Vector Expressions

Vector Expression

Description

The templated class vector_expression<E> is required to be a public base of all classes which model the Vector Expression concept.

Definition

Defined in the header expression_types.hpp.

Template parameters

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

Model of

None. Not a Vector Expression!

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.

Notes

The range, slice and project functions have been removed. Use the free functions defined in vector proxy instead.

Vector Container

Description

The templated class vector_container<C> is required to be a public base of all classes which model the Vector concept. This includes the class vector itself.

Definition

Defined in the header expression_types.hpp.

Template parameters

Parameter	Description	Default
`C`	The type of the vector container.

Model of

None. Not a Vector Expression OR Vector!

Type requirements

None.

Public base classes

vector_expression<C>

Members

Member	Description
`const container_type &operator () () const`	Returns a `const` reference of the container.
`container_type &operator () ()`	Returns a reference of the container.

Vector References

Reference

Description

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

Definition

Defined in the header vector_expression.hpp.

Template parameters

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

Model of

Vector Expression .

Type requirements

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

Public base classes

vector_expression<vector_reference<E> >

Members

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

Vector Operations

Unary Operation Description

Description

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

Definition

Defined in the header vector_expression.hpp.

Template parameters

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

Model of

Vector Expression .

Type requirements

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

Public base classes

vector_expression<vector_unary<E, F> >

Members

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

Unary Operations

Prototypes

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

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

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

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

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

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

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

Description

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

Definition

Defined in the header vector_expression.hpp.

Type requirements

E is a model of Vector Expression .

Preconditions

None.

Complexity

Linear depending from the size of the vector expression.

Examples

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

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

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

Binary Operation Description

Description

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

Definition

Defined in the header vector_expression.hpp.

Template parameters

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

Model of

Vector Expression .

Type requirements

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

Public base classes

vector_expression<vector_binary<E1, E2, F> >

Members

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

Binary Operations

Prototypes

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

    // (v1 + v2) [i] = v1 [i] + v2 [i]
    template<class E1, class E2>
    typename vector_binary_traits<E1, E2, scalar_plus<typename E1::value_type,
                                                       typename E2::value_type> >::result_type
    operator + (const vector_expression<E1> &e1,
                 const vector_expression<E2> &e2);

    // (v1 - v2) [i] = v1 [i] - v2 [i]
    template<class E1, class E2>
    typename vector_binary_traits<E1, E2, scalar_minus<typename E1::value_type,
                                                        typename E2::value_type> >::result_type
    operator - (const vector_expression<E1> &e1,
                 const vector_expression<E2> &e2);

Description

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

Definition

Defined in the header vector_expression.hpp.

Type requirements

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

Preconditions

e1 ().size () == e2 ().size ()

Complexity

Linear depending from the size of the vector expressions.

Examples

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

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

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

Binary Outer Operation Description

Description

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

Definition

Defined in the header matrix_expression.hpp.

Template parameters

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

Model of

Matrix Expression .

Type requirements

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

Public base classes

matrix_expression<vector_matrix_binary<E1, E2, F> >

Members

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

Binary Outer Operations

Prototypes

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

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

Description

outer_prod computes the outer product of two vector expressions.

Definition

Defined in the header matrix_expression.hpp.

Type requirements

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

Preconditions

None.

Complexity

Quadratic depending from the size of the vector expressions.

Examples

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

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

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

Scalar Vector Operation Description

Description

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

Definition

Defined in the header vector_expression.hpp.

Template parameters

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

Model of

Vector Expression .

Type requirements

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

Public base classes

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

Members

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

Scalar Vector Operations

Prototypes

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

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

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

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

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

Description

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

Definition

Defined in the header vector_expression.hpp.

Type requirements

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

Preconditions

None.

Complexity

Linear depending from the size of the vector expression.

Examples

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

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

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

Vector Reductions

Unary Reductions

Prototypes

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

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

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

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

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

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

Description

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

Definition

Defined in the header vector_expression.hpp.

Type requirements

E is a model of Vector Expression .

Preconditions

None.

Complexity

Linear depending from the size of the vector expression.

Examples

#include <boost/numeric/ublas/vector.hpp>

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

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

Binary Reductions

Prototypes

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

    // inner_prod (v1, v2) = sum (v1 [i] * v2 [i])
    template<class E1, class E2>
    typename vector_scalar_binary_traits<E1, E2, vector_inner_prod<typename E1::value_type,
                                                                   typename E2::value_type,
                                                                   typename promote_traits<typename E1::value_type,
                                                                                           typename E2::value_type>::promote_type> >::result_type
    inner_prod (const vector_expression<E1> &e1,
                const vector_expression<E2> &e2);

    template<class E1, class E2>
    typename vector_scalar_binary_traits<E1, E2, vector_inner_prod<typename E1::value_type,
                                                                   typename E2::value_type,
                                                                   typename type_traits<typename promote_traits<typename E1::value_type,
                                                                                                                typename E2::value_type>::promote_type>::precision_type> >::result_type
    prec_inner_prod (const vector_expression<E1> &e1,
                     const vector_expression<E2> &e2);

Description

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

Definition

Defined in the header vector_expression.hpp.

Type requirements

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

Preconditions

e1 ().size () == e2 ().size ()

Complexity

Linear depending from the size of the vector expressions.

Examples

#include <boost/numeric/ublas/vector.hpp>

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

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

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