# Vector Expressions

## Vector Expression

#### Description

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

#### Definition

Defined in the header vector_expression.hpp.

#### Template parameters

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

#### Model of

None.

#### Type requirements

None.

#### Public base classes

None.

#### Members

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

## Vector References

### Constant Reference

#### Description

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

#### Definition

Defined in the header vector_expression.hpp.

#### Template parameters

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

#### Model of

Vector Expression .

#### Type requirements

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

#### Public base classes

`vector_expression<vector_const_reference<E> >`

#### Members

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

### 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.

Last revised: 1/15/2003