    // Dispatcher
    template<class E1, class E2>
    typename matrix_matrix_binary_traits<typename E1::value_type, E1,
    typename E2::value_type, E2>::result_type
    prod (const matrix_expression<E1> &e1,
    const matrix_expression<E2> &e2);

    template<class E1, class E2>
    typename matrix_matrix_binary_traits<typename type_traits<typename E1::value_type>::precision_type, E1,
    typename type_traits<typename E2::value_type>::precision_type, E2>::result_type
    prec_prod (const matrix_expression<E1> &e1,
    const matrix_expression<E2> &e2,
    unknown_orientation_tag);

    // Dispatcher
    template<class E1, class E2>
    typename matrix_matrix_binary_traits<typename type_traits<typename E1::value_type>::precision_type, E1,
    typename type_traits<typename E2::value_type>::precision_type, E2>::result_type
    prec_prod (const matrix_expression<E1> &e1,
    const matrix_expression<E2> &e2);

    template<class M, class E1, class E2>
    M
    prod (const matrix_expression<E1> &e1,
    const matrix_expression<E2> &e2);

    template<class M, class E1, class E2>
    M
    prec_prod (const matrix_expression<E1> &e1,
    const matrix_expression<E2> &e2);

    Description

    prod computes the product of the matrix expressions. prec_prod computes the double precision product of the matrix expressions.

    Definition

    Defined in the header matrix_expression.hpp.

    Type requirements

  • E1 is a model of Matrix Expression .
  • E2 is a model of Matrix Expression .
  • Preconditions

  • e1 ().size2 () == e2 ().size1 ()
  • Complexity

    Cubic depending from the size of the matrix expression.

    Examples

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

    int main () {
    using namespace boost::numeric::ublas;
    matrix<double> m1 (3, 3), m2 (3, 3);
    for (unsigned i = 0; i < std::min (m1.size1 (), m2.size1 ()); ++ i)
    for (unsigned j = 0; j < std::min (m1.size2 (), m2.size2 ()); ++ j)
    m1 (i, j) = m2 (i, j) = 3 * i + j;

    std::cout << prod (m1, m2) << std::endl;
    }

    Triangular Solvers

    Prototypes

        template<class E1, class E2>
    struct matrix_matrix_solve_traits {
    typedef typename promote_traits<typename E1::value_type, typename E2::value_type>::promote_type promote_type;
    typedef matrix<promote_type> result_type;
    };

    template<class E1, class E2>
    void inplace_solve (const matrix_expression<E1> &e1,
    E2 &e2,
    lower_tag,
    matrix_tag);
    template<class E1, class E2>
    void inplace_solve (const matrix_expression<E1> &e1,
    E2 &e2,
    upper_tag,
    matrix_tag);
    template<class E1, class E2>
    void inplace_solve (const matrix_expression<E1> &e1,
    E2 &e2,
    unit_lower_tag,
    matrix_tag);
    template<class E1, class E2>
    void inplace_solve (const matrix_expression<E1> &e1,
    E2 &e2,
    unit_upper_tag,
    matrix_tag);

    template<class E1, class E2, class C>
    typename matrix_matrix_solve_traits<E1, E2>::result_type
    solve (const matrix_expression<E1> &e1,
    const matrix_expression<E2> &e2,
    C);

    Description

    solve solves a linear equation for lower or upper (unit) triangular matrices.

    Definition

    Defined in the header triangular.hpp.

    Type requirements

  • E1 is a model of Matrix Expression .
  • E2 is a model of Matrix Expression .
  • Preconditions

  • e1 ().size1 () == e1 ().size2 ()
  • e1 ().size2 () == e2 ().size1 ()
  • Complexity

    Cubic depending from the size of the matrix expressions.

    Examples

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

    int main () {
    using namespace boost::numeric::ublas;
    matrix<double> m1 (3, 3), m2 (3, 3);
    for (unsigned i = 0; i < std::min (m1.size1 (), m2.size1 ()); ++ i)
    for (unsigned j = 0; j <= i; ++ j)
    m1 (i, j) = m2 (i, j) = 3 * i + j + 1;

    std::cout << solve (m1, m2, lower_tag ()) << 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